Physics 451 Quantum mechanics I Fall 2012 Dec 5, 2012 Karine Chesnel

Homework Quantum mechanics Last assignment HW 24 Thursday Dec , 5.16, 5.18, Final exam Wednesday Dec 12, am – 10am C 285

Class evaluation Please fill the class evaluation survey online Quiz 34: 5 points Quantum mechanics

Solids Quantum mechanics e-e- Bravais k-space Fermi surface Pb 5.15:Relation between E tot and E F Pb 5.16:Case of Cu: calculate E F, v F, T F, and P F

Free electron gas Quantum mechanics Bravais k-space Fermi surface Total energy contained inside the Fermi surface

Free electron gas Quantum mechanics Bravais k-space Fermi surface Solid Quantum pressure

Solids Quantum mechanics e-e- Bravais k-space Fermi surface Number of unit cells

Solids Quantum mechanics V(x) Dirac comb Bloch’s theorem

Solids Quantum mechanics V(x) Circular periodic condition x-axis “wrapped around”

Solids Quantum mechanics V(x) Solving Schrödinger equation 0 a

Solids Quantum mechanics V(x) Boundary conditions 0 a or

Solids Quantum mechanics V(x) Boundary conditions at x = 0 0 a Continuity of  Discontinuity of

Solids Quantum mechanics Quantization of k: Band structure Pb 5.18 Pb 5.19 Pb 5.21

Quiz 33 Quantum mechanics A. 1 B. 2 C. q D. Nq E. 2N In the 1D Dirac comb model how many electrons can be contained in each band?

Solids Quantum mechanics Quantization of k: Band structure E N states Band Gap Band (2e in each state) 2N electrons Conductor: band partially filled Semi-conductor: doped insulator Insulator: band entirely filled ( even integer)

Quiz 33 Quantum mechanics A. Conductor B. Insulator C. Semi-conductor A material has q=3 valence electrons / atoms. In which category will it fall according to the 1D dirac periodic potential model?

Final Review Quantum mechanics What to remember?

Quantum mechanics Wave function and expectation values “Operator” x “Operator” p

Quantum mechanics Time-independent Schrödinger equation Here The potential is independent of time Stationary state General state

Review I Quantum mechanics Infinite square well Quantization of the energy x 0a Ground state Excited states

Quantum mechanics Harmonic oscillator x V(x) Operator position Operator momentum

Review I Quantum mechanics 4. Harmonic oscillator Ladder operators: Raising operator: Lowering operator:

Quiz 35 Quantum mechanics A. B. C. D. E. 0 What is the result of the operation ?

Quantum mechanics Square wells and delta potentials V(x) x Bound states E < 0 Scattering States E > 0 Symmetry considerations Physical considerations

Quantum mechanics Square wells and delta potentials Ch 2.6 Continuity at boundaries Delta functions Square well, steps, cliffs… is continuous is continuous except where V is infinite     m dx d         is continuous

Quantum mechanics Scattering state 0 AF B x Reflection coefficient Transmission coefficient The delta function well/ barrier “Tunneling”

Formalism Quantum mechanics Linear transformation (matrix) Operators Wave function Vector Observables are Hermitian operators is an eigenvector of Q is an eigenvalue of Q

Quantum mechanics Eigenvectors & eigenvalues To find the eigenvalues: We get a N th polynomial in : characteristic equation Find the N roots Spectrum Find the eigenvectors

Quantum mechanics The uncertainty principle Finding a relationship between standard deviations for a pair of observables Uncertainty applies only for incompatible observables Position - momentum

Quantum mechanics The uncertainty principle Energy - time Special meaning of  t Derived from the Heisenberg’s equation of motion

Quiz 33 Quantum mechanics A. B. C. D. E. Which one of these commutation relationships is not correct?

Quantum mechanics Schrödinger equation in spherical coordinates The angular equation The radial equation x y z r

Quantum mechanics The hydrogen atom Quantization of the energy Bohr radius

Quantum mechanics The hydrogen atom Energies levels Spectroscopy Energy transition Rydberg constant E 0 E1E1 E2E2 E3E3 E4E4 Lyman Balmer Paschen

Quantum mechanics The angular momentum eigenvectors x y z r Spherical harmonics are the eigenfunctions

Quantum mechanics The spin

Quantum mechanics Adding spins S Possible values for S when adding spins S 1 and S 2 : Clebsch- Gordan coefficients

Periodic table Quantum mechanics Filling the shells 22 6

Periodic table Quantum mechanics

Solids Quantum mechanics Bravais k-space Fermi surface e-e- Free electron gas theory Crystal Bloch’s theory

Quantum mechanics Thank you for your participation! And Merry Christmas! Good luck for finals