Perturbation Theory - Lecture Notes - Quantum Mechanics I | PHY 389K, Study notes of Quantum Mechanics

Download Perturbation Theory - Lecture Notes - Quantum Mechanics I | PHY 389K and more Study notes Quantum Mechanics in PDF only on Docsity! Perturbation Theory In section 1 we will give some descriptions of how and why approximation methods are used in solving problems in quantum mechanics. In section 2, the fundamental formulas of perturbation theory are derived. 1.1 Stationary Perturbation Theory and Its Underly- ing Assumptions 1.1.1 Introduction The description of any physical system is only approximate. ‘It is true, of course, that it is possible to solve some models exactly, but in turn these models only approximately describe real physical systems. For example, the harmonic oscillator describes a vibrating diatomic molecule only when the vibrations are not too violent and anharmonic forces are neglible. In a similar vein, it is possible to find the energy levels of the hydrogen atom after the nonrelativistic Hamiltonian is known, but the nonrelativistic Hamiltonian or Schrodinger equation is only approximate. To take into account the spin of the electron, the Pauli equation or Pauli Hamiltonian must be used to describe the electron. The Dirac equation is a relativistic equation that describes a spin 1/2 particle. To take into account. spin, relativistic corrections, and the fact that electromagnetic signals travel between the proton and electron at the speed of light, the Bethe-Salpeter equation can be employed. Even when the electron is traveling at speeds far less than that of light, the electron interacts electromagnetically with itself, causing small but experimentally detectable deviations from the energy level calculated from the Pauli equation or Pauli Hamiltonian. ‘Therefore if a physicist wishes to calculate numbers that can be compared with experimental values, approximations are inevitable. The most frequently used approxiamation method in quantum mechanics is based upon splitting the Hamiltonian into two parts. H=Ho+, (1.1) 2 CHAPTER 1. PERTURBATION THEORY This splitting is done such that Ho is the Hamiltonian of a known system whose eigen- value equation can be solved exactly, such as the oscillator or rotator. It is often called the free or interaction free Hamiltonian. H, describes some small interaction which perturbs the interaction free system slightly. This means the physical system with the exact Hamiltonian HT is similar to the sytem described by the free Hamiltonian Hp. To explain what it means that the quantum physical systems with H and Ho are similar, we first consider the rotator of section II[.4 in the presence of a magnetic field. This is a situation for which one does not need perturbation theory. 1.1.2. Magnetic Moment Operators The Hamiltonian of the interaction free system is the rotator Hamiltonian, Hy = 4J? of (II1.2.11). To conjecture the interaction Hamiltonian, Hy, for the interaction with an external magnetic field B, we follow the usual procedure and start with the energy of the corresponding classical system in a magnetic field and make the transition to the quantum system by replacing the classical observables by their corresponding quantum operators. The magnetic field acts on a magnetic dipole moment ji. The energy of a classical system with a dipole magnetic moment. ji in a magnetic field B is given by E=-ji-B. (1.2) The magnetic moment of a rotating or spinning object with angular momentum 7 is in general given by fe ted (1.3) where q is the electric charge and m the mass of Lhe abject. The factor g, called the Landé factor or pyromagnetic ratio, is a parameter which is specific for the physical object. For example, if the physical system is a mass-point of mass m with charge g moving in a circular orbit with orbital angular momentum J, then the magnetic moment is given by (1.3) with g=1: é Ome If the physical system is a classical relativistic spinning particle with angular momentum # about its center of mass and with no intrinsic structure then one can show that g = 2 (sect. IX.3a). ‘The electron with g = —¢ = —1.602 « 10-"C is a nearly structurless relativistic particle and its Landé factor is nearly getectron * 2 (precisely Geteceron = 2(14 2 —0.328(2.)*) due to QED corrections), thus i= (14) His 7 Dias? (1.5) where m, and § are the mass and spin of the electron, respectively. A proton has intrinsic structure and its magnetic moment is given by Fetectron & 2( Towoton % (2+ 9.80) ———F (1.8) 1.1, STATIONARY PERTURBATION THEORY AND ITS UNDERLYING ASSUMPTIONSS The eigenvectors of this Hamiltonian are already known to us. They are the eigenvectors \j.da) Of the ¢.3.c.0. J*, Jy found in section IIL3. The eigenvalues of the free Hamiltonian Hy are Lise ER = i+) (1.20) The eigenvalues of the exact Hamiltonian Hj, js) = Ej, 33), are immediately obtained by applying H of (1.19) to |j, js). The eigenvalues of H depend not only on j like the #Y, but also jg: hea . E= Ej, = apiG +1)+ a5 Bais. (1.21) Each eigenvalue E? has a 27 + 1-fold degeneracy. This means to each E? correspond 2j +1 eigenvectors |j, j3) or a 2j + 1 dimensional space R?. ‘The eigenvalues £;;, of H are not degenerate; to each £;;, corresponds one eigenvector |j, 5) or the one-dimensional space Ri Therefore to each degenerate energy level of Hy correspond (2j + 1) energy levels of H which merge into ER when the external magnetic field is switched off, i.e. when Bs — 0. £, EE. a ‘fds=—j+1 a) Eo (1.22) Ejjyas ‘The splitting of the energy levels in a magnetic field is called Zeeman splitting (or the Zeeman effect) and it occurs for atoms, molecules and any other rotationally invariant system that. is placed into a magnetic field which breaks the symmetry. If we had used the Hamiltonian (1.16) instead of (1.19), then the eigenvectors of Hy and Jy could not have been eigenvectors of H, because for H of (1.16), [H, Js] #0. In order to obtain eigenvectors of (1.16), we would have to use instead of |j, j3) a basis system |j, j,) of eigenvectors of the c.s.c.o, Jy, 7 - Sa J,, Where 7 is the unit vector in the direction of the magnetic field and proceed in the same way as above using |j, j,) in place of |¥, j4). However H and Ho of (1.19) and (1.17) (or (1.16) and (1.17) ) do not represent the general case of the exact and free Hamiltonian, because they have another special relationship with each other, which in general is not fulfilled, namely |H, Ho] = 0. For a general quantum system(not the rotator) in a magnetic field with the interaction Hamiltonian (1.18) this is not fulfilled since the free Hamiltonian Hp is in general not the rotator Hamiltonian (1.16) (or any Hp = f(J*)). In the general case the exact and free Hamiltonian do not commute, [H, Ho] # 0, as is the case for atoms and molecules in a magnetic field, for which Hp would not be given by a function f(J*). But Hp could still be a spherically symmetric Hamiltonian, i.e. fulfill [Ho, J;] = 0. These systems will also have Zeeman splitting, however the eigenvalues Ej js cannot be calculated as in (1.21), though the physical situation is very similar. If H and Ho do not commute one has to use a procedure called perturbation theory. We now consider this general case. 6 CHAPTER 1. PERTURBATION THEORY 1.1.4 Perturbation Theory We have two Hamiltonians Hy and H = Ho + Hy. Except for very special cases (such as the one dimensional harmonic oscillator) one has in addition to the energy quantum number other quantum numbers( such as jg in the case of the rotator). Then we have to select other operators, which together with Ho, form a c.s.c.o. and whose eigenvalues are known. We denote this set of operators B), Bo,...,By-; collectively by B and their eigenvalues(quantum numbers) 6), be,...,by—1 we denote by b. It is of great advantage if the same set of operators 8 also forma c.s.c.o. with the exact Hamiltonian H, ie. if (HW, B) =0. Then we have two c.s.c.0.’s consisting of N operators Ho, By, By, ..; Bu-1 which we abbreviate Hy, B (1.23) A, By, Bo,...,By—1 which we abbreviate H, 8 (1.24) If this is not the case, ie. if one or more of the B; do not commute with H then one tries to find another system of operators Ay, Az,...,Ay—-1 = A which commute with Hp as well as H. This we will discuss below. The two c.s.c.o.’s (1.23) and (1.24) can be treated as if they were c.s.c.o.'s that consists of N=2 operators, Hp, B and H, B, respectively. We have the following situation. The physical system described by (1.23) has solutions (ie. eigenvectors |E%,0) = |EG,d1,b2,....by—1) and eigenvalues E8,b = EB, by, ba, , bw that are completely known. We want. to find the eigenvectors |B, b >= | Ey by, bo, bon and eigenvalues H,b = E,b),bo,...,8n-1 of the physical system described by (1.24) and express them in terms of the solutions for the free systems. For the special case that [EH], Ho] = 0 as in (1.19), one can choose |B, b >= |E°,b) since the eigenvectors of H are those of Ho. For the case that [H, Hp] # 0, one must use perturbation theory to calculate the unknown eigenvalues E = E,(2) and eigenvectors |Z, > of H from the known eigenvalues E® and eigenvectors |}, 6) of Ho. The situation to which one applies perturbation theory is otherwise very similar to the situation depicted in (1.22). One has energy levels £° of the interaction free system which are degenerate (except for the very special case that B has only one eigenvalue or the observable B is not part of the system). To each of the energy levels E® corresponds a finite or infinite number of energy levels B,,, = By(b) of the system with interaction Hy, such that for zero interaction (i.c. 1, 4 0 in some manner) all these energy levels, E,.. merge into the same energy value E}. Ena En pont Bs? a (1.25) Ege However, in distinction to case (1.22), to each BE, corresponds a vector |#,,,b > different from the known eigenvector | £0, 6) of the c.s.c.o. (1.23). These vectors have to be calculated under the condition that 1.1, STATIONARY PERTURBATION T'HEORY AND ITS UNDERLYING ASSUMPTIONS7 [En sb > |.E8,8) for H,) 20 V b= 8, b?,...,0" (finite or infinite) (1.26) Thus in the typical situation to which perturbation theory applies, there are several (finite or infinite) known eigenvectors |£®,b) belonging to the one known eigenvalue EQ of Hp. They span the finite or infinite dimensional space of eigenvectors of Hy with the same eigenvalue E%. The perturbation H, then splits the energy level into the sublevel E,,, (the degeneracy of the eigenvalue E? is removed). Often- as is the case of the rotator in the magnetic field- the degeneracy stems from the symmetries in the unperturbed Hamiltonian, Ho. The perturbation Hamiltonian, H,, then breaks the symmetry, thereby removing the degeneracy’. In order for the scheme (1.25) and (1.26) to make physical sense the splitting inside a multiplet (ie. between the £,, for a given value of n and different values of 6) should be small compared to the splitting between different multiplets (i.c. between £,, and Ly, with n' #n). This means the interaction operator must be “small” in a certain mathematical sense (e.g. |(B2, b|Hy|£°,,0')| << |E°, — B9|), How to precisely formulate the mathematical conditions on the operators Hj and H such that (1.25) and (1.26) result and such that the perturbation procedure converges is a difficult mathematical problem. Physicists use the procedure to calculate numbers and hope that H, has been chosen right so that the perturbation series somehow converges. Before we describe in the next section how the unknown eigenvalues E and the eigen- vectors |B, > of H are determined in terms of the known eigenvalues £® and known eigenvectors [£9, b) of Ho by an approximated perturbation series, we want to discuss what to do in the case that [Ho, 8] = 0 but (H, B] 4 0. In this case we could still use b as labels for the eigenvectors of H, but these eigenvectors of H are not eigenvectors of B and the label b in |£(b) > would only be an “approximate” quantum number defined by the limit of (1.26). (This is what is often done in scattering theory for the continuous eigenvalues E of H where (b) are the asymptotic momenta 7 with z = E® = eigenvalue of Ho). For H and Ho with discrete spectra it is better if one does not use |(b) +, but tries to find an operator A, or a system of operators Aj, do,...,An—1, which together with H forms a C.8.6.0. [H, A] =0 (1.27) and which also has the property that (Ho, A] = 0. (1.28) One then obtains a new system of basis vectors of the interaction free system |E2, a}, which are eigenvectors of the c.s.c.o. Hy, A, by a linear transformation from the known free There is a very special case, e.g. the one-dimensional harmonic oscillator, which is of little practical value, in which there is no degeneracy and there is only one eigenvector |F8) of Ho for each value of E®, Then no splitting can occur and there is also only one eigenvalue ©, of H, which is shifted slightly from E}. This case is usually discussed in most textbooks under the heading of “nondegenerate perturbation theory." 10 CHAPTER 1, PERTURBATION THEORY For case (a) we seek eigenvectors |F,,b > of the “exact” Hamiltonian H H\Eng,b = EnplEnmb >, (1.45) that satisfy |Ea.tsb > [ER 8) for Hy 0 (1.46) The energy eigenvalues E® are degenerate, ie. they do not depend upon b, but the eigenval- ues of A are functions of b, H,(b) = Bay, and to a given value of n correspond m eigenvalues Engts-.+;2nom as shown in (1.25). For case (b) we seek eigenvectors |, > of the “exact” Hamiltonian A\E, += BE, >, (1.47) that satisly |E, >—+ |2) for H; +0 (1.48) (If H commuted with some other operator, the eigenvectors |E, » would be labeled by additional quantum numbers which are omitted here because they are not of consequence for our discussion.) Because, for this case, the energy eigenvalues of Ho are nondegenerate, the eigenvector of Ho that is the best. approximation to |E,, > is the eigenvector of Ho, |B), with the same value of the quantum number n. Since this is a special case of (a) we will not discuss it any further. The perturbation calculation is begun by considering the following matrix element: (ER, b|H|Eno,b >= (Eq, b|Ho + Hil Ena. > « (1.49) where E, = E,(b) = Ey». ‘The matrix element on the left-hand side of (1.49) is simplified using the fact that |E,,,, > is, by definition, an eigenvector of H with (unknown) eigenvalue E,,s The first matrix element on the right-hand side of (1.49) is evaluated using the hermiticity of Hq and (1.42). We thus obtain for (1.49) En p( Bo), b| Eno. b >= ELAES,, bl Bp, b> +(BRs, b|Hi|En pb > . (1.50) Since the perturbation is assumed to be small, the energy eigenvalues E,,y do not differ by much from the value £2. For the same reason, the eigenvectors |#,5,b > and |£°, are in a certain way also close to each other. This in particular means that for n’ = n the matrix element (E°,_,,, |En,0 > (BS, blB2,b) = 1. Therefore we can divide (1.50) by this matrix element and obtain one of the two fundamental equations in stationary perturbation theory: Ein,b, b> Eqp = ES + ne = El + Ang. (1.51) This means that the exact energy values E,,, are equal to the energy value of the interaction free system £° plus a small perturbation which depends upon the quantum numbers b. Ay» is the energy shift from the unperturbed energy value E® of the interaction free system to 1.2, STATIONARY PERTURBATION SERIES 11 the energy values #,,,, of the system with interaction. It is given by the matrix element of the “small” interaction Hamiltonian H,, which needs to be calculated. Tf n! & nin (1.50), then the energy difference (Z,,5— £2. fe is not a all foe like A,,y but will for every b be substantially different from zera, (Ey,»— E,) (£8 — £8.) # 0. Therefore one can divide (1.50) by (Fu, — BR) ee b|Ay | En, (£9, b)Bn4,8 = ca <, To derive the second fundamental equation of stationary perturbation theory, we expand the “exact” eigenvector |E,.6 > of H in terms of the eigenvectors |E2.,6) of Hy. The eigenvectors |Z°,b) of the ¢.3.c.o. Hy, Bi,..-,By—1 = Ho, B form a complete basis system of the space . of state vectors. Thus every vector ¢ ¢ H can be expanded in terms of this basis system. for n xn’. (1.52) O= SCE UME Ye) (1.53) a In particular each eigenvector @ = | Fy, > (with given value n, 6) of thec.s.c.o. H,By,...,Byi1 = H, B can be expanded as in (1.53) [Enpob = 9 [ER BK ES, bl Enos b = dle LP)CER b|Emasd > (1.54) nib The second equality holds because (BY ,U| Bap, b >= 0 for bl #5 (1.55) since |Z, 0') and |Hy,»; 6) are eigenvectors of the same (system of) hermitian operator(s) B with different eigenvalues. Therefore in the expansion on the right hand side of (1.54), only vectors |#2,b) with the same value as on the left hand side appear and we can write (1.54) as [Bais b= xX |B2,,B)(B8| Eno, >= |E2, 0) (E8, b]E ny, b > + > | BR, BED, bl Enos 6). nya (1.56) For a given value of n the terms with n! = n and na! # n in (1.56) are of quite different magnitude, since the matrix elements (E2, b| Eng. b ee (ES, o|B8, by = 1 (1.57) (EM b| Bay, b >a (BS, bl Be,b) = 0 for n' #n. (1.58) Therefore one writes the eigenvector |,,»,5) of H as the corresponding eigenvector | £2, b) of Hp plus a (infinite) sum of terms of much smaller magnitude. To obtain the traditional form we add and subtract |E®, 6) to the first term in (1.56) and insert (1.52) in the second term (the finite sum over mn’) of (1.56). Then (1.56) becomes 5 <b En, b> 2 be =lE ii 7a, yy Ee |En.as0 = |E2, b) + ((E3, o|Bng, b> —1)|B8, 6) + So (88 E,-E, (89) n'én 12 CHAPTER 1. PERTURBATION THEORY Except for the first term of the right hand side, the coefficients of the eigenvectors |E%, 6) and |£°,) are all of small magnitude due to (1.67) and (1.58). Equations (1.51) and (1.59) are the two fundamental equations in stationary perturbation theory, they are exact equations for the energy eigenvalues E,,, and eigenvectors | Bip.y,b > of H, respectively. But the unknown quantities £4 and |E,», - appear on both sides of these equations, and these equations must be solved for these unknown quantities. We do this hy successively inserting the left-hand sides into the right-hand sides. In carrying out this iterative procedure, it is important to keep in mind that H, is a small perturbation, Thus, the second term on the right-hand side of (1.51) is small compared with the first term. Similarly, the second term and the sum over n! on the right-hand side of (1.59) have small coefficients compared with the first term |Z®, b) Thus the zeroth-order approximation for the eigenvalues £,,., and eigenvector |Bn4, b>, denoted by ES and |#,4,6)0) respectively, is obtained by setting A, = 0 in (1.51) and (1.59): Buy? Boy = Ep (1.60) |Eno.0 + [Erp bY) = |ER, B). (1.61) Since the terms on the right-hand side of (1.51) and (1.59) that involve By or | Fp 9,6 > are small, we obtain the first-order approximations to &, and |Ey»,b > by ae their zeroth-order approximations (1.60), (1.61) on the right-hand sides of (1.61) and (1.59), respectively. Denoting the first-order approximation for energy by Ee), (1.51) a (Bh BLA [Ena bY (ER, b| Ena, bY Substituting (1.60), (1.61) and using the orthogonality relation (1.44) for the eigenvectors \£%, b), this becomes EG) = Bo + (1.62) BM) = E8 + (EY, bl Hy|E8, b) = Bo + AO} (1.63) To first order the energy shift, All of the degenerate energy level E® of the free or unperturbed Hamiltonian equals the matrix element of the perturbation Hamiltonian Hy between the unperturbed, known eigenstates |B2, b). Denoting the first-order approximation for energy eigenvectors by |E,,)" and using (1.60) and (1.61 in (1.59) we obtain the first order approxirnation to the exact eigenvectors 5 - |, b| Ay |B, b)O [= 188,04 (WER Bg 1) — 1) 8) + > ey a) BREET 6 a'fn be ieggt Again using equations (1.60),(1-61) for the zeroth-order approximations and the normaliza- tion (1.44), the above equation takes the form a |Eqp)) = EY, b) + Sie, gy Sa PE Oh (ER. bLAN LER, b) mom (1.65) nin