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Quantum Mechanics II Quantum Mechanics II. Janos Polonyi University of Strasbourg (Dated: September 8, 2021) Contents I. Perturbation expansion 4 A. Stationary perturbations 5 1. First order 6 2. Second order 6 3. Degenerate perturbation 7 B. Time dependent perturbations 9 C. Non-exponential decay rate 12 D. Quantum Zeno-effect 15 E. Time-energy uncertainty principle 16 F. Fermi’s golden rule 17 II. Rotations 18 A. Finite translations 18 B. Finite rotations 19 C. Euler angles 21 D. Summary of the angular momentum algebra 22 E. Rotational multiplets 23 F. Wigner’s D matrix 24 G. Invariant integration over rotations 26 H. Spherical harmonics 28 III. Addition of angular momentum 31 A. Additive observables and quantum numbers 31 1. Momentum 31 2. Angular momentum 32 2 B. System of two particles 32 IV. Selection rules 38 A. Tensor operators 38 B. Orthogonality relations 39 C. Wigner-Eckart theorem 41 V. Symmetries in quantum mechanics 43 A. Representation of symmetries 44 B. Unitary and anti-unitary symmetries 45 C. Space inversion 48 D. Time inversion 50 VI. Relativistic corrections to the hydrogen atom 54 A. Scale dependence of physical laws 54 B. Hierarchy of scales in QED 56 C. Unperturbed, non-relativistic dynamics 58 D. Fine structure 59 1. Relativistic corrections to the kinetic energy 59 2. Darwin term 60 3. Spin-orbit coupling 61 E. Hyperfine structure 64 F. Splitting of the fine structure degeneracy 65 1. n = 1 65 2. n = 2 66 VII. Identical particles 68 A. A macroscopic quantum effect 68 B. Fermions and bosons 69 C. Occupation number representation 73 D. Exchange interaction 74 VIII. Potential scattering 75 A. Cross section 75 B. Optical theorem 77 3 C. Lippmann-Schwinger equation 78 D. Partial waves 81 1. Scattering amplitude 84 2. Phase shift 85 3. Low energy scattering 87 4. Bound states 88 5. Resonances 89 IX. Outlook 90 A. Measurement theory 91 B. (In)Determinism 93 C. Non-locality I. Einsein-Podolksi-Rosen experiment 94 D. Non-locality II. Bell inequality 95 E. Contextuality 96 F. Instead of conclusion 97 A. Density matrix 98 1. Gleason theorem 98 2. Properties 100 3. Composite systems 101 4. Physical origin 102 a. Loss of classical information 102 b. Entangled states 104 c. Relative states 106 B. Quantum anomalies 107 1. Singular time dependence 108 2. Quantization rules in polar coordinates 109 4 Quantum mechanics is usually taught on four different levels: 1. Basic ideas, simple examples for a one dimensional particle, particle in spherical potential 2. More realistic, three dimensional cases with few particles 3. Several particles, relativistic effects (Quantum Field Theory) 4. Fundamental issues, challenges, paradoxes and interpretation of the quantum world This lecture note belongs to level 2. You should be warned when embarking level 1 to leave your expectations, intuitions behind and start from scratch. There is no other discipline thought at Universities where one has to ask the students at the half of their university time to follow such a strange attitude. Now, at level 2., you should keep the experience of your first encounter with quantum mechanics, whatever confusing and unsatisfactory they were, this is the time to develop further your knowledge. But please keep your dissatisfaction about the subject in mind and remember it later when you can try to improve our understanding and maneuverability in the quantum world. It is not clear how far we are from a better understanding of quantum mechanics. We may miss only few elements or we may even have to change completely our mathematics. But it is sure that we have to improve our way we look at Nature. Until this happens try to understand why don’t you understand quantum mechanics. I. PERTURBATION EXPANSION The Hamiltonian, being a hermitian operator, can always be brought into a diagonal form by using an appropriate basis where the Schr¨odinger equation, and the dynamics of a quantum system is trivial. But this simplicity is naturally misleading because the basis where the dynamics is simple is usually highly non-trivial in terms of the usual observables. A systematic approximation scheme, the perturbation expansion, is based on the assumption that the Hamiltonian can be written as a sum of two hermitian operators, H = H0 + gH1, one is stationary and easily diagonalizable and the other is assumed to be weak, a condition to be specified later. We treat g as a small parameter and organize the solution of the Schr¨odinger equation as a power series in g. 5 A. Stationary perturbations We assume first that H0 is non-degenerate and H1 is time independent and seek the spectrum and the stationary states, thee eigenvalues and the eigenvectors, H ψ = E ψ . For this end we | ni n| ni assume that the unknown eigenvector and eigenvalues can be expanded in the small parameter, ψ = ψ(0) + g ψ(1) + g2 ψ(2) + , | ni | n i | n i | n i · · · E = E(0) + gE(1) + g2E(2) + , (1) n n n n · · · which gives after inserting into the eigenvalue equation 0 = g0 H ψ(0) E(0) ψ(0) 0| n i− n | n i +g H ψ(1) + H ψ(0) E(1) ψ(0) E(0) ψ(1) 0| n i 1| n i− n | n i− n | n i +g2 H ψ(2) + H ψ(1) E(2) ψ(0) E(1) ψ(1) E(0) ψ(2) + (2) 0| n i 1| n i− n | n i− n | n i− n | n i · · · We can now consider the different orders one-by-one, 0 (0) (0) (0) g : H ψn = E ψ O 0| i n | n i (0) (1) (1) (0) (g) : (H En ) ψn = (E H ) ψ O 0 − | i n − 1 | n i 2 (0) (2) (1) (1) (2) (0) g : (H En ) ψn = (E H ) ψ + E ψ O 0 − | i n − 1 | n i n | n i k (0) (k) (1) (k 1) (k) (0) g : (H En ) ψn = (E H ) ψ − + + E ψ (3) O 0 − | i n − 1 | n i · · · n | n i (0) The zeroth-order equation shows that the vectors ψn , the eigenvectors of a non-degenerate | i (0) (0) Hermitian operator are orthogonal, ψm ψn = δ . The higher order equations can not be h | i mn (k) solved in a unique manner. In fact, to solve say the last equation for ψn one multiplies the | i (0) 1 equation by (H En ) . However the latter operator has a non-trivial null-space (the null-space 0 − − of an operator A is a linear subspace, consistsing of the vectors turned into zero by the operator in question, A 0 = 0) where its inverse does not exist. To overcome this difficulty observe that if the | i (k) vector ψn represents a solution of the k-th order eigenvalue equation then another solution can | i (k) (0) (0) (k) be found by adding to it a vector from the null-spaces, ψn + c ψn . We choose c = ψn ψn | i | i −h | i (0) (k) which is equivalent with imposing the condition ψn ψn = δ , to render the solution unique and h | i k,0 well defined by definition. This method of solving the equations fails if the spectrum is continuous. In that case one adds an infinitesimal imaginary term to the free Hamiltonian, H H + iǫ, with 0 → 0 ǫ 0, c.f. eq. (364). → 6 1. First order (1) (0) One supposes the form ψn = c ψ and the projection of the second equation of (3) | i k n,k| k i (0) on ψ gives P h k | (E(0) E(0))c = E(1)δ H , k − n n,k n k,n − 1kn H1kn (0) (0) , k = n En E 6 cn,k = − k 0 k = n, (1) En = H1nn, (4) (0) (0) where H = ψm H ψn . 1mn h | 1| i 2. Second order (2) (0) (0) We now write ψn = d ψ and project the third equation of (3) on ψ , | i k n,k| k i h k | P (E(0) E(0))d = E(1)c + E(2)δ H c (5) k − n n,k n n,k n k,n − 1kℓ n,ℓ Xℓ The case k = n, using cn,n = 0 gives H 2 E(2) = 1nℓ (6) n (0)| | (0) En E Xℓ − ℓ and we find H1nnH1kn + 1 H1kℓH1ℓn , k = n (0) (0) 2 (0) (0) ℓ (0) (0) − (En E ) En E En E 6 dn,k = − k − k − ℓ (7) P 0 k = n. These kind of corrections are small as long as g ψ(0) H ψ(0) E(0) h n | 1| n i ≪ n g ψ(0) H ψ(0) E(0) E(0) . (8) |h k | 1| n i| ≪ | n − k | The first equation indicates that perturbation should be smaller than H0 in the diagonal, a con- ditions which is compatible with classical expectations. But the second inequality states an upper bound for the off-diagonal matrix elements of the perturbation which is a genuine quantum phe- nomenon. 7 Though the inequalities (8) can be satisfied for sufficiently small g the perturbation series are usually non-convergent in quantum mechanics due to the piling up of the higher order contributions. Consider for instance an anharmonic oscillators, defined by the Hamiltonian p2 mω2 g H = + x2 + x4. (9) 2m 2 4! The basic assumption of perturbation expansion is that the eigenvalues and the eigenvectors are analytical at g = 0. But this is usually not the case because analyticity shows up on a complex plane rather than on a line.
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