Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel

Class- evaluation Phys 452 Please take some time to submit your class evaluation on line by April 15 Quiz 35 on Wed April 13 5 points

Class schedule Phys 452 Today, April 11 : Finish the presentations Final Review Wed, April 13 : Practice final test Review the chosen problem with your group Type the solution Decide on who is going to present Quiz 36: 10 pts

Class scores W 2011 Phys 452 Homework (40%) Quizzes (10%) Midterms (30%) Final (20%) Class average 86.7 Tentative grade scale A93 A-89 B+85 B80 B-75 C+70 C60 C-55 D+50 D45 D-40 Prepare well for the final!!

Final test Phys 452 Where: classroom C247 Time – limited: 3 hours Comprehensive Closed book, closed notes Bring a calculator Tuesday, April 19 2pm - 5 pm

Final test Phys Time-independent perturbation theory 2. Hydrogen fine structure, Zeeman effect 3. Variational principle 4. WKB approximation 5. Emission, absorption 6. Scattering Choose 5 out of 6 problems Similar to

Techniques to solve for the allowed energies Phys The perturbation theory 2. The variational principle 3. The WKB approximation Quiz 34 a For which of these techniques you need to have a first estimate of the eigenstates? A.1 B.2 C.3 D.1 & 2 E.All of them

Perturbation theory Phys 452 Unperturbed states Building the true states and true energies to some order zero- order first- order second- order

Non-degenerate Perturbation theory First-order correction Phys 452 Energy State

Non-degenerate Perturbation theory Second-order correction Phys 452 Energy Only works if the energies are non-degenerate

Degenerate perturbation theory Phys 452 General method Start with an ortho-normal basis of the unperturbed states If the state is degenerate: build Diagonalize W : the eigenvalues are If the state is non-degenerate:

Phys 452 The fine structure of hydrogen Motion of the electron Coulomb interaction between e - and nucleus Bohr’s energies

Quiz 34 b Phys 452 What kind of interaction is at the origin of the spin-orbit coupling effect? A. An interaction between the spins of two electrons located at different orbits B. The spin of an electron interacting with the spin of the nucleus C. The spin of an electron interacting with field created by its angular momentum D. The spin of an electron interacting with the field created by another electron’s angular momentum E. An interaction between the spins of two electrons located in the same orbit

Phys 452 The fine structure of hydrogen Bohr’s energy E = Relativistic correction + Fine structure Spin-orbit coupling + S “ Classical view” B e+e+ e-e-

Phys 452 The fine structure of hydrogen Bohr’s energy E = Relativistic correction + Fine structure Spin-orbit coupling +

Phys 452 The fine structure of hydrogen Bohr’s energy E = New relevant quantum numbers: n, l, s, j and m j + Zeeman effect+ Fine structure ?

Phys 452 Zeeman effect “Classical view” e-e- S L B ext Weak-field Strong field Intermediate field Fine structure dominates Zeeman effect dominates Comparing:and

Phys 452 Zeeman effect Weak -field e-e- S L B ext Good eigenstates: with Lande factor:

Phys 452 Zeeman effect e-e- S L B ext Strong -field Good eigenstates:

Variational principle Phys 452 Ground state Expectation value on any normalized function  Hamiltonian Many particles Schrödinger Equation… … very hard to solve! ???

Quiz 34 c / d Phys 452 With the variational principle, we are guaranteed to find out the ground state A. TRUE B. FALSE What are we basically adjusting in the variational principle? A. The Hamiltonian B. The wave function C. Both

Variational principle The method: Phys 452 Define your system, and the Hamiltonian H Pick a normalized wave function  Calculate You get an estimate of ground state energy Minimize

The ground state of Helium Phys 452 He atom 2 particles system Kinetic energy Interaction with proton Electron- electron interaction Zero-order Hamiltonian H 0 Perturbation

The ground state of Helium Phys 452 He atom Use the variational principle to account for screening effects

The ground state of Helium Phys 452 He atom Use the variational principle to account for screening effects

Hydrogen molecule ion H 2 + Phys 452 Energy Minimization Presence of a minimum: Evidence of bonding Equilibrium separation distance:

Phys 452 The WKB approximation V(x) E Classical region (E>V) Non-classical region (E<V) Non-classical region (E<V) Turning points

Phys 452 The WKB approximation Excluding the turning points: where

Phys 452 Tunneling trough a barrier V(x) x V0V0 A B F -a+a Transmission coefficient

Quiz 34 e Phys 452 A. The transmission coefficient through the barrier depends on E, V and a B. The transmission coefficient increases when a decreases for a given E and V C. The transmission coefficient increases when V decreases for a given E and a D. The transmission coefficient increases when E decreases for a given V and a E. The particle has some chances to be reflected by the barrier if V>E A particle with an incident energy E is approaching a barrier of potential V and width a. Which one of these statements does NOT apply?

Phys 452 The WKB approximation V(x) E Classical region (E>V) Non-classical region (E<V) Patching – upward slope Linear approximation Patching region Overlap 1 Overlap 2 X=0

Phys 452 The WKB approximation General expression for the wave function Patching – upward slope

Phys 452 The WKB approximation Connection formulas Potential with no walls Potential with 2 walls Potential with 1 wall

Dynamical systems Phys 452 V depends on time General solution: Probability to measure the energy E n :

Two- level systems Time- dependent perturbation Phys 452 Sinusoidal perturbation Probability of transition:  P(  ) for a given time t Resonance effect

Emission and absorption of a radiation Phys 452 y z x with Transition rate

Quiz 36e Phys 452 A. It is the same as the transition rate for absorption B. It is opposite to the transition rate for absorption C. It is inverse of the transition rate for absorption D. It adds up with the spontaneous emission rate to cancel the absorption’s one What can we say about the transition rate of a stimulated emission ?

Emission and absorption: Einstein coefficients Phys 452 : stimulated absorption rate : stimulated emission rate : spontaneous emission rate Thermal equilibrium Analogy with Planck’s blackbody formula Boltzman distribution of particles Excited state lifetime

Emission and absorption Selection rules Phys 452 E Electric Dipole transitions

Adiabatic approximation Phys 452 General solution Adiabatic approx Geometric phase Dynamic phase with Berry’s phase

Radiation zone intermediate zone Phys 452 Scattering Develop the solution in terms of spherical harmonics, Solution to Coulomb potential Scattering zone

Phys 452 Scattering Partial wave analysis Connecting all three regions and expressing the Global wave function in spherical coordinates Total cross-section Rayleigh’s formula To be determined by the type of potential + boundary conditions Scattered waves

Phys 452 Scattering Phase - shifts Scattering factor Scattering Cross-section

Phys 452 Born approximation Scattering vector Plane wave Spherical wave

Phys 452 Born approximation Case of spherical potential Low energy approximation

Phys 452 Compton scattering Furthermore, we can evaluate the cross-section: We retrieve the conservation laws: Quantum theory