By Ben Simons
Quantum mechanics underpins numerous huge topic components inside physics
and the actual sciences from excessive strength particle physics, sturdy country and
atomic physics via to chemistry. As such, the topic is living on the core
of each physics programme.
In the next, we checklist an approximate “lecture by means of lecture” synopsis of
the various issues handled during this path.
1 Foundations of quantum physics: evaluation after all constitution and
organization; short revision of old historical past: from wave mechan-
ics to the Schr¨odinger equation.
2 Quantum mechanics in a single size: Wave mechanics of un-
bound debris; strength step; capability barrier and quantum tunnel-
ing; sure states; oblong good; !-function capability good; Kronig-
Penney version of a crystal.
3 Operator tools in quantum mechanics: Operator methods;
uncertainty precept for non-commuting operators; Ehrenfest theorem
and the time-dependence of operators; symmetry in quantum mechan-
ics; Heisenberg illustration; postulates of quantum thought; quantum
4 Quantum mechanics in additional than one size: inflexible diatomic
molecule; angular momentum; commutation family members; elevating and low-
ering operators; illustration of angular momentum states.
5 Quantum mechanics in additional than one size: valuable po-
tential; atomic hydrogen; radial wavefunction.
6 movement of charged particle in an electromagnetic ﬁeld: Classical
mechanics of a particle in a ﬁeld; quantum mechanics of particle in a
ﬁeld; atomic hydrogen – general Zeeman impact; diamagnetic hydrogen and quantum chaos; gauge invariance and the Aharonov-Bohm impact; unfastened electrons in a magnetic ﬁeld – Landau levels.
7-8 Quantum mechanical spin: historical past and the Stern-Gerlach experi-
ment; spinors, spin operators and Pauli matrices; concerning the spinor to
spin path; spin precession in a magnetic ﬁeld; parametric resonance;
addition of angular momenta.
9 Time-independent perturbation thought: Perturbation sequence; ﬁrst and moment order enlargement; degenerate perturbation conception; Stark impression; approximately loose electron model.
10 Variational and WKB process: floor kingdom strength and eigenfunc tions; program to helium; excited states; Wentzel-Kramers-Brillouin method.
11 exact debris: Particle indistinguishability and quantum statis-
tics; area and spin wavefunctions; outcomes of particle statistics;
ideal quantum gases; degeneracy strain in neutron stars; Bose-Einstein
condensation in ultracold atomic gases.
12-13 Atomic constitution: Relativistic corrections; spin-orbit coupling; Dar-
win constitution; Lamb shift; hyperﬁne constitution; Multi-electron atoms;
Helium; Hartree approximation and past; Hund’s rule; periodic ta-
ble; coupling schemes LS and jj; atomic spectra; Zeeman effect.
14-15 Molecular constitution: Born-Oppenheimer approximation; H2+ ion; H2
molecule; ionic and covalent bonding; molecular spectra; rotation; nu-
clear facts; vibrational transitions.
16 box idea of atomic chain: From debris to ﬁelds: classical ﬁeld
theory of the harmonic atomic chain; quantization of the atomic chain;
17 Quantum electrodynamics: Classical conception of the electromagnetic
ﬁeld; conception of waveguide; quantization of the electromagnetic ﬁeld and
18 Time-independent perturbation idea: Time-evolution operator;
Rabi oscillations in point platforms; time-dependent potentials – gen-
eral formalism; perturbation idea; surprising approximation; harmonic
perturbations and Fermi’s Golden rule; moment order transitions.
19 Radiative transitions: Light-matter interplay; spontaneous emis-
sion; absorption and prompted emission; Einstein’s A and B coefficents;
dipole approximation; choice ideas; lasers.
20-21 Scattering thought I: fundamentals; elastic and inelastic scattering; method
of particle waves; Born approximation; scattering of exact particles.
22-24 Relativistic quantum mechanics: background; Klein-Gordon equation;
Dirac equation; relativistic covariance and spin; loose relativistic particles
and the Klein paradox; antiparticles and the positron; Coupling to EM
ﬁeld: gauge invariance, minimum coupling and the relationship to non- relativistic quantum mechanics; ﬁeld quantization.
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Additional info for Advanced Quantum Physics
1 Rigid diatomic molecule As a pilot example let us consider the quantum mechanics of a rigid diatomic molecule with nuclear masses m1 and m2 , and (equilibrium) bond length, Re (see figure). Since the molecule is rigid, its coordinates can be specified by 2 r2 its centre of mass, R = m1mr11 +m +m2 , and internal orientation, r = r2 − r1 (with |r| = Re ). 1) ˆ = −i ∇R and L ˆ =r×p ˆ denotes the angular momentum associated where P with the internal degrees of freedom. Since the internal and centre of mass degrees of freedom separate, the wavefunction can be factorized as ψ(r, R) = eiK·R Y (r), where the first factor accounts for the free particle motion of the body, and the second factor relates to the internal angular degrees of freedom.
The ground state wavefunction is therefore spherically symmetric, and the function w(ρ) = w0 is just a constant. Hence u(ρ) = ρe−ρ w0 and the actual radial wavefunction is this divided by r, and of course suitably normalized. To write the wavefunction in terms of r, we need to find κ. 2 1 With this definition, the energy levels can then be expressed as En = − 4π1 0 (Ze) 2a0 n2 . Moving on to the excited states: for n = 2, we have a choice: either the radial function w(ρ) can have one term, as before, but now the angular momentum = 1 (since n = k + + 1); or w(ρ) can have two terms (so k = 1), and = 0.
4. ATOMIC HYDROGEN 40 To further simplify the wave equation, it is convenient to introduce the dimensionless variable ρ = κr, leading to the equation ∂ρ2 u(ρ) = 1− 2ν ( + 1) + ρ ρ2 u(ρ) , where (for reasons which will become apparent shortly) we have introduced Ze2 κ . Notice that in transforming from r the dimensionless parameter ν = 4π 0 2E to the dimensionless variable ρ, the scaling factor depends on energy, so will be different for different energy bound states! Consider now the behaviour of the wavefunction near the origin.