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Lectures on Advanced Quantum Mechanics Lectures on Advanced Quantum Mechanics M. Zirnbauer Institut f¨urTheoretische Physik Universit¨atzu K¨oln WS 10/11, WS 17/18 Contents 1 Scattering theory 4 1.1 Preliminaries . 4 1.2 Types of scattering process . 5 1.3 Observables . 6 1.3.1 Classical scattering . 7 1.4 Lippmann-Schwinger equation . 8 1.4.1 Born approximation . 9 1.4.2 Examples . 9 1.5 Scattering by a centro-symmetric potential (partial waves) . 11 1.5.1 Optical theorem . 13 1.5.2 Example: scattering from a hard ball . 14 1.5.3 Asymptotics of the spherical Bessel functions . 15 1.6 Time-dependent scattering theory . 16 1.6.1 Example: potential scattering d =1...................... 18 1.6.2 Scattering by a centro-symmetric potential in d = 3 . 21 ⊂ y 1.6.3 On the condition range(W−) domain(W+) . 22 1.7 Time reversal and scattering . 23 1.7.1 Some linear algebra . 26 1.7.2 T -invariant scattering for spin 0 and 1=2................... 28 2 Relativistic quantum mechanics: Dirac equation 29 2.1 Motivation . 29 2.2 Dirac equation . 30 2.3 Relativistic formulation . 31 2.4 Non-relativistic reduction . 32 2.5 Enter the electromagnetic field . 33 2.6 Continuity equation . 35 1 2.7 Clifford algebra . 36 2.8 Spinor representation . 38 2.8.1 Tensor product . 42 2.9 Transforming scalar and vector fields . 42 2.10 Infinitesimal spin transformations . 43 2.11 Spin group . 46 2.12 Relativistic covariance of the Dirac equation . 47 2.13 Discrete symmetries of the Dirac equation . 49 2.13.1 Pin group and parity transformation . 49 2.13.2 Anti-unitary symmetries of the Dirac equation . 50 2.13.3 Parity violation by the weak interaction . 51 2.14 Dirac electron in a Coulomb potential . 53 2.14.1 Using rotational symmetry . 53 2.14.2 Reduction by separation of variables . 55 2.14.3 Solution of radial problem . 56 3 Second quantization 58 3.1 Bosons and fermions . 58 3.2 Weyl algebra and Clifford algebra . 60 3.3 Representation on Fock space . 63 3.4 Fock space scalar product . 66 3.5 Second quantization of one- and two-body operators . 67 3.6 Second quantization of Dirac's theory . 70 3.6.1 Motivation: problems with the Dirac equation . 70 3.6.2 Stable second quantization . 71 3.6.3 The idea of hole theory . 73 3.6.4 Mode expansion of the Dirac field . 74 3.6.5 Charge conjugation symmetry . 77 3.7 Quantization of the electromagnetic field . 79 3.7.1 Review and consolidation . 79 3.7.2 More preparations . 82 3.7.3 Symplectic structure . 83 3.7.4 Complex structure . 84 3.7.5 Fock space: multi-photon states . 86 3.7.6 Casimir energy . 87 3.7.7 Mode expansion for the torus geometry . 88 3.8 Matter-field interaction . 90 3.9 γ-decay of excited states . 91 2 4 Invariants of the rotation group SO3 92 4.1 Motivation . 92 4.2 Basic notions of representation theory . 93 4.2.1 Borel-Weil Theorem. 96 4.3 Invariant tensors . 97 4.3.1 Invariant tensors of degree 2 . 98 4.3.2 Example: SO3-invariant tensors of degree 2 . 100 4.3.3 Example: SU2-invariant tensors of degree 2 . 102 4.3.4 Invariant tensors of degree 3 . 105 4.4 The case of SU2 : Wigner 3j-symbol, Clebsch-Gordan coefficient . 107 4.5 Integrals of products of Wigner D-functions . 112 4.6 Tensor operators, Wigner-Eckart Theorem . 115 5 Dirac quantization condition 116 5.1 Dirac's argument . 117 5.2 The treatment of Wu & Yang . 120 5.3 Special case: monopole charge n = 2 . 121 5.3.1 Tangent bundle . 121 5.3.2 Complex structure . 123 5.3.3 Covariant derivative . 124 5.4 Other Cases (µ =6 2h=e) ................................ 127 5.5 Lesson learned . 129 3 1 Scattering theory Literature. A good fraction of the material of this chapter, in particular the introduction and the section on time-dependent scattering, is taken from the book by W.O. Amrein, J.M. Jauch and K.B. Sinha, Scattering Theory in Quantum Mechanics [Benjamin, Reading (Mass.), 1977]. 1.1 Preliminaries The formalism of scattering theory plays an important role in physics, as it allows to predict experimental observations from the fundamental interactions postulated by theory. A typical setup, say (i), for a scattering experiment looks as follows: Thus there is a source for a beam of particles which, after preparation of the beam kinematics, hit a target and then get observed in a detecting device. Note that thick targets cause multiple scattering events; while these might be hard to analyze in the case of an amorphous target, they lead to characteristic interference patterns and thus structural information in the case of a crystal (for the target). Next, let us look at the symmetric situation, (ii), of a scattering experiment with two colliding beams: In this setting one wants to place the detector far from the collision zone for good angular reso- lution. One does not place the detector in the forward direction of an incoming beam. Remark. In the case of scattering of massive particles, the situation (i) can be reduced to the situation (ii) by a change of inertial reference frame. Here comes a list of large accelerators: • Fermilab Chicago, Tevatron (1983; p + p; ca. 2 TeV; top quark discovered; 6 km) • Stanford Linear Accelerator (`SLAC'; 3 km) • Brookhaven RHIC (relativistic heavy ion collider; quark-gluon plasma) 4 • DESY Hamburg (HERA p + e; PETRA e+ + e−) • CERN Geneva (LEP; LHC = Large Hadron Collider, 17 TeV?) 1.2 Types of scattering process We briefly introduce some nomenclature: 1. Elastic scattering is schematically depicted as a + b −! a + b : In this case all energy is returned to the state of motion. 2. In the case of inelastic scattering, a + b −! a0 + b0 ; internal degrees of freedom of the particles get excited (e.g., rotational or vibrational degrees of freedom). 3. In the general category of rearrangement scattering, the identity of the scattered particles is altered: a + b −! c + d + e + ::: An example is the break-up of the deuteron into its constituents (namely, one proton and one neutron) during a collision with another particle. A second example is nucleon-nucleon scattering, where one has the possibilities plus various other \channels" not shown here. (The theoretical analysis of rearrangement scattering requires the formalism of multichannel scattering, which will not be treated in this lecture course.) 4. By resonance scattering one means a process that involves the formation and subsequent decay of an unstable (but possibly very long-lived) intermediate state. 5. In scattering theory one may also consider the decay of an unstable particle (a −! b+c+:::). For example, a free neutron decays into a proton, an electron, and an anti-neutrino: n −! p + e +ν ¯e : 5 1.3 Observables Here are some quantities that are observable by scattering experiments: • the (differential) scattering cross section; • the life time of an unstable particle; the resonance width; • branching ratios in multichannel scattering processes. We now introduce the differential scattering cross section, assuming the situation (i) of Sect. 1.1. Let the (probability) current density I of the incoming particles be homogeneous and directed along the z-axis for a Cartesian coordinate system fx; y; zg: I = i dxdy : The physical dimension of the constant i is particle number [i] = : area × time p At large distances r = x2 + y2 + z2 from the scattering region, the current density J of the outgoing (scattered) particles becomes ~r J = j sin θ dθ dϕ ; j = j(Ω) ; Ω = (point on the unit sphere); r where the polar angle θ and the azimuthal angle ϕ relate to the axis of the incoming beam (here, the z-axis). The coefficient j has the physical dimension particle number [j] = : solid angle × time The differential scattering cross section is then defined as the ratio dσ j = (not quite the usual derivative): dΩ i Its physical dimension is [ ] dσ = area : dΩ The total scattering cross section is obtained by integrating over all solid angles (dΩ ≡ sin θ dθ dϕ): Z Z dσ 1 σtot = dΩ = J: dΩ i S2 Remark. In these lecture notes we will be concerned with the scattering of quantum mechanical particles. We mention in passing that the notions of differential and total scattering cross section already make sense in the setting of classical mechanics. For example, the total cross section for 2 hard balls (as target particles) of radius R is σtot = πR in the classical limit. We now state a practical recipe by which to calculate the differential scattering cross section for potential scattering of Schr¨odingerparticles with energy E = ~2~k2=2m and wave vector ~k. 6 Prescription. Find the solution of Schr¨odinger'sequation H = E with the asymptotic form ikr j~rj=r!1 e (~r) −! ei~k·~r + f (Ω) ; k = j~kj ; E r where fE(Ω) = fE(~r=r) is called the scattering amplitude. Then the differential cross section is dσ = jf (Ω)j2 : dΩ E Problem. Verify this formula from the definition of dσ=dΩ. 1.3.1 Classical scattering Although our topic here is quantum scattering, let us briefly touch on the differential scattering cross section for classical scattering. In the classical case, one has a scattering map S : R2 ! S2, which is determined by the classical dynamics and sends any impact parameter (x; y) 2 R2 of the incoming particles to the corresponding direction of motion ~r=r 2 S2 of the outgoing particles.
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