Relativistic Quantum Mechanics of a Charged Particle in an Electromagnetic Field, Summaries of Quantum Mechanics

Download Relativistic Quantum Mechanics of a Charged Particle in an Electromagnetic Field and more Summaries Quantum Mechanics in PDF only on Docsity! Chapter 10 Relativistic Quantum Mechanics In this Chapter we will address the issue that the laws of physics must be formulated in a form which is Lorentz–invariant, i.e., the description should not allow one to differentiate between frames of reference which are moving relative to each other with a constant uniform velocity ~v. The transformations beween such frames according to the Theory of Special Relativity are described by Lorentz transformations. In case that ~v is oriented along the x1–axis, i.e., ~v = v1x̂1, these transformations are x1′ = x1 − v1t√ 1 − ( v1 c )2 , t′ = t − v1 c2 x1√ 1 − ( v1 c )2 , x′2 = x2 ; x′3 = x3 (10.1) which connect space time coordinates (x1, x2, x3, t) in one frame with space time coordinates (x′1, x ′ 2, x ′ 3, t ′) in another frame. Here c denotes the velocity of light. We will introduce below Lorentz-invariant differential equations which take the place of the Schrödinger equation of a par- ticle of mass m and charge q in an electromagnetic field [c.f. (refeq:ham2, 8.45)] described by an electrical potential V (~r, t) and a vector potential ~A(~r, t) i~ ∂ ∂t ψ(~r, t) = [ 1 2m ( ~ i ∇ − q c ~A(~r, t) )2 + qV (~r, t) ] ψ(~r, t) (10.2) The replacement of (10.2) by Lorentz–invariant equations will have two surprising and extremely important consequences: some of the equations need to be formulated in a representation for which the wave functions ψ(~r, t) are vectors of dimension larger one, the components representing the spin attribute of particles and also representing together with a particle its anti-particle. We will find that actually several Lorentz–invariant equations which replace (10.2) will result, any of these equations being specific for certain classes of particles, e.g., spin–0 particles, spin–1 2 particles, etc. As mentioned, some of the equations describe a particle together with its anti-particle. It is not possible to uncouple the equations to describe only a single type particle without affecting nega- tively the Lorentz invariance of the equations. Furthermore, the equations need to be interpreted as actually describing many–particle-systems: the equivalence of mass and energy in relativistic formulations of physics allows that energy converts into particles such that any particle described will have ‘companions’ which assume at least a virtual existence. Obviously, it will be necessary to begin this Chapter with an investigation of the group of Lorentz transformations and their representation in the space of position ~r and time t. The representation 287 288 Relativistic Quantum Mechanics in Sect. 10.1 will be extended in Sect. 10.4 to cover fields, i.e., wave functions ψ(~r, t) and vectors with functions ψ(~r, t) as components. This will provide us with a general set of Lorentz–invariant equations which for various particles take the place of the Schrödinger equation. Before introduc- ing these general Lorentz–invariant field equations we will provide in Sects. 10.5, 10.7 a heuristic derivation of the two most widely used and best known Lorentz–invariant field equations, namely the Klein–Gordon (Sect. 10.5) and the Dirac (Sect. 10.7) equation. 10.1 Natural Representation of the Lorentz Group In this Section we consider the natural representation of the Lorentz group L, i.e. the group of Lorentz transformations (10.1). Rather than starting from (10.1), however, we will provide a more basic definition of the transformations. We will find that this definition will lead us back to the transformation law (10.1), but in a setting of representation theory methods as applied in Secti. 5 to the groups SO(3) and SU(2) of rotation transformations of space coordinates and of spin. The elements L ∈ L act on 4–dimensional vectors of position– and time–coordinates. We will denote these vectors as follows xµ def= (x0, x1, x2, x3) (10.3) where x0 = ct describes the time coordinate and (x1, x2, x3) = ~r describes the space coordinates. Note that the components of xµ all have the same dimension, namely that of length. We will, henceforth, assume new units for time such that the velocity of light c becomes c = 1. This choice implies dim(time) = dim(length). Minkowski Space Historically, the Lorentz transformations were formulated in a space in which the time component of xµ was chosen as a purely imaginary number and the space components real. This space is called the Minkowski space. The reason for this choice is that the transformations (10.1) leave the quantity s2 = (x0)2 − (x1)2 − (x2)2 − (x3)2 (10.4) invariant, i.e., for the transformed space-time–cordinates x′µ = (x′0, x′1, x′2, x′3) holds (x0)2 − (x1)2 − (x2)2 − (x3)2 = (x′0)2 − (x′1)2 − (x′2)2 − (x′3)2 . (10.5) One can interprete the quantity √ −s2 as a distance in a 4–dimensional Euclidean space if one chooses the time component purely imaginary. In such a space Lorentz transformations corre- spond to 4-dimensional rotations. Rather than following this avenue we will introduce Lorentz transformations within a setting which does not require real and imaginary coordinates. The Group of Lorentz Transformations L = O(3,1) The Lorentz transformations L describe the relationship between space-time coordinates xµ of two reference frames which move relative to each other with uniform fixed velocity ~v and which might be reoriented relative to each other by a rotation around a common origin. Denoting by xµ the 10.1: Natural Representation of the Lorentz Group 291 Classification of Lorentz Transformations We like to classify now the elements of L = O(3,1). For this purpose we consider first the value of detL. A statement on this value can be made on account of property (10.14). Using detAB = detAdetB and detAT = detA yields (detL)2 = 1 or detL = ±1 . (10.23) One can classify Lorentz transformations according to the value of the determinant into two distinct classes. A second class property follows from (10.14) which we employ in the formulation (10.12). Consid- ering in (10.12) the case ρ = 0, σ = 0 yields( L0 0 )2 − ( L1 0 )2 − ( L2 0 )2 − ( L3 0 )2 = 1 . (10.24) or since (L1 0)2 + (L2 0)2 + (L3 0)2 ≥ 0 it holds (L0 0)2 ≥ 1. From this we can conclude L0 0 ≥ 1 or L0 0 ≤ −1 , (10.25) i.e., there exist two other distinct classes. Properties (10.23) and (10.25) can be stated as follows: The set of all Lorentz transformations L is given as the union L = L↑+ ∪ L ↓ + ∪ L ↑ − ∪ L ↓ − (10.26) where L↑+, L ↓ +, L ↑ −, L ↓ − are disjunct sets defined as follows L↑+ = {L,L ∈ O(3, 1), detL = 1, L0 0 ≥ 1} ; (10.27) L↓+ = {L,L ∈ O(3, 1), detL = 1, L0 0 ≤ −1} ; (10.28) L↑− = {L,L ∈ O(3, 1), detL = −1, L0 0 ≥ 1} ; (10.29) L↓− = {L,L ∈ O(3, 1), detL = −1, L0 0 ≤ −1} . (10.30) It holds g ∈ L and −11 ∈ L as one can readily verify testing for property (10.14). One can also verify that one can write L↑− = gL↑+ = L↑+g ; (10.31) L↓+ = −L↑+ ; (10.32) L↓− = −gL↑+ = −L↑+g (10.33) where we used the definition aM = {M1,∃M2,M2 ∈ M,M1 = aM2}. The above shows that the set of proper Lorentz transformations L↑+ allows one to generate all Lorentz transformations, except for the trivial factors g and −11. It is, hence, entirely suitable to investigate first only Lorentz transformations in L↑+. We start our investigation by demonstrating that L↑+ forms a group. Obviously, L↑+ contains 11. We can also demonstrate that for A,B ∈ L↑+ holds C = AB ∈ L↑+. For this purpose we consider the value of C0 0 = A0 µB µ 0 = ∑3 j=1A 0 jB j 0 + A0 0B 0 0. Schwartz’s inequality yields 3∑ j=1 A0 jB j 0 2 ≤ 3∑ j=1 ( A0 j )2 3∑ j=1 ( Bj 0 )2 . (10.34) 292 Relativistic Quantum Mechanics From (10.12) follows (B0 0)2 − ∑3 j=1(Bj 0)2 = 1 or ∑3 j=1(Bj 0)2 = (B0 0)2 − 1. Similarly, one can conclude from (10.21) ∑3 j=1(A0 j)2 = (A0 0)2 − 1. (10.34) provides then the estimate 3∑ j=1 A0 jB j 0 2 ≤ [ (A0 0)2 − 1 ] [ (B0 0)2 − 1 ] < (A0 0)2 (B0 0)2 . (10.35) One can conclude, therefore, | ∑3 j=1A 0 jB j 0| < A0 0B 0 0. Since A0 0 ≥ 1 and B0 0 ≥ 1, obviously A0 0B 0 0 ≥ 1. Using the above expression for C0 0 one can state C0 0 > 0. In fact, since the group property of O(3,1) ascertains CTgC = g it must hold C0 0 ≥ 1. The next group property of L↑+ to be demonstrated is the existence of the inverse. For the inverse of any L ∈ L↑+ holds (10.16). This relationship shows (L−1)0 0 = L0 0, from which one can conclude L−1 ∈ L↑+. We also note that the identity operator 11 has elements 11µν = δµν (10.36) where we defined1 δµν = { 1 for µ = ν 0 for µ 6= ν (10.37) It holds, 110 0 =≥ 1 and, hence, 11 ∈ L↑+. Since the associative property holds for matrix multipli- cation we have verified that L↑+ is indeed a subgroup of SO(3,1). L↑+ is called the subgroup of proper, orthochronous Lorentz transformations. In the following we will consider solely this subgroup of SO(3,1). Infinitesimal Lorentz transformations The transformations in L↑+ have the property that they are continously connected to the identity 11, i.e., these transformations can be parametrized such that a continuous variation of the parameters connects any element of L↑+ with 11. This property will be exploited now in that we consider first transformations in a small neighborhood of 11 which we parametrize by infinitesimal parameters. We will then employ the Lie group properties to generate all transformations in L↑+. Accordingly, we consider transformations Lµν = δµν + εµν ; εµν small . (10.38) For these transformations, obviously, holds L0 0 > 0 and the value of the determinant is close to unity, i.e., if we enforce (10.14) actually L0 0 ≥ 1 and det L = 1 must hold. Property (10.14) implies ( 11 + εT ) g ( 11 + ε ) = g (10.39) where we have employed the matrix form ε defined as in (10.13). To order O(ε2) holds εT g + g ε = 0 . (10.40) 1It should be noted that according to our present definition holds δµν = gµρδ ρ ν and, accordingly, δ00 = 1 and δ11 = δ22 = δ33 = −1. 10.1: Natural Representation of the Lorentz Group 293 Using (10.15) one can conclude εT = −g εg (10.41) which reads explicitly ε00 ε10 ε20 ε30 ε01 ε11 ε21 ε31 ε02 ε12 ε22 ε32 ε03 ε13 ε23 ε33  =  −ε00 ε01 ε02 ε03 ε10 −ε11 −ε12 −ε13 ε20 −ε21 −ε22 −ε23 ε30 −ε31 −ε32 −ε33  . (10.42) This relationship implies εµµ = 0 ε0j = εj0 , j = 1, 2, 3 εjk = − εkj , j, k = 1, 2, 3 (10.43) Inspection shows that the matrix ε has 6 independent elements and can be written ε(ϑ1, ϑ2, ϑ3, w1, w2, w3) =  0 −w1 −w2 −w3 −w1 0 −ϑ3 ϑ2 −w2 ϑ3 0 −ϑ1 −w3 −ϑ2 ϑ1 0  . (10.44) This result allows us now to define six generators for the Lorentz transformations(k = 1, 2, 3) Jk = ε(ϑk = 1, other five parameters zero) (10.45) Kk = ε(wk = 1, other five parameters zero) . (10.46) The generators are explicitly J1 =  0 0 0 0 0 0 0 0 0 0 0 −1 0 0 1 0  ; J2 =  0 0 0 0 0 0 0 1 0 0 0 0 0 −1 0 0  ; J3 =  0 0 0 0 0 0 −1 0 0 1 0 0 0 0 0 0  (10.47) K1 =  0 −1 0 0 −1 0 0 0 0 0 0 0 0 0 0 0  ; K2 =  0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 0  ; K3 =  0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0  (10.48) These commutators obey the following commutation relationships [ Jk, J` ] = εk`m Jm (10.49) [ Kk, K` ] = − εk`m Jm [ Jk, K` ] = εk`m Km . The operators also obey ~J · ~K = J1J1 + J2J2 + J3J3 = 0 (10.50) 296 Relativistic Quantum Mechanics 10.2 Scalars, 4–Vectors and Tensors In this Section we define quantities according to their behaviour under Lorentz transformations. Such quantities appear in the description of physical systems and statements about transformation properties are often extremely helpful and usually provide important physical insight. We have encountered examples in connection with rotational transformations, namely, scalars like r =√ x2 1 + x2 2 + x2 3, vectors like ~r = (x1, x2, x3)T , spherical harmonics Y`m(r̂), total angular momentum states of composite systems like Y`m(`1, `2|r̂1, r̂2) and, finally, tensor operators Tkm. Some of these quantities were actually defined with respect to representations of the rotation group in function spaces, not in the so-called natural representation associated with the 3–dimensional Euclidean space E3. Presently, we have not yet defined representations of Lorentz transformations beyond the ‘natural’ representation acting in the 4–dimensional space of position– and time–coordinates. Hence, our definition of quantities with special properties under Lorentz transformations presently is confined to the natural representation. Nevertheless, we will encounter an impressive example of physical properties. Scalars The quantities with the simplest transformation behaviour are so-called scalars f ∈ R which are invariant under transformations, i.e., f ′ = f . (10.65) An example is s2 defined in (10.4), another example is the rest mass m of a particle. However, not any physical property f ∈ R is a scalar. Counterexamples are the energy, the charge density, the z–component x3 of a particle, the square of the electric field | ~E(~r, t)|2 or the scalar product ~r1 · ~r2 of two particle positions. We will see below how true scalars under Lorentz transformations can be constructed. 4-Vectors The quantities with the transformation behaviour like that of the position–time vector xµ defined in (10.3) are the so-called 4–vectors aµ. These quantites always come as four components (a0, a1, a2, a3)T and transform according to a′ µ = Lµνa ν . (10.66) Examples of 4-vectors beside xµ are the momentum 4-vector pµ = (E, ~p) , E = m√ 1 − ~v 2 , ~p = m~v√ 1 − ~v 2 (10.67) the transformation behaviour of which we will demonstrate further below. A third 4-vector is the so-called current vector Jµ = (ρ, ~J) (10.68) where ρ(~r, t) and ~J(~r, t) are the charge density and the current density, respectively, of a system of charges. Another example is the potential 4-vector Aµ = (V, ~A) (10.69) where V (~r, t) and ~A(~r, t) are the electrical and the vector potential of an electromagnetic field. The 4-vector character of Jµ and of Aµ will be demonstrated further below. 10.2: Scalars, 4–Vectors and Tensors 297 Scalar Product 4-vectors allow one to construct scalar quantities. If aµ and bµ are 4-vectors then aµgµνb ν (10.70) is a scalar. This property follows from (10.66) together with (10.12) a′ µ gµνb ′ν = LµρgµνL ν σa ρbσ = aρgρσb σ (10.71) Contravariant and Covariant 4-Vectors It is convenient to define a second class of 4-vectors. The respective vectors aµ are associated with the 4-vectors aµ, the relationship being aµ = gµνa ν = (a0,−a1,−a2,−a3) (10.72) where aν is a vector with transformation behaviour as stated in (10.66). One calls 4-vectors aµ covariant and 4-vectors aµ contravariant. Covariant 4-vectors transform like a′µ = gµνL ν ρg ρσaσ (10.73) where we defined gµν = gµν . (10.74) We like to point out that from definition (10.72) of the covariant 4-vector follows aµ = gµνaν . In fact, one can employ the tensors gµν and gµν to raise and lower indices of Lµν as well. We do not establish here the consistency of the ensuing notation. In any case one can express (10.73) a′µ = Lµ σaσ . (10.75) Note that according to (10.17) Lµσ is the transformation inverse to Lσµ. In fact, one can express [(L−1)T ]µν = (L−1)νµ and, accordingly, (10.17) can be written (L−1)νµ = Lµ ν . (10.76) The 4-Vector ∂µ An important example of a covariant 4-vector is the differential operator ∂µ = ∂ ∂xµ = ( ∂ ∂t , ∇ ) (10.77) The transformed differential operator will be denoted by ∂′µ def= ∂ ∂x′µ . (10.78) To prove the 4-vector property of ∂µ we will show that gµν∂ν transforms like a contravariant 4- vector, i.e., gµν∂′ν = Lµρg ρσ∂σ. We start from x′µ = Lµνx ν . Multiplication (and summation) of x′µ = Lµνx ν by Lρσgρµ yields, using (10.12), gσνxν = Lρσgρµx ′µ and gµσgσν = δµν , xν = gνσLρσgρµx ′µ . (10.79) This is the inverse Lorentz transformation consistent with (10.16). We have duplicated the expres- sion for the inverse of Lµν to obtain the correct notation in terms of covariant, i.e., lower, and 298 Relativistic Quantum Mechanics contravariant, i.e., upper, indices. (10.79) allows us to determine the connection between ∂µ and ∂′µ. Using the chain rule of differential calculus we obtain ∂′µ = 3∑ ν=0 ∂xν ∂x′µ ∂ ∂xν = gνσLρσgρµ∂ν = Lµ ν ∂ν (10.80) Multiplication by gλµ (and summation over µ) together with gλµgρµ = δλρ yields gλµ∂′µ = Lλσg σν∂ν , (10.81) i.e., ∂µ does indeed transform like a covariant vector. d’Alembert Operator We want to construct now a scalar differential operator. For this purpose we define first the contravariant differential operator ∂µ = gµν∂ν = ( ∂ ∂t , −∇ ) . (10.82) Then the operator ∂µ∂ µ = ∂2 t − ∇2 (10.83) is a scalar under Lorentz transformations. In fact, this operator is equal to the d’Alembert operator which is known to be Lorentz-invariant. Proof that pµ is a 4-vector We will demonstrate now that the momentum 4-vector pµ defined in (10.67) transforms like (10.66). For this purpose we consider the scalar differential (dτ)2 = dxµdxµ = (dt)2 − (d~r)2 (10.84) It holds ( dτ dt )2 = 1 − (~v)2 (10.85) from which follows d dτ = 1√ 1 − ~v 2 d dt . (10.86) One can write p0 = E = m√ 1 − ~v 2 = m√ 1 − ~v 2 dt dt . (10.87) The remaining components of pµ can be written, e.g., p1 = mv1 √ 1 − ~v 2 = m√ 1 − ~v 2 dx1 dt . (10.88) One can express then the momentum vector pµ = m√ 1 − ~v 2 dxµ dt = m d dτ xµ . (10.89) 10.3: Relativistic Electrodynamics 301 The relationships (10.103, 10.104) establishe the transformation behaviour of ~E(~r, t) and ~B(~r, t). In a new frame of reference holds F ′µν = Lµα L ν β F αβ (10.106) In case that the Lorentz transformation Lµν is given by (10.62) or, equivalently, by (10.63), one obtains F ′µν =  0 −Ex −Ey−v1Bz√ 1−v2 1 −Ez+v1By√ 1−v2 1 Ex 0 −Bz−v1Ey√ 1−v2 1 By+v1Ez√ 1−v2 1 Ey−v1Bz√ 1−v2 1 Bz−v1Ey√ 1−v2 1 0 −Bx Ez+v1By√ 1−v2 1 −By+v1Ez√ 1−v2 1 Bx 0  (10.107) Comparision with F ′µν =  0 −E′x −E′y −E′z E′x 0 −B′z B′y E′y B′z 0 −B′x E′z −B′y B′x 0  (10.108) yields then the expressions for the transformed fields ~E′ and ~B′. The results can be put into the more general form ~E′‖ = ~E‖ , ~E′⊥ = ~E⊥ + ~v × ~B√ 1 − ~v 2 (10.109) ~B′‖ = ~B‖ , ~B′⊥ = ~B⊥ − ~v × ~E√ 1 − ~v 2 (10.110) where ~E‖, ~B‖ and ~E⊥, ~B⊥ are, respectively, the components of the fields parallel and perpendicular to the velocity ~v which determines the Lorentz transformation. These equations show that under Lorentz transformations electric and magnetic fields convert into one another. Maxwell Equations in Lorentz-Invariant Form One can express the Maxwell equations in terms of the tensor Fµν in Lorentz-invariant form. Noting ∂µF µν = ∂µ∂ µAν − ∂µ∂ νAµ = ∂µ∂ µAν − ∂ν∂µA µ = ∂µ∂ µAν , (10.111) where we used (10.105) and (10.97), one can conclude from (10.102) ∂µF µν = 4π Jν . (10.112) One can readily prove that this equation is equivalent to the two inhomogeneous Maxwell equations (8.1, 8.2). From the definition (10.105) of the tensor Fµν one can conclude the property ∂σFµν + ∂µF νσ + ∂νF σµ = 0 (10.113) which can be shown to be equivalent to the two homogeneous Maxwell equations (8.3, 8.4). 302 Relativistic Quantum Mechanics Lorentz Force One important property of the electromagnetic field is the Lorentz force acting on charged particles moving through the field. We want to express this force through the tensor Fµν . It holds for a particle with 4-momentum pµ as given by (10.67) and charge q dpµ dτ = q m pν F µν (10.114) where d/dτ is given by (10.86). We want to demonstrate now that this equation is equivalent to the equation of motion (8.5) where ~p = m~v/ √ 1− v2. To avoid confusion we will employ in the following for the energy of the particle the notation E = m/ √ 1− v2 [see (10.87)] and retain the definition ~E for the electric field. The µ = 0 component of (10.114) reads then, using (10.104), dE dτ = q m ~p · ~E (10.115) or with (10.86) dE dt = q E ~p · ~E . (10.116) From this one can conclude, employing (10.93), 1 2 dE2 dt = 1 2 d~p 2 dt = q ~p · ~E (10.117) This equation follows, however, also from the equation of motion (8.5) taking the scalar product with ~p ~p · d~p dt = q~p · ~E (10.118) where we exploited the fact that according to ~p = m~v/ √ 1− v2 holds ~p ‖ ~v. For the spatial components, e.g., for µ = 1, (10.114) reads using (10.103) dpx dτ = q m ( EEx + pyBz − pzBy ) . (10.119) Employing again (10.86) and (10.67), i.e., E = m/ √ 1− v2, yields dpx dt = q [ Ex + (~v × ~B)x ] (10.120) which is the x-component of the equation of motion (8.5). We have, hence, demonstrated that (10.114) is, in fact, equivalent to (8.5). The term on the r.h.s. of (10.120) is referred to as the Lorentz force. Equation (10.114), hence, provides an alternative description of the action of the Lorentz force. 10.4: Function Space Representation of Lorentz Group 303 10.4 Function Space Representation of Lorentz Group In the following it will be required to decribe the transformation of wave functions under Lorentz transformations. In this section we will investigate the transformation properties of scalar functions ψ(xµ), ψ ∈ C∞(4). For such functions holds in the transformed frame ψ′(Lµνxν) = ψ(xµ) (10.121) which states that the function values ψ′(x′µ) at each point x′µ in the new frame are identical to the function values ψ(xµ) in the old frame taken at the same space–time point xµ, i.e., taken at the pairs of points (x′µ = Lµνx ν , xµ). We need to emphasize that (10.121) covers solely the transformation behaviour of scalar functions. Functions which represent 4-vectorsor other non-scalar entities, e.g., the charge-current density in case of Sect. 10.3 or the bi-spinor wave function of electron-positron pairs in Sect. 10.7, obey a different transformation law. We like to express now ψ′(x′µ) in terms of the old coordinates xµ. For this purpose one replaces xµ in (10.121) by (L−1)µνx ν and obtains ψ′(xµ) = ψ((L−1)µνx ν) . (10.122) This result gives rise to the definition of the function space representation ρ(Lµν) of the Lorentz group (ρ(Lµν)ψ)(xµ) def= ψ((L−1)µνx ν) . (10.123) This definition corresponds closely to the function space representation (5.42) of SO(3). In analogy to the situation for SO(3) we seek an expression for ρ(Lµν) in terms of an exponential operator and transformation parameters ~ϑ, ~w, i.e., we seek an expression which corresponds to (10.51) for the natural representation of the Lorentz group. The resulting expression should be a generalization of the function space representation (5.48) of SO(3), in as far as SO(3,1) is a generalization (rotation + boosts) of the group SO(3). We will denote the intended representation by L(~ϑ, ~w) def= ρ(Lµν(~ϑ, ~w)) = ρ ( e ~ϑ· ~J + ~w· ~K ) (10.124) which we present in the form L(~ϑ, ~w) = exp ( ~ϑ · ~J + ~w · ~K ) . (10.125) In this expression ~J = (J1,J2,J3) and ~K = (K1,K2,K3) are the generators of L(~ϑ, ~w) which correspond to the generators Jk and Kk in (10.47), and which can be constructed following the procedure adopted for the function space representation of SO(3). However, in the present case we exclude the factor ‘−i’ [cf. (5.48) and (10.125)]. Accordingly, one can evaluate Jk as follows Jk = lim ϑk→0 1 ϑ1 [ ρ ( eϑkJk ) − 11 ] (10.126) and Kk Kk = lim wk→0 1 w1 [ ρ ( ewkKk ) − 11 ] . (10.127) 306 Relativistic Quantum Mechanics Inserting these results into the definition of K3 in (10.128) yields K3 = x0∂3 + x3∂0 = ∂ ∂Ω . (10.144) The action of the exponential operator (10.137) on a function f(Ω) ∈ C∞(1) is then that of a shift operator L(w3) f(Ω) = exp ( w3 ∂ ∂Ω ) f(Ω) = f(Ω + w3) . (10.145) 10.5 Klein–Gordon Equation In the following Sections we will provide a heuristic derivation of the two most widely used quan- tum mechanical descriptions in the relativistic regime, namely the Klein–Gordon and the Dirac equations. We will provide a ‘derivation’ of these two equations which stem from the historical de- velopment of relativistic quantum mechanics. The historic route to these two equations, however, is not very insightful, but certainly is short and, therefore, extremely useful. Further below we will provide a more systematic, representation theoretic treatment. Free Particle Case A quantum mechanical description of a relativistic free particle results from applying the correspon- dence principle, which allows one to replace classical observables by quantum mechanical operators acting on wave functions. In the position representation the correspondence principle states E =⇒ Ê = −~ i ∂t ~p =⇒ ~̂p = ~ i ∇ (10.146) which, in 4-vector notation reads pµ =⇒ p̂µ = i~(∂t,∇) = i~∂µ ; pµ =⇒ p̂µ = i(∂t,−∇) = i~∂µ . (10.147) Applying the correspondence principle to (10.92) one obtains the wave equation − ~2∂µ∂µ ψ(xν) = m2 ψ(xν) (10.148) or ( ~ 2 ∂µ∂µ + m2 ) ψ(xν) = 0 . (10.149) where ψ(xµ) is a scalar, complex-valued function. The latter property implies that upon change of reference frame ψ(xµ) transforms according to (10.121, 10.122). The partial differential equation (10.151) is called the Klein-Gordon equation. In the following we will employ so-called natural units ~ = c = 1. In these units the quantities energy, momentum, mass, (length)−1, and (time)−1 all have the same dimension. In natural units the Klein–Gordon equation (10.151) reads( ∂µ∂ µ + m2 ) ψ(xµ) = 0 (10.150) 10.5: Klein–Gordon Equation 307 or ( ∂2 t − ∇2 + m2 ) ψ(xµ) = 0 . (10.151) One can notice immeadiately that (10.150) is invariant under Lorentz transformations. This follows from the fact that ∂µ∂µ and m2 are scalars, and that (as postulated) ψ(xµ) is a scalar. Under Lorentz transformations the free particle Klein–Gordon equation (10.150) becomes( ∂′µ∂ ′µ + m2 ) ψ′(x′µ) = 0 (10.152) which has the same form as the Klein–Gordon equation in the original frame. Current 4-Vector Associated with the Klein-Gordon Equation As is well-known the Schrödinger equation of a free particle i∂tψ(~r, t) = − 1 2m ∇2 ψ(~r, t) (10.153) is associated with a conservation law for particle probability ∂tρS(~r, t) + ∇ ·~jS(~r, t) = 0 (10.154) where ρS(~r, t) = ψ∗(~r, t)ψ(~r, t) (10.155) describes the positive definite probability to detect a particle at position ~r at time t and where ~jS(~r, t) = 1 2mi [ψ∗(~r, t)∇ψ(~r, t) − ψ(~r, t)∇ψ∗(~r, t) ] (10.156) describes the current density connected with motion of the particle probability distribution. To derive this conservation law one rewrites the Schrödinger equation in the form (i∂t − 1 2m∇ 2)ψ = 0 and considers Im [ ψ∗ ( i∂t − 1 2m ∇2 ) ψ ] = 0 (10.157) which is equivalent to (10.154). In order to obtain the conservation law connected with the Klein–Gordon equation (10.150) one considers Im [ ψ∗ ( ∂µ∂ µ + m2 ) ψ ] = 0 (10.158) which yields ψ∗∂2 t ψ − ψ∂2 t ψ ∗ − ψ∗∇2ψ + ψ∇2ψ∗ = ∂t (ψ∗∂tψ − ψ∂tψ ∗) + ∇ · (ψ∇ψ∗ − ψ∗∇ψ) = 0 (10.159) which corresponds to ∂tρKG(~r, t) + ∇ ·~jKG(~r, t) = 0 (10.160) where ρKG(~r, t) = i 2m (ψ∗(~r, t)∂tψ(~r, t) − ψ(~r, t)∂tψ∗(~r, t) ) ~jKG(~r, t) = 1 2mi (ψ∗(~r, t)∇ψ(~r, t) − ψ(~r, t)∇ψ∗(~r, t) ) . (10.161) 308 Relativistic Quantum Mechanics This conservation law differs in one important aspect from that of the Schrödinger equation (10.154), namely, in that the expression for ρKG is not positive definite. When the Klein-Gordon equation had been initially suggested this lack of positive definiteness worried physicists to a degree that the Klein–Gordon equation was rejected and the search for a Lorentz–invariant quantum mechanical wave equation continued. Today, the Klein-Gordon equation is considered as a suitable equation to describe spin–0 particles, for example pions. The proper interpretation of ρKG(~r, t), it had been realized later, is actually that of a charge density, not of particle probability. Solution of the Free Particle Klein–Gordon Equation Solutions of the free particle Klein–Gordon equation are ψ(xµ) = N e−ipµx µ = N ei(~p0·~r−Eot) . (10.162) Inserting this into the Klein–Gordon equation (10.151) yields( E2 o − ~p 2 0 − m2 ) ψ(~r, t) = 0 (10.163) which results in the expected [see (10.93] dispersion relationship connecting E0, ~p0, m E2 0 = m2 + ~p 2 o . (10.164) The corresponding energy is Eo(~po, ±) = ± √ m2 + ~p 2 o (10.165) This result together with (10.162) shows that the solutions of the free particle Klein-Gordon e- quation (10.150) are actually determined by ~po and by the choice of sign ±. We denote this by summarizing the solutions as follows( ∂µ∂ µ + m2 ) ψo(~p, λ|xµ) = 0 (10.166) ψo(~p, λ|xµ) = Nλ,p e i(~p·~r−λEo(~p)t) Eo(~p) = √ m2 + ~p 2 o , λ = ± The spectrum of the Klein–Gordon equation (10.150) is a continuum of positive energies E ≥ m, corresponding to λ = +, and of negative energies E ≤ −m, corresponding to λ = −. The density ρKG(~p, λ) associated with the corresponding wave functions ψo(~p, λ|xµ) according to (10.161) and (10.166) is ρKG(~p, λ) = λ Eo(~p) m ψ∗o(~p, λ|xµ)ψo(~p, λ|xµ) (10.167) which is positive for λ = + and negative for λ = −. The proper interpretation of the two cases is that the Klein–Gordon equation describes particles as well as anti-particles; the anti-particles carry a charge opposite to that of the associated particles, and the density ρKG(~p, λ) actually describes charge density rather than probability. Generating a Solution Through Lorentz Transformation A particle at rest, i.e., with ~p = 0, according to (??) is decribed by the ~r–independent wave function ψo(~p = 0, λ|xµ) = N e−iλmt , λ = ± . (10.168) 10.6: Klein–Gordon Equation with Electromagnetic Field 311 particular, larger than |i∂tΨ/Ψ| and alrger than qV , i.e., | i∂tΨ Ψ | << m , |q V | << m . (10.183) The term on the l.h.s. of (10.181) can then be approximated as follows: (i∂t − qV )2e−imtΨ = (i∂t − qV ) (me−imtΨ + e−imti∂tΨ − qV e−imtΨ) = m2e−imtΨ + me−imti∂tΨ − qV e−imtΨ +me−imti∂tΨ − e−imt∂2Ψ − qV e−imti∂tΨ −me−imtqVΨ − e−imti∂tqVΨ + q2V 2e−imtΨ ≈ m2e−imtΨ − 2mqV e−imtΨ − 2me−imti∂tΨ (10.184) where we neglected all terms which did not contain factors m. The approximation is justified on the ground of the inequalities (10.183). The Klein-Gordon equation (10.181) reads then i ∂t Ψ(~r, t) = [ [~̂p − q ~A(~r, t)]2 2m + qV (~r, t) ] Ψ(~r, t) (10.185) This is, however, identical to the Schrödinger equation (10.2) of a non-relativistic spin-0 particle moving in an electromagnetic field. Pionic Atoms To apply the Klein–Gordon equation (10.181) to a physical system we consider pionic atoms, i.e., atoms in which one or more electrons are replaced by π− mesons. This application demonstrates that the Klein–Gordon equation describes spin zero particles, e.g., spin-0 mesons. To ‘manufacture’ pionic atoms, π− mesons are generated through inelastic proton–proton scattering p + p −→ p + p + π− + π+ , (10.186) then are slowed down, filtered out of the beam and finally fall as slow pions onto elements for which a pionic variant is to be studied. The process of π− meson capture involves the so-called Auger effect, the binding of a negative charge (typically an electron) while at the same time a lower shell electron is being emitted π− + atom −→ (atom − e− + π−) + e− . (10.187) We want to investigate in the following a description of a stationary state of a pionic atom involving a nucleus with charge +Ze and a π− meson. A stationary state of the Klein–Gordon equation is described by a wave function ψ(xµ) = ϕ(~r ) e−iεt . (10.188) Inserting this into (10.181) yields (we assume now that the Klein–Gordon equation describes a particle with mass mπ and charge −e) for qV (~r, t) = −Ze2 r and ~A(~r, t) ≡ 0[( ε + Ze2 r )2 + ∇2 − m2 π ] ϕ(~r ) = 0 . (10.189) 312 Relativistic Quantum Mechanics Because of the radial symmetry of the Coulomb potential we express this equation in terms of spherical coordinates r, θ, φ. The Laplacian is ∇2 = 1 r ∂2 r r + 1 r2sin2θ ∂θsinθ∂θ + 1 r2sin2θ ∂2 φ = 1 r ∂2 r r − L̂2 r2 . (10.190) With this expression and after expanding (ε + Ze2 r )2 one obtains( d2 dr2 − L̂2 − Z2e4 r2 + 2εZe2 r + ε2 − m2 π ) r φ(~r) = 0 . (10.191) The operator L̂2 in this equation suggests to choose a solution of the type ϕ(~r ) = R`(r) r Y`m(θ, φ) (10.192) where the functions Y`m(θ, φ) are spherical harmonics, i.e., the eigenfunctions of the operator L̂2 in (10.191) L̂2 Y`m(θ, φ) = ` (` + 1)Y`m(θ, φ) . (10.193) (10.192) leads then to the ordinary differential equation( d2 dr2 − `(`+ 1) − Z2e4 r2 + 2εZe2 r + ε2 − m2 π ) R`(r) = 0 . (10.194) Bound state solutions can be obtained readily noticing that this equation is essentially identical to that posed by the Coulomb problem (potential −Ze2 r ) for the Schrödinger equation( d2 dr2 − `(`+ 1) r2 + 2mπZe 2 r + 2mπE ) R`(r) = 0 (10.195) The latter problem leads to the well-known spectrum En = − mπ (Ze2)2 2n2 ;n = 1, 2, . . . ; ` = 0, 1, . . . n− 1 . (10.196) In this expression the number n′ defined through n′ = n − ` − 1 (10.197) counts the number of nodes of the wave function, i.e., this quantity definitely must be an integer. The similarity of (10.194) and (10.195) can be made complete if one determines λ such that λ(`) (λ(`) + 1) = ` (` + 1) − Z2e4 . (10.198) The suitable choice is λ(`) = −1 2 + √ (` + 1 2 )2 − Z2e4 (10.199) 10.6: Klein–Gordon Equation with Electromagnetic Field 313 and one can write (10.194)( d2 dr2 − λ(`) (λ(`) + 1 ) r2 + 2εZe2 r + ε2 − m2 π ) R`(r) = 0 . (10.200) The bound state solutions of this equation should correspond to ε values which can be obtained from (10.196) if one makes the replacement E −→ ε2 − m2 π 2mπ , ` −→ λ(`) , e2 −→ e2 ε mπ . (10.201) One obtains ε2 − m2 π 2mπ = − mπ Z 2e4 ε2 m2 π 2 (n′ + λ(`) + 1)2 . (10.202) Solving this for ε (choosing the root which renders ε ≤ mπ, i.e., which corresponds to a bound state) yields ε = mπ√ 1 + Z2e4 (n′+λ(`)+1 )2 ; n′ = 0, 1, . . . ; ` = 0, 1, . . . . (10.203) Using (10.197, 10.199) and definingEKG = ε results in the spectrum EKG(n, `,m) = mπ√ 1 + Z2e4 (n− `− 1 2 + √ (`+ 1 2 )2−Z2e4)2 n = 1, 2, . . . ` = 0, 1, . . . , n− 1 m = −`,−`+ 1, . . . ,+` (10.204) In order to compare this result with the spectrum of the non-relativistic hydrogen-like atom we expand in terms of the fine structure constant e2 to order O(ε8). Introducing α = Z2e4 and β = `+ 1 2 (10.204) reads 1√ 1 + α (n−β+ √ β2−α)2 (10.205) and one obtains the series of approximations 1√ 1 + α (n−β+ √ β2−α)2 ≈ 1√ 1 + α (n− α 2β +O(α2))2 ≈ 1√ 1 + α n2− α β n+O(α2) 316 Relativistic Quantum Mechanics From this follows that γ0 has real eigenvalues ±1 and γj , j = 1, 2, 3 has imaginary eigenvalues ±i. Accordingly, one can impose the condition γ0 is hermitian ; γj , j = 1, 2, 3 are anti-hermitian . (10.215) For µ 6= ν (10.213) reads γµγν = −γνγµ , (10.216) i.e., the γµ are anti-commuting. From this one can conclude for the determinants of γµ det(γµγν) = det(−γνγµ) = (−1)ddet(γνγµ) = (−1)ddet(γµγν) . (10.217) Obviously, as long as det(γµ) 6= 0 the dimension d of the square matrices γµ must be even so that (−1)d = 1. For d = 2 there exist only three anti-commuting matrices, namely the Pauli matrices σ1, σ2, σ3 for which, in fact, holds ( σj )2 = 11 ; σjσk = −σkσj for j 6= k . (10.218) The Pauli matrices allow one, however, to construct four matrices γµ for the next possible dimension d = 4. A proper choice is γ0 = ( 11 0 0 −11 ) ; γj = ( 0 σj −σj 0 ) , (10.219) Using property (10.218) of the Pauli matrices one can readily prove that condition (10.213) is satisfied. We will argue further below that the choice f γµ, except for similarity trasnformations, is unique. The Dirac Equation Altogether we have shown that the Klein–Gordon equation can be factorized formally ( iγµ∂µ + m ) ( iγµ∂µ − m ) Ψ(xµ) = 0 (10.220) where Ψ(xµ) represents a 4-dimensional wave function, rather than a scalar wave function. From this equation one can conclude that also the following should hold ( iγµ∂µ − m ) Ψ(xµ) = 0 (10.221) which is the celebrated Dirac equation. The Adjoint Dirac Equation The adjoint equation is Ψ†(xµ) ( i(γµ)† ← ∂µ + m ) = 0 (10.222) where we have defined Ψ† = (ψ∗1, ψ ∗ 2, ψ ∗ 3, ψ ∗ 4) and where ← ∂µ denotes the differential operator ∂µ operating to the left side, rather than to the right side. One can readily show using the hermitian 10.7: Dirac Equation 317 property of the Pauli matrices (γ0)† = γ0 and (γj)† = −γj for j = 1, 2, 3 which, in fact, is implied by (10.215). This property can also be written (γµ)† = γ0γµγ0 . (10.223) Inserting this into (10.222) and multiplication from the right by γ0 yields the adjoint Dirac equation Ψ†(xµ) γ0 ( iγµ ← ∂µ + m ) = 0 . (10.224) Similarity Transformations of the Dirac Equation - Chiral Representation The Dirac equation can be subject to any similarity transformation defined through a non-singular 4× 4–matrix S. Defining a new representation of the wave function Ψ̃(xµ) Ψ̃(xµ) = SΨ(xµ) (10.225) leads to the ‘new’ Dirac equation ( iγ̃µ∂µ − m ) Ψ̃(xµ) = 0 (10.226) where γ̃µ = S γµS−1 (10.227) A representation often adopted beside the one given by (10.222, 10.219) is the socalled chiral representation defined through Ψ̃(xµ) = SΨ(xµ) ; S = 1√ 2 ( 11 11 11 −11 ) (10.228) and γ̃0 = ( 0 11 11 0 ) ; γ̃j = ( 0 −σj σj 0 ) , j = 1, 2, 3 . (10.229) The similarity transformation (10.227) leaves the algebra of the Dirac matrices unaffected and commutation property (10.213) still holds, i.e., [γ̃µ, γ̃ν ]+ = 2 gµν . (10.230) Exercise 10.7.1: Derive (refeq:Dirac-intro20a) from (10.213), (10.227). Schrödinger Form of the Dirac Equation Another form in which the Dirac equation is used often results from multiplying (10.221) from the left by γ0 ( i ∂t + i ~̂α · ∇ − β̂ m ) Ψ(~r, t) = 0 (10.231) 318 Relativistic Quantum Mechanics where~̂α has the three components αj , j = 1, 2, 3 and β̂ = ( 11 0 0 −11 ) ; α̂j = ( 0 σj σj 0 ) , j = 1, 2, 3 . (10.232) This form of the Dirac equation is called the Schrödinger form since it can be written in analogy to the time-dependent Schrödinger equation i∂tΨ(xµ) = HoΨ(xµ) ; Ho = ~̂α · ~̂p + β̂ m . (10.233) The eigenstates and eigenvalues ofH correspond to the stationary states and energies of the particles described by the Dirac equation. Clifford Algebra and Dirac Matrices The matrices defined through dj = iγj , j = 1, 2, 3 ; d4 = γ0 (10.234) satisfy the anti-commutation property djdk + dkdj = { 2 for j = k 0 for j 6= k (10.235) as can be readily verified from (10.213). The associative algebra generated by d1 . . . d4 is called a Clifford algebra C4. The three Pauli matrices also obey the property (10.235) and, hence, form a Clifford algebra C3. The representations of Clifford algebras Cm are well established. For example, in case of C4, a representation of the dj ’s is d1 = ( 0 1 1 0 )⊗( 1 0 0 1 ) ; d2 = ( 1 0 0 −1 )⊗( 0 1 1 0 ) d3 = ( 0 i −i 0 )⊗( 1 0 0 1 ) ; d4 = ( 1 0 0 −1 )⊗( 0 i −i 0 ) (10.236) where ‘⊗’ denotes the Kronecker product between matrices, i.e., the matrix elements of C = A⊗B are Cjk,`m = Aj`Bkm. The Clifford algebra C4 entails a subgroup G4 of elements ± dj1dj2 · · · djs , j1 < j2 < · · · < js s ≤ 4 (10.237) which are the ordered products of the operators ±11 and d1, d2, d3, d4. Obviously, any product of the dj ’s can be brought to the form (10.237) by means of the property (10.235). There are (including the different signs) 32 elements in G4 which we define as follows Γ±1 = ±11 Γ±2 = ±d1 , Γ±3 = ±d2 , Γ±4 = ±d3 , Γ±5 = ±d4 Γ±6 = ±d1d2 , Γ±7 = ±d1d3 , Γ±8 = ±d1d4 , Γ±9 = ±d2d3 Γ±10 = ±d2d4 , Γ±11 = ±d3d4 Γ±12 = ±d1d2d3 , Γ±13 = ±d1d2d4 , Γ±14 = ±d1d3d4 , Γ±15 = ±d2d3d4 Γ±16 = ±d1d2d3d4 (10.238) 10.8: Lorentz Invariance of the Dirac Equation 321 from which we can conclude σµνεµν = −σµνενµ = −σνµεµν , i.e., it must hold σµν = −σνµ . (10.250) One can readily show expanding SS−1 = 11 to first order in εµν that for the inverse infinitesimal transformation holds S−1(εµν) = 11 + i 4 σµνε µν (10.251) Inserting (10.248, 10.251) into (10.246) results then in a condition for the generators σµν − i 4 (σαβγµ − γµσαβ ) εαβ = ενµγν . (10.252) Since six of the coefficients εαβ can be chosen independently, this condition can actually be expressed through six independent conditions. For this purpose one needs to express formally the r.h.s. of (10.252) also as a sum over both indices of εαβ . Furthermore, the expression on the r.h.s., like the expression on the l.h.s., must be symmetric with respect to interchange of the indices α and β. For this purpose we express ενµγν = 1 2 εαµγα + 1 2 εβµγβ = 1 2 εαβ δµβγα + 1 2 εαβ δµαγβ = 1 2 εαβ ( δµβγα − δµαγβ ) . (10.253) Comparing this with the l.h.s. of (10.252) results in the condition for each α, β [σαβ, γµ ]− = 2i ( δµβγα − δµαγβ ) . (10.254) The proper σαβ must be anti-symmetric in the indices α, β and operate in the same space as the Dirac matrices. In fact, a solution of condition (10.254) is σαβ = i 2 [ γα, γβ ]− (10.255) which can be demonstrated using the properties (10.213, 10.216) of the Dirac matrices. Exercise 7.5: Show that the σαβ defined through (10.255) satisfy condition (10.254). Algebra of Generators of Bispinor Transformation We want to construct the bispinor Lorentz transformation by exponentiating the generators σµν . For this purpose we need to verify that the algebra of the generators involving addition and mul- tiplication is closed. For this purpose we inspect the properties of the generators in a particular representation, namely, the chiral representation introduced above in Eqs. (10.228, 10.229). In this representation the Dirac matrices γ̃µ = (γ̃0,−~̃γ) are γ̃0 = ( 0 11 11 0 ) ; γ̃j = ( 0 σj −σj 0 ) , j = 1, 2, 3 . (10.256) 322 Relativistic Quantum Mechanics One can readily verify that the non-vanishing generators σ̃µν are given by (note σ̃µν = −σ̃νµ, i.e. only six generators need to be determined) σ̃0j = i 2 [γ̃0, γ̃j ] = ( −iσj 0 0 iσj ) ; σ̃jk = [γ̃j , γ̃k] = εjk` ( σ` 0 0 σ` ) . (10.257) Obviously, the algebra of these generators is closed under addition and multiplication, since both operations convert block-diagonal operators( A 0 0 B ) (10.258) again into block-diagonal operators, and since the algebra of the Pauli matrices is closed. We can finally note that the closedness of the algebra of the generators σµν is not affected by similarity transformations and that, therefore, any representation of the generators, in particular, the representation (10.255) yields a closed algebra. Finite Bispinor Transformation The closedness of the algebra of the generators σµν defined through (10.248) allows us to write the transformation S for any, i.e., not necessarily infinitesimal, εµν in the exponential form S = exp ( − i 4 σµνε µν ) . (10.259) We had stated before that the transformation S is actually determined through the Lorentz trans- formation Lµν . One should, therefore, be able to state S in terms of the same parameters ~w and ~ϑ as the Lorentz transformation in (10.51). In fact, one can express the tensor εµν through ~w and ~ϑ using εµν = εµρg ρν and the expression (10.44) εµν =  0 −w1 −w2 −w3 w1 0 ϑ3 −ϑ2 w2 −ϑ3 0 ϑ1 w3 ϑ2 −ϑ1 0  (10.260) Inserting this into (10.259) yields the desired connection between the Lorentz transformation (10.51) and S. In order to construct an explicit expression of S in terms of ~w and ~ϑ we employ again the chiral representation. In this representation holds − i 4 σ̃µνε µν = − i 2 (σ̃01ε 01 + σ̃02ε 02 + σ̃03ε 03 + σ̃12ε 12 + σ̃13ε 13 + σ̃23ε 23) = 1 2 ( (~w − i~ϑ) · ~σ 0 0 −(~w + i~ϑ) · ~σ ) . (10.261) We note that this operator is block-diagonal. Since such operator does not change its block-diagonal form upon exponentiation the bispinor transformation (10.259) becomes in the chiral representation S̃(~w, ~ϑ) = ( e 1 2 (~w− i~ϑ)·~σ 0 0 e− 1 2 (~w+ i~ϑ)·~σ ) (10.262) This expression allows one to transform according to (10.241) bispinor wave functions from one frame of reference into another frame of reference. 10.8: Lorentz Invariance of the Dirac Equation 323 Current 4-Vector Associated with Dirac Equation We like to derive now an expression for the current 4-vector jµ associated with the Dirac equation which satisfies the conservation law ∂µj µ = 0 . (10.263) Starting point are the Dirac equation in the form (10.221) and the adjoint Dirac equation (10.224). Multiplying (10.221) from the left by Ψ†(xµ)γ0, (10.224) from the right by Ψ(xµ), and addition yields Ψ†(xµ)γ0 ( iγµ∂µ + iγµ ← ∂µ ) Ψ(xµ) = 0 . (10.264) The last result can be written ∂µ Ψ†(xν)γ0γµΨ(xν) = 0 , (10.265) i.e., the conservation law (10.263) does hold, in fact, for jµ(xµ) = (ρ, ~j) = Ψ†(xµ)γ0γµΨ(xµ) . (10.266) The time-like component ρ of jµ ρ(xµ) = Ψ†(xµ)Ψ(xµ) = 4∑ s=1 |ψs(xµ)|2 (10.267) has the desired property of being positive definite. The conservation law (10.263) allows one to conclude that jµ must transform like a contravariant 4-vector as the notation implies. The reason is that the r.h.s. of (10.263) obviously is a scalar under Lorentz transformations and that the left hand side must then also transform like a scalar. Since ∂µ transforms like a covariant 4-vector, jµ must transform like a contravariant 4-vector. This transformation behaviour can also be deduced from the transformation properties of the bispinor wave function Ψ(xµ). For this purpose we prove first the relationship S−1 = γ0 S†γ0 . (10.268) We will prove this property in the chiral representation. Obviously, the property applies then in any representation of S. For our proof we note first S̃−1(~w, ~ϑ) = S̃(−~w,−~ϑ) = ( e− 1 2 (~w− i~ϑ)·~σ 0 0 e 1 2 (~w+ i~ϑ)·~σ ) (10.269) One can readily show that the same operator is obtained evaluating γ̃0S̃†(~w, ~ϑ)γ̃0 = ( 0 11 11 0 ) ( e 1 2 (~w+ i~ϑ)·~σ 0 0 e− 1 2 (~w− i~ϑ)·~σ ) ( 0 11 11 0 ) . (10.270) We conclude that (10.268) holds for the bispinor Lorentz transformation. We will now determine the relationship between the flux j′ µ = Ψ′†(x′µ)γ0γµΨ′(x′) (10.271) 326 Relativistic Quantum Mechanics For our further characterization we will deal with the positive and negative energy solutions [cf. (10.281)] separately. For the positive energy solution, i.e., the solution for ε = +E(~p), we present φo through the normalized vector φo = ( u1 u2 ) = u ∈ C 2 , u†u = |u1|2 + |u2|2 = 1 . (10.285) The corresponding free Dirac particle is then described through the wave function Ψ(~p,+|xµ) = N+(~p) ( u ~σ·~p E(~p) +m u ) ei(~p·~r− εt) , ε = +E(~p) . (10.286) Here N+(~p) is a constant which will be chosen to satisfy the normalization condition Ψ†(~p,+) γ0 Ψ†(~p,+) = 1 , (10.287) the form of which will be justified further below. Similarly, we present the negative energy solution, i.e., the solution for ε = −E(~p), through χo given by χo = ( u1 u2 ) = u ∈ C2 , u†u = |u1|2 + |u2|2 = 1 . (10.288) corresponding to the wave function Ψ(~p,−|xµ) = N−(~p) ( −~σ·~p E(~p) +m u u ) ei(~p·~r− εt) , ε = −E(~p) . (10.289) Here N−(~p) is a constant which will be chosen to satisfy the normalization condition Ψ†(~p,+) γ0 Ψ†(~p,+) = −1 , (10.290) which differs from the normalization condition (10.287) in the minus sign on the r.h.s. The form of this condition and of (10.287) will be justified now. First, we demonstrate that the product Ψ†(~p,±)γ0Ψ(~p,±), i.e., the l.h.s. of (10.287, 10.290), is invariant under Lorentz transformations. One can see this as follows: Let Ψ(~p,±) denote the solution of a free particle moving with momentum ~p in the laboratory frame, and let Ψ(0,±) denote the corresponding solution of a particle in its rest frame. The connection between the solutions, according to (10.241), is Ψ(~p,±) = SΨ(0,±) , where S is given by (10.259). Hence, Ψ†(~p,±) γ0 Ψ(~p,±) = ( Ψ†(0,±)S† ) γ0 SΨ(0,±) = Ψ†(0,±) ( γ0 S−1 γ0 ) γ0 SΨ(0,±) = Ψ†(0,±) γ0 Ψ(0,±). (10.291) Note that we have used that, according to (10.268), S−1 = γ0S†γ0 and, hence, S† = γ0S−1γ0. We want to demonstrate now that the signs on the r.h.s. of (10.287, 10.290) should differ. For this purpose we consider first the positive energy solution. Employing (10.286) for ~p = 0 yields, using γ0 as given in (10.219) and u†u = 1 [c.f. (10.285)], Ψ†(0,+) γ0 Ψ(0,+) = |N+(0)|2 (u†, 0) γ0 ( u 0 ) = |N+(0)|2 . (10.292) 10.9: Solutions of the Free Particle Dirac Equation 327 The same calculation for the negative energy wave function as given in (10.289) yields Ψ†(0,−) γ0 Ψ(0,−) = |N−(0)|2 (0, u†) γ0 ( 0 u ) = − |N−(0)|2 . (10.293) Obviously, this requires the choice of a negative side on the r.h.s. of (10.290) to assign a positive value to |N−(0)|2. We can also conclude from our derivation N±(0) = 1 . (10.294) We want to determine now N±(~p) for arbitrary ~p. We consider first the positive energy solution. Condition (10.287) written explicitly using (10.286) is N 2 +(~p) ( (u∗)T , [ ~σ · ~p E(~p) + m u∗ ]T) γo ( u ~σ·~p E(~p) +m u ) = 1 (10.295) Evaluating the l.h.s. using γ0 as given in (10.219) yields N 2 +(~p) [ u†u − u† (~σ · ~p)2 (E(~p) + m)2 u ] = 1 . (10.296) Replacing (~σ · ~p)2 by ~p 2 [c.f. (5.234)] and using the normalization of u in (10.285) results in N 2 +(~p) [ 1 − ~p 2 (E(~p) + m)2 ] = 1 (10.297) from which follows N+(~p) = √ (m + E(~p) )2 (m + E(~p) )2 − ~p 2 . (10.298) Noting (m + E(~p) )2 − ~p 2 = m2 − ~p 2 + 2mE(~p) + E2(~p) = 2(m + E(~p))m (10.299) the normalization coefficient (10.298) becomes N+(~p) = √ m + E(~p) 2m . (10.300) This result completes the expression for the wave function (10.286). Exercise 7.6: Show that the normalization condition N ′2+ (~p) ( (u∗)T , [ ~σ · ~p E(~p) + m u∗ ]T) ( u ~σ·~p E(~p) +m u ) = 1 (10.301) yields the normalization coefficient N ′+(~p) = √ m + E(~p) 2 E(~p) . (10.302) 328 Relativistic Quantum Mechanics We consider now the negative energy solution. Condition (10.290) written explicitly using (10.289) is N 2 −(~p) ([ −~σ · ~p E(~p) + m u∗ ]T , (u∗)T ) γo ( −~σ·~p E(~p) +m u u ) = −1 (10.303) Evaluating the l.h.s. yields N 2 −(~p) [ u† (~σ · ~p)2 (E(~p) + m)2 u − u†u ] = −1 . (10.304) This condition is, however, identical to the condition (10.296) for the normalization constant N+(~p) of the positive energy solution. We can, hence, conclude N−(~p) = √ m + E(~p) 2m (10.305) and, thereby, have completed the determination for the wave function (10.289). The wave functions (10.286, 10.289, 10.300) have been constructed to satisfy the free particle Dirac equation (10.275). Inserting (10.286) into (10.275) yields Ho Ψ(~p, λ|xµ) = λE(~p) Ψ(~p, λ|xµ) , (10.306) i.e., the wave functions constructed represent eigenstates of Ho. The wave functions are also eigenstates of the momentum operator i∂µ, i.e., i∂µΨ(~p, λ|xµ) = pµ Ψ(~p, λ|xµ) (10.307) where pµ = (ε,−~p). This can be verified expressing the space–time factor of Ψ(~p, λ|xµ) in 4-vector notation, i.e., exp[i(~p · ~r − εt)] = exp(ipµxµ). Helicity The free Dirac particle wave functions (10.286, 10.289) are not completely specified, the two com- ponents of u indicate another degree of freedom which needs to be defined. This degree of freedom describes a spin–1 2 attribute. This attribute is the so-called helicity, defined as the component of the particle spin along the direction of motion. The corresponding operator which measures this observable is Λ = 1 2 σ · ~̂p |~̂p| . (10.308) Note that ~̂p represents here an operator, not a constant vector. Rather than considering the observable (10.307) we investigate first the observable due to the simpler operator ~σ · ~̂p. We want to show that this operator commutes with Ho and ~̂p to ascertain that the free particle wave function can be simultaneously an eigenvector of all three operators. The commutation property [~σ · ~̂p, p̂j ] = 0 , j = 1, 2, 3 is fairly obvious. The property [~σ · ~̂p,Ho] = 0 follows from (10.233) and from the two identities( 11 0 0 −11 )( ~σ 0 0 ~σ ) · ~̂p − ( ~σ 0 0 ~σ ) · ~̂p ( 11 0 0 −11 ) = 0 (10.309) 10.9: Solutions of the Free Particle Dirac Equation 331 Ψ̃(p = 0, +, −1 2 |t) = 1√ 2  0 1 0 1  e−imt , Ψ̃(p = 0, −, 1 2 |t) = 1√ 2  1 0 −1 0  e+imt , Ψ̃(p = 0, −, −1 2 |t) = 1√ 2  0 1 0 −1  e+imt . (10.322) The reader can readily verify that transformation of these solutions to the Dirac representationsas defined in (10.228) yields the corresponding solutions (10.311, 10.313) in the p → 0 limit. This correspondence justifies the characterization ±,±1 2 of the wave functions stated in (10.322). The solutions (10.322) can be written in spinor form 1√ 2 ( φo χo ) e∓imt , φo , χo ∈ {( 1 0 ) , ( 0 1 )} (10.323) Transformation (10.262) for a boost in the x3–direction, i.e., for ~w = (0, 0, w3), yields for the exponential space–time dependence according to (10.174, 10.176) ∓imt → i ( p3x 3 ∓ Et ) (10.324) and for the bispinor part according to (10.262)( φo χo ) → ( e 1 2 w3σ3 0 0 e− 1 2 w3σ3 ) ( φo χo ) = ( e 1 2 w3σ3 φo e− 1 2 w3σ3 χo ) . (10.325) One should note that φo, χo are eigenstates of σ3 with eigenvalues ±1. Applying (10.324, 10.325) to (10.323) should yield the solutions for non-vanishing momentum p in the x3–direction. For the resulting wave functions in the chiral representation one can use then a notation corresponding to that adopted in (10.311) Ψ̃(p(w3)ê3, +, +1 2 |~r, t) = 1√ 2  e 1 2 w3 ( 1 0 ) e− 1 2 w3 ( 1 0 )  ei(px 3−Ept) Ψ̃(p(w3)ê3, +, −1 2 |~r, t) = 1√ 2  e− 1 2 w3 ( 0 1 ) e 1 2 w3 ( 0 1 )  ei(px 3−Ept) 332 Relativistic Quantum Mechanics Ψ̃(p(w3)ê3, −, +1 2 |~r, t) = 1√ 2  e 1 2 w3 ( 1 0 ) −e− 1 2 w3 ( 1 0 )  ei(px 3 +Ept) Ψ̃(p(w3)ê3, −, −1 2 |~r, t) = 1√ 2  e− 1 2 w3 ( 0 1 ) − e 1 2 w3 ( 0 1 )  ei(px 3 +Ept) (10.326) where according to (10.61) p(w3) = m sinhw3. Transformation to the Dirac representation by means of (10.228) yields Ψ(p(w3)ê3, +, +1 2 |~r, t) =  coshw3 2 ( 1 0 ) sinhw3 2 ( 1 0 )  ei(px 3−Ept) Ψ(p(w3)ê3, +, −1 2 |~r, t) =  coshw3 2 ( 0 1 ) − sinhw3 2 ( 0 1 )  ei(px 3−Ept) Ψ(p(w3)ê3, −, +1 2 |~r, t) =  sinhw3 2 ( 1 0 ) coshw3 2 ( 1 0 )  ei(px 3 +Ept) Ψ(p(w3)ê3, −, −1 2 |~r, t) =  − sinhw3 2 ( 0 1 ) coshw3 2 ( 0 1 )  ei(px 3 +Ept) (10.327) Employing the hyperbolic function properties cosh x 2 = √ coshx + 1 2 , sinh x 2 = √ coshx − 1 2 , (10.328) the relationship (10.61) between the parameter w3 and boost velocity v3, and the expression (10.311) for Ep one obtains coshw3 2 = 1√ 2 √√√√√ 1 1 − v2 3 + 1 = 1√ 2 √√√√√1 + v2 3 1 − v2 3 + 1 = √√√√√ √ m2 + m2v2 3 1− v2 3 + m 2m = √ Ep + m 2m (10.329) 10.9: Invariance of Dirac Equation Revisited 333 and similarly sinh w3 2 = √ Ep − m 2m = p√ 2m (Ep + m) (10.330) Inserting expressions (10.329, 10.330) into (10.327), indeed, reproduces the positive energy wave functions (10.311) as well as the negative energy solutions (10.313) for −p. The change of sign for the latter solutions had to be expected as it was already noted for the negative energy solutions of the Klein–Gordon equation (10.168–10.176). Invariance of Dirac Equation Revisited At this point we like to provide a variation of the derivation of (10.243), the essential property stating the Lorentz–invariance of the Dirac equation. Actually, we will derive this equation only for infinitesimal transformations, which however, is sufficient since (1) it must hold then for any finite transformation, and since (2) the calculations following (10.243) considered solely the limit of infinitesimal transformations anyway. The reason why we provide another derivation of (10.243) is to familiarize ourselves with a formu- lation of Lorentz transformations of the bispinor wave finction Ψ(xµ) which treats the spinor and the space-time part of the wave function on the same footing. Such description will be essential for the formal description of Lorentz invariant wave equations for arbitray spin further below. In the new derivation we consider the particle described by the wave function transformed, but not the observer. This transformation, refered to as the active transformation, expresses the system in the old coordinates. The transformation is Ψ′(xµ) = S(Lηξ) ρ(Lηξ) Ψ(xµ) (10.331) where S(Lηξ) denotes again the transformation acting on the bispinor character of the wave function Ψ(xµ) and where ρ(Lηξ) denotes the transformation acting on the space-time character of the wave function Ψ(xµ). ρ(Lηξ) has been defined in (10.123) above and characterized there. Such transformation had been applied by us, of course, when we generated the solutions Ψ(~p, λ,Λ|xµ) from the solutions describing particles at rest Ψ(~p = 0, λ,Λ|t). We expect, in general, that if Ψ(xµ) is a solution of the Dirac equation that Ψ′(xµ) as given in (10.331) is a solution as well. Making this expectation a postulate allows one to derive the condition (10.243) and, thereby, the proper transformation S(Lηξ). To show this we rewrite the Dirac equation (10.221) using (10.331)( i S(Lηξ)γµS−1(Lηξ) ρ(Lηξ)∂µρ−1(Lηξ) − m ) Ψ′(xµ) = 0 (10.332) Here we have made use of the fact that S(Lηξ) commutes with ∂µ and ρ(Lηξ) commutes with γµ. The fact that any such Ψ′(xµ) is a solution of the Dirac equation allows us to conclude S(Lηξ)γµS−1(Lηξ) ρ(Lηξ)∂µρ−1(Lηξ) = γν∂ν (10.333) which is satisfied in case that the following conditions are met ρ(Lηξ)∂µρ−1(Lηξ) = Lνµ ∂ν ; S(Lηξ)γµS−1(Lηξ)Lνµ = γν . (10.334) 336 Relativistic Quantum Mechanics and assume that for the time-derivative of Φ and X holds∣∣∣∣∂tΦΦ ∣∣∣∣ << m , ∣∣∣∣∂tXX ∣∣∣∣ << m . (10.352) Using (10.350, 10.351) in (10.348, 10.349) yields i∂tΦ = ~σ · ~̂πX + qV Φ (10.353) i∂tX = ~σ · ~̂πΦ + qV X − 2mX . (10.354) The properties (10.346, 10.352) allow one to approximate (10.354) 0 ≈ ~σ · ~̂π Φ − 2m X . (10.355) and, accordingly, one can replace X in (10.353) by X ≈ ~σ · ~̂π 2m Φ (10.356) to obtain a closed equation for Φ i∂tΦ ≈ ( ~σ · ~̂π )2 2m Φ + qV Φ . (10.357) Equation (10.356), due to the m−1 factor, identifies X as the small component of the bi-spinor wave function which, henceforth, does not need to be considered anymore. Equation (10.357) for Φ can be reformulated by expansion of (~σ · ~̂π)2. For this purpose we employ the identity (5.230), derived in Sect. 5.7, which in the present case states (~σ · ~̂π)2 = ~̂π 2 + i ~σ · (~̂π × ~̂π) . (10.358) For the components of ~̂π × ~̂π holds( ~̂π × ~̂π ) ` = εjk` (πjπk − πkπj ) = εjk` [πj , π`] . (10.359) We want to evaluate the latter commutator. One obtains [πj , πk] = [ 1 i ∂j + qAj , 1 i ∂k + qAk] = [ 1 i ∂j , 1 i ∂k]︸ ︷︷ ︸ = 0 + q [Aj , 1 i ∂k] + q [ 1 i ∂j , Ak] + q2 [Aj , Ak]︸ ︷︷ ︸ = 0 = q i [Aj , ∂k] + q i [∂j , Ak] . (10.360) For an arbitrary function f(~r) holds ( [Aj , ∂k] + [∂j , Ak] ) f = ( ∂jAk − Ak∂j + Aj∂k − ∂kAj ) f . (10.361) 10.10: Dirac Particles in Electromagnetic Field 337 Using ∂jAk f = ((∂jAk)) f + Ak∂j f ∂kAj f = ((∂kAj)) f + Aj∂k f where ((∂j · · ·)) denotes confinement of the differential operator to within the brackets ((· · ·)), one obtains ( [Aj , ∂k] + [∂j , Ak] ) f = [ ((∂jAk)) − ((∂kAj)) ] f (10.362) or, using (10.360) and Aµ = (V, − ~A), [πj , πk] = q i (( ∂jAk − ∂kAj )) = −q i ( ∇× ~A ) ` εjk` = −q i B` εjk` (10.363) where we employed ~B(~r, t) = ∇ × ~A(~r, t) [see (8.6)]. Equations (10.344, 10.358, 10.359, 10.363) allow us to write (10.357) in the final form i∂tΦ(~r, t) ≈ [ [~̂p − q ~A(~r, t)]2 2m − q 2m ~σ · ~B(~r, t) + q V (~r, t) ] Φ(~r, t) (10.364) which is referred to as the Pauli equation. Comparision of (10.364) governing a two-dimensional wave function Φ ∈ C2 with the corresponding non-relativistic Schrödinger equation (10.2) governing a one-dimensional wave function ψ ∈ C, reveals a stunning feature: the Pauli equation does justice to its two-dimensional character; while agreeing in all other respects with the non-relativistic Schrödinger equation (10.2) it introduces the extra term q~σ · ~B Φ which describes the well-known interaction of a spin-1 2 particle with a magnetic field ~B. In other words, the spin-1 2 which emerged in the Lorentz-invariant theory as an algebraic necessity, does not leave the theory again when one takes the non-relativistic limit, but rather remains as a steady “guest” of non-relativistic physics with the proper interaction term. Let us consider briefly the consequences of the interaction of a spin-1 2 with the magnetic field. For this purpose we disregard the spatial degrees of freedom and assume the Schrödinger equation i∂tΦ(t) = q ~σ · ~B Φ(t) . (10.365) The formal solution of this equation is Φ(t) = e−iqt ~B·~σ Φ(0) . (10.366) Comparision of this expression with (5.222, 5.223) shows that the propagator in (10.366) can be interpreted as a rotation around the field ~B by an angle qtB, i.e., the interaction q~σ · ~B induces a precession of the spin-1 2 around the magnetic field. Dirac Particle in Coulomb Field - Spectrum We want to describe now the spectrum of a relativistic electron (q = −e) in the Coulomb field of a nucleus with charge Ze. The respective bispinor wave function Ψ(xµ) ∈ C4 is described as the stationary solution of the Dirac equation (10.343) for the vector potential Aµ = (−Ze 2 r , 0, 0, 0) . (10.367) 338 Relativistic Quantum Mechanics For the purpose of the solution we assume the chiral representation, i.e, we solve [ iγ̃µ(∂µ + iqAµ) − m ] Ψ̃(xµ) = 0 (10.368) where Ψ̃(xµ) and γ̃µ are defined in (10.228) and in (10.229), respectively. Employing πµ as defined in Table 10.1 one can write (10.368) ( γ̃µ πµ − m ) Ψ̃(xµ) = 0 . (10.369) For our solution we will adopt presently a strategy which follows closely that for the spectrum of pionic atoms in Sect. 10.6. For this purpose we ‘square’ the Dirac equation, multiplying (10.369) from the left by γνπν + m. This yields [ iγ̃µ(∂µ + iqAµ) + m ] [ iγ̃µ(∂µ + iqAµ) − m ] Ψ̃(xµ) = (γ̃µπ̂µ γ̃ν π̂ν − m2) Ψ̃(xµ) = 0 . (10.370) Any solution of (10.368) is also a solution of (10.370), but the converse is not necessarily true. However, once a solution Ψ̃(xµ) of (10.370) is obtained then [ iγ̃µ(∂µ + iqAµ) + m ] Ψ̃(xµ) (10.371) is a solution of (10.369). This follows from [ iγ̃µ(∂µ + iqAµ) + m ] [ iγ̃µ(∂µ + iqAµ) − m ] = [ iγ̃µ(∂µ + iqAµ) − m ] [ iγ̃µ(∂µ + iqAµ) + m ] (10.372) according to which follows from (10.370) [ iγ̃µ(∂µ + iqAµ) − m ] [ iγ̃µ(∂µ + iqAµ) + m ] Ψ̃(xµ) = 0 (10.373) such that we can conclude that (10.371), indeed, is a solution of (10.369). Equation (10.370) resembles closely the Klein-Gordon equation (10.180), but differs from it in an essential way. The difference arises from the term γ̃µπ̂µ γ̃ ν π̂ν in (10.370) for which holds γ̃µπ̂µ γ̃ ν π̂ν = 3∑ µ=0 (γ̃µ)2 π̂2 µ + ∑ µ,ν=1 µ 6=ν γ̃µγ̃ν π̂µπ̂ν . (10.374) The first term on the r.h.s. can be rewritten using, according to (10.230), (γ̃0)2 = 11 and (γ̃j)2 = −11, j = 1, 2, 3, 3∑ µ=0 (γ̃µ)2 π̂2 µ = π̂2 0 − ~̂π 2 . (10.375) Following the algebra that connected Eqs. (5.231), (5.232) in Sect. 5.7 one can write the second term in (10.374), noting from (10.230) γ̃µγ̃ν = − γ̃ν γ̃µ, µ 6= ν and altering ‘dummy’ summation 10.10: Dirac Particles in Electromagnetic Field 341 momentum operator L̂2 [cf. (6.151]. We select, therefore, a specific pair of total spin-orbital angular momentum quantum numbers j, m and expand φ±(~r) = h±(r) r Yjm(j − 1 2 , 1 2 |r̂) + g±(r) r Yjm(j + 1 2 , 1 2 |r̂) (10.390) Using ~σ · r̂Yjm(j ± 1 2 , 1 2 |r̂) = −Yjm(j ∓ 1 2 , 1 2 |r̂) (10.391) derived in Sect. 6.5 [c.f. (6.186)], property (6.151), which states that the states Yjm(j ± 1 2 , 1 2 |r̂) are eigenfunctions of L̂2, together with the orthonormality of these two states leads to the coupled differential equation[( ∂2 r + 2Ze2ε r + ε2 − m ) ( 1 0 0 1 ) (10.392) − 1 r2 ( (j − 1 2 )(j + 1 2 ) − Z2e4 ±iZe2 ±i Ze2 (j + 1 2 )(j + 3 2 ) − Z2e4 )]( h±(r) g±(r) ) = 0 . We seek to bring (10.392) into diagonal form. Any similarity transformation leaves the first term in (10.392), involving the 2× 2 unit matrix, unaltered. However, such transformation can be chosen as to diagonalize the second term. Since, in the present treatment, we want to determine solely the spectrum, not the wave functions, we require only the eigenvalues of the matrices B± = ( (j − 1 2 )(j + 1 2 ) − Z2e4 ±iZe2 ±i Ze2 (j + 1 2 )(j + 3 2 ) − Z2e4 ) , (10.393) but do not explicitly consider further the wavefunctions. Obviously, the eigenvalues are independent of m. The two eigenvalues of both matrices are identical and can be written in the form λ1(j) [λ1(j) + 1] and λ2(j) [λ2(j) + 1] (10.394) where λ1(j) = √ (j + 1 2 )2 − Z2e4 (10.395) λ2(j) = √ (j + 1 2 )2 − Z2e4 − 1 (10.396) Equation (10.392) reads then in the diagonal representation( ∂2 r − λ1,2(j)[λ1,2(j) + 1) r2 + 2εZe2 r + ε2 − m2 ) f1,2(r) = 0 (10.397) This equation is identical to the Klein-Gordon equation for pionic atoms written in the form (10.200), except for the slight difference in the expression of λ1,2(j) as given by (10.395, 10.396) and (10.199), namely, the missing additive term − 1 2 , the values of the argument of λ1,2(j) being j = 1 2 , 3 2 , . . . rather than ` = 0, 1, . . . as in the case of pionic atoms, and except for the fact that we have two sets of values for λ1,2(j), namely, λ1(j) and λ2(j).. We can, hence, conclude that the 342 Relativistic Quantum Mechanics spectrum of (10.397) is again given by eq. (10.203), albeit with some modifications. Using (10.395, 10.396) we obtain, accordingly, ε1 = m√ 1 + Z2e4 (n′ + 1 + √ (j+ 1 2 )2−Z2e4)2 ; (10.398) ε2 = m√ 1 + Z2e4 (n′ + √ (j+ 1 2 )2−Z2e4)2 ; (10.399) n′ = 0, 1, 2, . . . , j = 1 2 , 3 2 , . . . , m = −j,−j + 1, . . . , j where ε1 corresponds to λ1(j) as given in (10.395) and ε2 corresponds to λ2(j) as given in (10.396). For a given value of n′ the energies ε1 and ε2 for identical j-values correspond to mixtures of states with orbital angular momentum ` = j − 1 2 and ` = j + 1 2 . The magnitude of the relativistic effect is determined by Z2e4. Expanding the energies in terms of this parameter allows one to identify the relationship between the energies ε1 and ε2 and the non-relativistic spectrum. One obtains in case of (10.398, 10.399) ε1 ≈ m − mZ2e4 2 (n′ + j + 3 2 )2 + O(Z4e8) (10.400) ε2 ≈ m − mZ2e4 2 (n′ + j + 1 2 )2 + O(Z4e8) (10.401) n′ = 0, 1, 2, . . . , j = 1 2 , 3 2 , . . . , m = −j,−j + 1, . . . , j . These expressions can be equated with the non-relativistic spectrum. Obviously, the second term on the r.h.s. of these equations describe the binding energy. In case of non-relativistic hydrogen-type atoms, including spin- 1 2 , the stationary states have binding energies E = −mZ 2e4 2n2 , n = 1, 2, . . . ` = 0, 1, . . . , n− 1 m = −`,−`+ 1, . . . , ` ms = ± 1 2 . (10.402) In this expression n is the so-called main quantum number. It is given by n = n′ + ` + 1 where ` is the orbital angular momentum quantum number and n′ = 0, 1, . . . counts the nodes of the wave function. One can equate (10.402) with (10.400) and (10.401) if one attributes to the respective states the angular momentum quantum numbers ` = j + 1 2 and ` = j − 1 2 . One may also state this in the following way: (10.400) corresponds to a non-relativistic state with quantum numbers n, ` and spin-orbital angular momentum j = `− 1 2 ; (10.401) corresponds to a non-relativistic state with quantum numbers n, ` and spin-orbital angular momentum j = `+ 1 2 . These considerations are summarized in the following equations ED(n, `, j = `− 1 2 ,m) = m√ 1 + Z2e4 (n− `+ √ (`+1)2−Z2e4)2 ; (10.403) ED(n, `, j = `+ 1 2 ,m) = m√ 1 + Z2e4 (n− `− 1 + √ `2−Z2e4)2 ; (10.404) n = 1, 2, . . . ; ` = 0, 1, . . . , n− 1 ; m = −j,−j + 1, . . . , j 10.10: Dirac Particles in Electromagnetic Field 343 main orbital spin- non-rel. rel. spectr. quantum angular orbital binding binding notation number mom. ang. mom. energy / eV energy / eV n ` j Eq. (10.402) Eq. (10.405 ) 1s 1 2 1 0 1 2 -13.60583 -13.60601 2s 1 2 2 0 1 2 -3.40146 -3.40151 2p 1 2 2 1 1 2 ⇑ ⇑ 2p 3 2 2 1 3 2 ⇑ - 3.40147 3s 1 2 3 0 1 2 -1.51176 -1.551178 3p 1 2 3 1 1 2 ⇑ ⇑ 3p 3 2 3 1 3 2 ⇑ - 1.551177 3d 3 2 3 2 3 2 ⇑ ⇑ 3d 5 2 3 2 5 2 ⇑ - 1.551176 Table 10.2: Binding energies for the hydrogen (Z = 1) atom. Degeneracies are denoted by ⇑. The energies were evaluated with m = 511.0041 keV and e2 = 1/137.036 by means of Eqs. (10.402, 10.405). One can combine the expressions (10.403, 10.404) finally into the single formula ED(n, `, j,m) = m√ 1 + Z2e4 (n− j− 1 2 + √ (j+ 1 2 )2−Z2e4)2 n = 1, 2, . . . ` = 0, 1, . . . , n− 1 j = { 1 2 for ` = 0 `± 1 2 otherwise m = −j,−j + 1, . . . , j (10.405) In order to demonstrate relativistic effects in the spectrum of the hydrogen atom we compare in Table 10.2 the non-relativistic [cf. (10.402)] and the relativistic [cf. (10.405)] spectrum of the hydrogen atom. The table entries demonstrate that the energies as given by the expression (10.405) in terms of the non-relativistic quantum numbers n, ` relate closely to the corresponding non- relativistic states, in fact, the non-relativistic and relativistic energies are hardly discernible. The reason is that the mean kinetic energy of the electron in the hydrogen atom, is in the range of 10 eV, i.e., much less than the rest mass of the electron (511 keV). However, in case of heavier nuclei the kinetic energy of bound electrons in the ground state scales with the nuclear charge Z like Z2 such that in case Z = 100 the kinetic energy is of the order of the rest mass and relativistic effects become important. This is clearly demonstrated by the comparision of non-relativistic and relativistic spectra of a hydrogen-type atom with Z = 100 in Table 10.3. 346 Relativistic Quantum Mechanics The differential equations (10.411, 10.412) are four-dimensional with ~r-dependent wave functions. The arguments above allow one to eliminate the angular dependence by expanding Φ(~r) and X (~r) in terms of Yjm(j + 1 2 , 1 2 |r̂) and Yjm(j − 1 2 , 1 2 |r̂), i.e., Φ(~r) X (~r)  =  a(r) r Yjm(j + 1 2 , 1 2 |r̂) + b(r) r Yjm(j − 1 2 , 1 2 |r̂) c(r) r Yjm(j + 1 2 , 1 2 |r̂) + d(r) r Yjm(j − 1 2 , 1 2 |r̂)  . (10.417) In general, such expansion must include states with all possible j,m values. Presently, we consider the case that only states for one specific j,m pair contribute. Inserting (10.417) into (10.411, 10.412), using (10.415, 10.416), the orthonormality property (6.157), and multiplying by r results in the following two independent pairs of coupled differential equations i [ ∂r − j + 1 2 r ] d(r) + [ m + V (r) − ε ] a(r) = 0 i [ ∂r + j + 1 2 r ] a(r) + [−m + V (r) − ε ] d(r) = 0 (10.418) and i [ ∂r + j + 1 2 r ] c(r) + [ m + V (r) − ε ] b(r) = 0 i [ ∂r − j + 1 2 r ] b(r) + [−m + V (r) − ε ] c(r) = 0 . (10.419) Obviously, only a(r), d(r) are coupled and b(r), c(r) are coupled. Accordingly, there exist two independent solutions (10.417) of the form Φ(~r) X (~r)  =  i f1(r) r Yjm(j + 1 2 , 1 2 |r̂) − g1(r) r Yjm(j − 1 2 , 1 2 |r̂)  (10.420) ( Φ(~r) X (~r) ) =  i f2(r) r Yjm(j − 1 2 , 1 2 |r̂) − g2(r) r Yjm(j + 1 2 , 1 2 |r̂)  (10.421) where the factors i and −1 have been introduced for convenience. According to (10.418) holds for f1(r), g1(r) [ ∂r − j + 1 2 r ] g1(r) + [ ε − m − V (r) ] f1(r) = 0 [ ∂r + j + 1 2 r ] f1(r) − [ ε + m − V (r) ] g1(r) = 0 (10.422) 10.10: Dirac Particles in Electromagnetic Field 347 and for f2(r), g2(r) [ ∂r + j + 1 2 r ] g2(r) + [ ε − m − V (r) ] f2(r) = 0 [ ∂r − j + 1 2 r ] f2(r) − [ ε + m − V (r) ] g2(r) = 0 (10.423) Equations (10.422) and (10.423) are identical, except for the opposite sign of the term (j + 1 2 ); the equations determine, together with the appropriate boundary conditions at r = 0 and r → ∞, the radial wave functions for Dirac particles in the potential (10.409). Dirac Particle in Coulomb Field - Wave Functions We want to determine now the wave functions of the stationary states of hydrogen-type atoms which correspond to the energy levels (10.405). We assume the 4-vector potential of pure Coulomb type (10.367) which is spherically symmetric such that equations (10.422, 10.423) apply for V (r) = −Ze2/r. Equation (10.422) determines solutions of the form (10.420). In the non-relativistic limit, Φ in (10.420) is the large component and X is the small component. Hence, (10.422) corresponds to states Ψ(xµ) ≈  i f1(r) r Yjm(j + 1 2 , 1 2 |r̂) 0  , (10.424) i.e., to states with angular momentum ` = j + 1 2 . According to the discussion of the spectrum (10.405) of the relativistic hydrogen atom the corresponding states have quantum numbers n = 1, 2, . . . , ` = 0, 1, . . . , n − 1. Hence, (10.422) describes the states 2p 1 2 , 3p 1 2 , 3d 3 2 , etc. Similarly, (10.423), determining wave functions of the type (10.421), i.e., in the non-relativistic limit wave functions Ψ(xµ) ≈  i f2(r) r Yjm(j − 1 2 , 1 2 |r̂) 0  , (10.425) covers states with angular momentum ` = j−1 2 and, correspodingly the states 1s 1 2 , 2s 1 2 , 2p 3 2 , 3s 1 2 , 3p 3 2 , 3d 5 2 , etc. We consider first the solution of (10.422). The solution of (10.422) follows in this case from the same procedure as that adopted for the radial wave function of the non-relativistic hydrogen-type atom. According to this procedure, one demonstrates first that the wave function at r → 0 behaves as rγ for some suitable γ, one demonstrates then that the wave functions for r →∞ behaves as exp(−µr) for some suitable µ, and obtains finally a polynomial function p(r) such that rγexp(−µr)p(r) solves (10.422); enforcing the polynomial to be of finite order leads to discrete eigenvalues ε, namely, the ones given in (10.405). 348 Relativistic Quantum Mechanics Behaviour at r → 0 We consider first the behaviour of the solutions f1(r) and g1(r) of (10.422) near r = 0. We note that (10.422), for small r, can be written[ ∂r − j + 1 2 r ] g1(r) + Ze2 r f1(r) = 0 [ ∂r + j + 1 2 r ] f1(r) − Ze2 r g1(r) = 0 . (10.426) Setting f1(r) ∼ → 0 a rγ , g1(r) ∼ → 0 b rγ (10.427) yields γ b rγ−1 − (j + 1 2) b rγ−1 + Ze2 a rγ−1 = 0 γ a rγ−1 + (j + 1 2) a rγ−1 − Ze2 b rγ−1 = 0 . (10.428) or ( γ + (j + 1 2) −Ze2 Ze2 γ − (j + 1 2) ) ( a b ) = 0 . (10.429) This equation poses an eigenvalue problem (eigenvalue −γ) for proper γ values. One obtains γ = ± √ (j + 1 2)2 − Z2e4. The assumed r-dependence in (10.427) makes only the positive solution possible. We have, hence, determined that the solutions f1(r) and g1(r), for small r, assume the r-dependence in (10.427) with γ = √ (j + 1 2 )2 − Z2e4 . (10.430) Note that the exponent in (10.427), in case j+ 1 2)2 < Ze2, becomes imaginary. Such r-dependence would make the expectation value of the potential∫ r2dr ρ(~r) 1 r (10.431) infinite since, according to (10.266, 10.267, 10.420), for the particle density holds then ρ(~r) ∼ |rγ−1|2 = 1 r2 . (10.432) Behaviour at r → ∞ For very large r values (10.422) becomes ∂r g1(r) = − ( ε − m, ) f1(r) ∂r f1(r) = ( ε + m ) g1(r) (10.433) 10.10: Dirac Particles in Electromagnetic Field 351 Using (10.430) one can write this βs = s + γ − εZe2√ m2−ε2 s (s + 2γ) βs−1 . (10.454) Defining so = εZe2 √ m2 − ε2 − γ (10.455) one obtains βs = s− so s(s+ 2γ) βs−1 = (s− 1− so)(s− so) (s− 1)s (s− 1 + 2γ)(s+ 2γ) βs−2 ... = (1− so)(2− so) . . . (s− so) s! (2γ + 1)(2γ + 2) · · · (2γ + s) β0 (10.456) From (10.452) follows αs = j + 1 2 − mZe2√ m2−ε2 so (−so)(1− so)(2− so) . . . (s− so) s! (2γ + 1)(2γ + 2) · · · (2γ + s) β0 (10.457) One can relate the polynomials φ1(ρ) and φ2(ρ) defined through (10.448, 10.449) and (10.456, 10.457) with the confluent hypergeometric functions F (a, c;x) = 1 + a c x + a(a+ 1) c(c+ 1) x2 2! + . . . (10.458) or, equivalently, with the associated Laguerre polynomials L(α) n = F (−n, α+ 1, x) . (10.459) It holds φ1(ρ) = β0 j + 1 2 − mZe2√ m2−ε2 so ργ F (−so, 2γ + 1; ρ) (10.460) φ2(ρ) = β0 ρ γ F (1− so, 2γ + 1; ρ) . (10.461) In order that the wave functions remain normalizable the power series (10.448, 10.449) must be of finite order. This requires that all coefficients αs and βs must vanish for s ≥ n′ for some n′ ∈ N. The expressions (10.456) and (10.457) for βs and αs imply that so must then be an integer, i.e., so = n′. According to the definitions (10.430, 10.455) this confinement of so implies discrete values for ε, namely, ε(n′) = m√ 1 + Z2e4 (n′+ √ (j+ 1 2 )2−Z2e4)2 , n′ = 0, 1, 2, . . . (10.462) 352 Relativistic Quantum Mechanics This expression agrees with the spectrum of relativistic hydrogen-type atoms derived above and given by (10.405). Comparision with (10.405) allows one to identify n′ = n− j + 1 2 which, in fact, is an integer. For example, for the states 2p 1 2 , 3p 1 2 , 3d 3 2 holds n′ = 1, 2, 1. We can, hence, conclude that the polynomials in (10.461 for ε values given by (10.405) and the ensuing so values ( 10.455) are finite. Altogether we have determined the stationary states of the type (10.421) with radial wave functions f1(r), g1(r) determined by (10.438, 10.439), (10.442), and (10.460, 10.461). The coefficients β0 in (10.460, 10.461) are to be chosen to satisfy a normalization condition and to assign an overall phase. Due to the form (10.410) of the stationary state wave function the density ρ(xµ) of the states under consideration, given by expression (10.267), is time-independent. The normalization integral is then ∫ ∞ 0 r2dr ∫ π 0 sin θdθ ∫ 2π 0 dφ (|Φ(~r)|2 + |X (~r)|2) = 1 (10.463) where Φ and X , as in (10.421), are two-dimensional vectors determined through the explicit form of the spin-orbital angular momentum states Yjm(j± 1 2 , 1 2 |r̂) in (6.147, 6.148). The orthonormality properties (6.157, 6.158) of the latter states absorb the angular integral in (10.463) and yield [note the 1/r factor in (10.421)] ∫ ∞ 0 dr (|f1(r)|2 + |g1(r)|2) = 1 (10.464) The evaluation of the integrals, which involve the confluent hypergeometric functions in (10.460, 10.461), can follow the procedure adopted for the wave functions of the non-relativistic hydrogen atom and will not be carried out here. The wave functions (10.421) correspond to non-relativistic states with orbital angular momentum ` = j + 1 2 . They are described through quantum numbers n, j, ` = j + 1 2 ,m. The complete wave function is given by the following set of formulas Ψ(n, j, ` = j + 1 2 ,m|x µ) = e−iεt ( iF1(r)Yj,m(j + 1 2 , 1 2 |r̂) G1(r)Yj,m(j − 1 2 , 1 2 |r̂) ) (10.465) F1(r) = F−(κ|r) , G1(r) = F+(κ|r) , κ = j + 1 2 (10.466) where2 F±(κ|r) = ∓N (2µr)γ−1e−µr {[ (n′ + γ)m ε − κ ] F (−n′, 2γ + 1; 2µr) ± n′F (1− n′, 2γ + 1; 2µr) } (10.467) N = (2µ) 3 2 Γ(2γ + 1) √√√√ m∓ ε)Γ(2γ + n′ + 1) 4m (n′+γ)m ε ( (n′+γ)m ε − κ ) n′! (10.468) 2This formula has been adapted from ”Relativistic Quantum Mechanics” by W. Greiner, (Springer, Berlin, 1990), Sect. 9.6. 10.10: Dirac Particles in Electromagnetic Field 353 and µ = √ (m− ε)(m+ ε) γ = √ (j + 1 2 )2 − Z2e4 n′ = n − j − 1 2 ε = m√ 1 + Z2e4 (n′+γ)2 . (10.469) We want to consider now the stationary states of the type (10.421) which, in the non-relativistic limit, become Ψ(xµ) ≈ e−1εt  i f2(r) r Yjm(j − 1 2 , 1 2 |r̂) 0  . (10.470) Obviously, this wavefunction has an orbital angular momentum quantum number ` = j − 1 2 and, accordingly, describes the complementary set of states 1s 1 2 , 2s 1 2 , 2p 3 2 , 3s 1 2 , 3p 3 2 , 3d 5 2 , etc. not not covered by the wave functions given by (10.465–10.469). The radial wave functions f2(r) and g2(r) in (10.421) are governed by the radial Dirac equation (10.423) which differs from the radial Dirac equation for f1(r) and g1(r) solely by the sign of the terms (j + 1 2)/r. One can verify, tracing all steps which lead from (10.422) to (10.469) that the following wave functions result Ψ(n, j, ` = j − 1 2 ,m|x µ) = e−iεt ( iF2(r)Yj,m(j − 1 2 , 1 2 |r̂) G2(r)Yj,m(j + 1 2 , 1 2 |r̂) ) (10.471) F2(r) = F−(κ|r) , G2(r) = F+(κ|r) , κ = − j − 1 2 (10.472) where F±(κ|r) are as given in (10.467–10.469). We have, hence, obtained closed expressions for the wave functions of all the stationary bound states of relativistic hydrogen-type atoms.