Lecture 22 Relativistic Quantum Mechanics, Lecture notes of Quantum Mechanics

Download Lecture 22 Relativistic Quantum Mechanics and more Lecture notes Quantum Mechanics in PDF only on Docsity! Lecture 22 Relativistic Quantum Mechanics Background Why study relativistic quantum mechanics? 1 Many experimental phenomena cannot be understood within purely non-relativistic domain. e.g. quantum mechanical spin, emergence of new sub-atomic particles, etc. 2 New phenomena appear at relativistic velocities. e.g. particle production, antiparticles, etc. 3 Aesthetically and intellectually it would be profoundly unsatisfactory if relativity and quantum mechanics could not be united. Background When does relativity intrude on QM? 1 When Ekin ! mc2, i.e. p ! mc 2 From uncertainty relation, !x!p > h, this translates to a length !x > h mc = "c the Compton wavelength. 3 for massless particles, "c =", i.e. relativity always important for, e.g., photons. Relativistic quantum mechanics: outline 1 Special relativity (revision and notation) 2 Klein-Gordon equation 3 Dirac equation 4 Quantum mechanical spin 5 Solutions of the Dirac equation 6 Relativistic quantum field theories 7 Recovery of non-relativistic limit Special relativity (revision and notation) Space-time is specified by a 4-vector A contravariant 4-vector x = (xµ) # (x0, x1, x2, x3) # (ct, x) transformed into covariant 4-vector xµ = gµ!x! by Minkowskii metric (gµ!) = (gµ!) = ! ""# 1 $1 $1 $1 $ %%& , gµ!g!" = gµ " # #µ ", Scalar product: x · y = xµyµ = xµy!gµ! = xµyµ Special relativity (revision and notation) Lorentz group splits up into four components: 1 Every LT maps time-like vectors (x2 > 0) into time-like vectors 2 Orthochronous transformations "0 0 > 0, preserve forward/backward sign 3 Proper: det " = 1 (as opposed to $1) 4 Group of proper orthochronous transformation: L" + – subgroup of Lorentz group – excludes time-reversal and parity T = ! ""# $1 1 1 1 $ %%& , P = ! ""# 1 $1 $1 $1 $ %%& 5 Remaining components of group generated by L# $ = TL" +, L" $ = PL" +, L# + = TPL" +. Special relativity (revision and notation) 1 Special relativity requires theories to be invariant under LT or, more generally, Poincaré transformations: xµ & "µ !x! + aµ 2 Generators of proper orthochronous transformations, " ' L" +, can be reached by infinitesimal transformations "µ ! = #µ ! + $µ ! , $µ ! ( 1 gµ!"µ #"! $ = g#$ + $#$ + $$# + O($2) ! = g#$ i.e. $#$ = $$$#, $#$ has six independent components L" + has six independent generators: three rotations and three boosts 3 covariant and contravariant derivative, chosen s.t. %µxµ = 1 %µ = % %xµ = + 1 c % %t ,) , , %µ = % %xµ = + 1 c % %t ,$) , . 4 d’Alembertian operator: %2 = %µ%µ = 1 c2 %2 %t2 $)2 Relativistic quantum mechanics: outline 1 Special relativity (revision and notation) 2 Klein-Gordon equation 3 Dirac equation 4 Quantum mechanical spin 5 Solutions of the Dirac equation 6 Relativistic quantum field theories 7 Recovery of non-relativistic limit Klein-Gordon equation Alternatively, apply quantization to energy-momentum invariant: E 2 = p2c2 + m2c4, $!2%2 t & = - $!2c2)2 + m2c4 . & Setting kc = 2' "c = mc ! , leads to Klein-Gordon equation, - %2 + k2 c . & = 0 Klein-Gordon equation is local and manifestly Lorentz covariant. Invariance of & under rotations means that, if valid at all, Klein-Gordon equation limited to spinless particles But can |&|2 be interpreted as probability density? Klein-Gordon equation: Probabilities Probabilities? Take lesson from non-relativistic quantum mechanics: &% Schrodinger eqn. ) *' (+ i!%t + !2)2 2m , & = 0, c.c.) *' ( & + $i!%t + !2)2 2m , &% = 0 i.e. %t |&|2 $ i ! 2m ) · (&%)& $ &)&%) = 0 cf. continuity relation – conservation of probability: %t( +) · j = 0 ( = |&|2, j = $i ! 2m (&%)& $ &)&%) Klein-Gordon equation: Probabilities Applied to KG equation: &% + 1 c2 %2 t $)2 + k2 c , & = 0 !2%t (&%%t& $ &%t& %)$ !2c2) · (&%)& $ &)&%) = 0 cf. continuity relation – conservation of probability: %t( +) · j = 0. ( = i ! 2mc2 (&%%t& $ &%t& %) , j = $i ! 2m (&%)& $ &)&%) With 4-current jµ = ((c , j), continuity relation %µjµ = 0. i.e. Klein-Gordon density is the time-like component of a 4-vector. Lecture 23 Relativistic Quantum Mechanics: Dirac equation Relativistic quantum mechanics: outline 1 Special relativity (revision and notation) 2 Klein-Gordon equation 3 Dirac equation 4 Quantum mechanical spin 5 Solutions of the Dirac equation 6 Relativistic quantum field theories 7 Recovery of non-relativistic limit Dirac Equation Dirac placed emphasis on two constraints: 1 Relativistic equation must be first order in time derivative (and therefore proportional to %µ = (%t/c ,))). 2 Elements of wavefunction must obey Klein-Gordon equation. Dirac’s approach was to try to factorize Klein-Gordon equation: (%2 + m2)& = 0 (where henceforth we set ! = c = 1) ($i!!%! $m)(i!µ%µ $m)& = 0 i.e. with p̂µ = i%µ (!µp̂µ $m) & = 0 Dirac Equation: ! matrices i%t& = Ĥ&, Ĥ = " · p̂ + )m, " = !0!, ) = !0 Hermiticity of Ĥ assured if "† = ", and )† = ), i.e. (!0!)† # !†!0† = !0!, and !0† = !0 So we obtain the defining properties of Dirac ! matrices, !µ† = !0!µ!0, {!µ, !!} = 2gµ! Since space-time is four-dimensional, ! must be of dimension at least 4+ 4 – & has at least four components. However, 4-component wavefunction & does not transform as 4-vector – it is known as a spinor (or bispinor). Dirac Equation: ! matrices !µ† = !0!µ!0, {!µ, !!} = 2gµ! From the defining properties, there are several possible representations of ! matrices. In the Dirac/Pauli representation: !0 = + I2 0 0 $I2 , , ! = + 0 # $# 0 , # – Pauli spin matrices +i+j = #ij + i,ijk+k , +i † = +i e.g., +1 = + 0 1 1 0 , , +2 = + 0 $i i 0 , , +3 = + 1 0 0 $1 , So, in Dirac/Pauli representation, " = !0! = + 0 # # 0 , , ) = !0 = + I2 0 0 $I2 , Dirac Equation: conjugation, density and current (!µp̂µ $m) & = 0, !µ† = !0!µ!0 Applying complex conjugation to Dirac equation [(!µp̂µ $m)&]† = &† 3 $i!† µ,$ % µ $m 4 = 0, &†,$% µ # (%µ&)† Since (!0)2 = I, we can write, 0 = &†!0 ' () * &̄ ($i !0!† µ ' () * !µ!0 ,$ % µ $m!0) = $&̄ 3 i!µ,$% µ + m 4 !0 Introducing Feynman ‘slash’ notation -a # !µaµ, obtain conjugate form of Dirac equation &̄(i ,$ -% + m) = 0 Relativistic covariance &!(x !) = S(")&(x), S(")!!S$1(") = ("$1)! µ!µ But how do we determine S(")? For an infinitesimal (i.e. proper orthochronous) LT "µ ! = #µ ! + $µ ! , ("$1)µ ! = #µ ! $ $µ ! + · · · (recall that generators, $µ! = $$!µ, are antisymmetric). This allows us to form the Taylor expansion of S("): S(") # S(I + $) = S(I)'()* I + + %S %$ , µ!' () * $ i 4 #µ! $µ! + O($2) where #µ! = $#!µ (follows from antisymmetry of $) is a matrix in bispinor space, and $µ! is a number. Relativistic covariance S(") = I$ i 4 #µ!$µ! + · · · , S$1(") = I + i 4 #µ!$µ! + · · · Requiring that S(")!!S$1(") = ("$1)! µ!µ, a little bit of algebra (see problem set/handout) shows that matrices #µ! must obey the relation, [#µ&, !! ] = 2i - !µ#! & $ !&#! µ . This equation is satisfied by (exercise) ##$ = i 2 [!#, !$] In summary, under set of infinitesimal Lorentz transformation, x ! = "x , where " = I + $, relativistic covariance of Dirac equation demands that wavefunction transforms as &!(x !) = S(")& where S(") = I$ i 4#µ!$µ! + O($2) and #µ! = i 2 [!µ, !! ]. Relativistic covariance S(") = I$ i 4 #µ!$µ! + · · · “Finite” transformations (i.e. non-infinitesimal) generated by S(") = exp 5 $ i 4 ##$$#$ 6 , $#$ = "#$ $ g#$ 1 Transformations involving unitary matrices S("), where S†S = I translate to spatial rotations. 2 Transformations involving Hermitian matrices S("), where S† = S translate to Lorentz boosts. So what?? What are the consequences of relativistic covariance? Angular momentum and spin Altogether, combining components of transformation, &!(x !) = I$ i-n · S)*'( S(") &(x"$1x !)' () * (I$ i-n · L̂)&(x !) . (I$ i-n · (S + L̂))&(x !) we obtain &!(x !) = S(")&("$1x !) . (1$ i-n · Ĵ)&(x !) where Ĵ = L̂ + S represents total angular momentum. Intrinsic contribution to angular momentum known as spin. [Si ,Sj ] = i,ijkSk , (Si ) 2 = 1 4 for each i Dirac equation is relativistic wave equation for spin 1/2 particles. Parity So far we have only dealt with the subgroup of proper orthochronous Lorentz transformations, L" +. Taking into account Parity, Pµ! = ! ""# 1 $1 $1 $1 $ %%& relativistic covariance demands S(")!!S$1(") = ("$1)! µ!µ S$1(P)!0S(P) = !0, S$1(P)! iS(P) = $! i achieved if S(P) = !0e i(, where . denotes arbitrary phase. But since P2 = I, e i( = 1 or $1 &!(x !) = S(P)&("$1x !) = /!0&(Px !) where / = ±1 — intrinsic parity of the particle Lecture 24 Relativistic Quantum Mechanics: Solutions of the Dirac equation Free particle solutions of Dirac Equation &(x) = e$ip·xu(p) = e$iEt+ip·xu(p) What about bispinor amplitude, u(p)? In Dirac/Pauli representation, !0 = + I2 $I2 , , ! = + # $# , the components of the bispinor obeys the condition, (!µpµ $m)u(p) = + p0 $m $# · p # · p $p0 $m , u(p) = 0 i.e. bispinor elements: u(p) = + 0 / , , 8 (p0 $m)0 = + · p/ + · p0 = (p0 + m)/ Free particle solutions of the Dirac Equation u(p) = + 0 / , , 8 (p0 $m)0 = + · p/ + · p0 = (p0 + m)/ Consistent when (p0)2 = p2 + m2 and / = # · p p0 + m 0 u(r)(p) = N(p) 9 1(r) # · p p0 + m 1(r) : where 1(r) are a pair of orthogonal two-component vectors with index r = 1, 2, and N(p) is normalization. Helicity: Eigenvalue of spin projected along direction of motion 1 2 # · p |p|1 (±) # S · p |p|1 (±) = ±1 2 1(±) e.g. if p = p ê3, 1(+) = (1, 0), 1($) = (0, 1) Free particle solutions of the Dirac Equation So, general positive energy plane wave solution written in eigenbasis of helicity, &(±) p (x) = N(p)e$ip·x ! # 1(±) ± |p| p0 + m 1(±) $ & But how to deal with the problem of negative energy states? Must we reject the Dirac as well as the Klein-Gordon equation? In fact, the existence of negative energy states provided the key that led to the discovery of antiparticles. To understand why, let us consider the problem of scattering from a potential step... Klein paradox and antiparticles 1 + r = t (1), p E + m (1$ r) = p! E ! + m t (2) From (2), 1$ r = 2t where 2 = p! p (E + m) (E ! + m) Together with (1), (1 + 2)t = 2 t = 2 1 + 2 , 1 + r 1$ r = 1 2 , r = 1$ 2 1 + 2 Interpret solution by studying vector current: j = &̄!& = &†"& j3 = &†*3&, *3 = !0!3 = + +3 +3 , Klein paradox and antiparticles j3 = &† + +3 +3 , & (Up to overall normalization) the incident, transmitted and reflected currents given by, j (i)3 = - 1 0 p E+m 0 . + 0 +3 +3 0 , ! ""# 1 0 p E+m 0 $ %%& = 2p E + m , j (t)3 = 1 E ! + m (p! + p!%)|t|2, j (r)3 = $ 2p E + m |r |2 where we note that, depending on height of the potential, p! may be complex (cf. NRQM). Klein paradox and antiparticles 2 = p! p E + m E ! + m Therefore, ratio of reflected/transmitted to incident currents, j (r)3 j (i)3 = $|r |2 = $ ;;;; 1$ 2 1 + 2 ;;;; 2 j (t)3 j (i)3 = |t|2 (p! + p!%) 2p E + m E ! + m = 4 |1 + 2|2 1 2 (2 + 2%) = 2(2 + 2%) |1 + 2|2 From which we can confirm current conservation, j (i)3 = j (r)3 + j (t)3 : 1 + j (r)3 j (i)3 = |1 + 2|2 $ |1$ 2|2 |1 + 2|2 = 2(2 + 2%) |1 + 2|2 = j (t)3 j (i)3 Klein paradox and antiparticles Negative energy states Existence of antiparticles suggests redefinition of plane wave states with E < 0: Dirac particles are, in fact, fermions and Pauli exclusion applies. Dirac vacuum corresponds to infinite sea of filled negative energy states. When V > 2m the potential step is in a precarious situation: It becomes energetically favourable to create particle/antiparticle pairs – cf. vacuum instability. Incident beam stimulates excitation of a positive energy particle from negative energy sea leaving behind positive energy “hole” – an antiparticle. Klein paradox and antiparticles cf. creation of electron-positron pair vacuum due to high energy photon. Klein paradox and antiparticles Therefore, for E < 0, we should set p0 = + 7 p2 + m2 and &(x) = e+ip·xv(p) where (-p + m)v(p) = 0 (N.B. “+”) v (r)(p) = N(p) 9 # · p p0 + m 1(r) 1(r) : But Dirac equation was constructed on premace that & associated with “single particle” (cf. Schrödinger equation). However, for V > 2m, theory describes creation of particle/antiparticle pairs. & must be viewed as a quantum field capable of describing an indefinite number of particles!! In fact, Dirac equation must be viewed as field equation, cf. wave equation for harmonic chain. As with chain, quantization of theory leads to positive energy quantum particles (cf. phonons). Allows reinstatement of Klein-Gordon theory as a relativistic theory for scalar (spin 0 particles)... Quantization of Dirac field Dirac equation associated with Lagrangian density, L = &̄ (i!µ%µ $m)&, i.e. %)̄L = (i!µ%µ $m) & = 0 With momentum ' = %)̇L = i&̄!0 = i&†, Hamiltonian density H = '&̇ $ L = &̄i!0%0& $ L = &̄ ($i! ·)+ m) & Once again, we can follow using canonical quantization procedure, promoting fields to operators – but, in this case, one must impose equal time anti-commutation relations, {&̂#(x, t), '̂(x!, t)} # &̂#(x, t)'̂$(x!, t) + '̂$(x, t)&̂#(x!, t) = i#3(x$ x!)##$ Quantization of Dirac field Turning to Fourier space (with k0 # $k = 2 k2 + m2) &(x) = 2= r=1 < d3k (2')32$k 1 ar (k)u(r)(k)e$ik·x + b†r (k)v (r)(k)e ik·x 2 with equal time anti-commutation relations (hallmark of fermions!) > ar (k), a†s (k !) ? = > br (k), b†s (k !) ? = (2')32$k#rs# 3(k$ k!) > a†r (k), a†s (k !) ? = > b†r (k), b†s (k !) ? = 0 which accommdates Pauli exclusion a†r (k)2 = 0(!), obtain Ĥ = 2= r=1 < d3k (2')32$k $k / a†r (k)ar (k) + b†r (k)br (k) 0 Physically a(k)u(r)(k)e$ik·x annihilates +ve energy fermion particle (helicity r), and b†(k)v (r)(k)e ik·x creates a +ve energy antiparticle. Low energy limit of the Dirac equation Previously, we have explored the relativistic (fine-structure) corrections to the hydrogen atom. At the time, we alluded to these as the leading relativistic contributions to the Dirac theory. In the following section, we will explore how these corrections emerge from relativistic formulation. But first, we must consider interaction of charged particle with electromagnetic field. As with non-relativistic quantum mechanics, interaction of Dirac particle of charge q (q = $e for electron) with EM field defined by minimal substitution, pµ %$& pµ $ qAµ, where Aµ = (.,A), i.e. (-p $ q -A$m)& = 0 Low energy limit of the Dirac equation &b = 1 2mc2 + W $ V c# · (p̂$ qA)&a Taking into account the leading order (in v/c) correction (with A = 0 for simplicity), we have &b . 1 2mc2 + 1$ W $ V 2mc2 , c# · p̂&a Then substituted into the second bispinor equation (and taking into account correction from normalization) we find Ĥ . p̂2 2m + V $ p̂4 8m3c2 ' () * k.e. + 1 2m2c2 S · ()V )+ p̂ ' () * spin$orbit coupling + !2 8m2c2 ()2V ) ' () * Darwin term Synopsis: (mostly revision) Lectures 1-4ish 1 Foundations of quantum physics: †Historical background; wave mechanics to Schrödinger equation. 2 Quantum mechanics in one dimension: Unbound particles: potential step, barriers and tunneling; bound states: rectangular well, #-function well; †Kronig-Penney model . 3 Operator methods: Uncertainty principle; time evolution operator; Ehrenfest’s theorem; †symmetries in quantum mechanics; Heisenberg representation; quantum harmonic oscillator; †coherent states. 4 Quantum mechanics in more than one dimension: Rigid rotor; angular momentum; raising and lowering operators; representations; central potential; atomic hydrogen. † non-examinable *in this course*. Synopsis: Lectures 5-10 5 Charged particle in an electromagnetic field: Classical and quantum mechanics of particle in a field; normal Zeeman e%ect; gauge invariance and the Aharonov-Bohm e%ect; Landau levels, †Quantum Hall e%ect. 6 Spin: Stern-Gerlach experiment; spinors, spin operators and Pauli matrices; spin precession in a magnetic field; parametric resonance; addition of angular momenta. 7 Time-independent perturbation theory: Perturbation series; first and second order expansion; degenerate perturbation theory; Stark e%ect; nearly free electron model. 8 Variational and WKB method: Variational method: ground state energy and eigenfunctions; application to helium; †Semiclassics and the WKB method. † non-examinable *in this course*.