QM Reminder

1. QM Reminder

2. C Nave @ gsu.edu http://hyperphysics.phy-astr.gsu.edu/hbase/quacon.html#quacon

3. Outline • Postulates of QM • Picking Information Out of Wavefunctions • Expectation Values • Eigenfunctions & Eigenvalues • Where do we get wavefunctions from? • Non-Relativistic • Relativistic • What good-looking Ys look like • Techniques for solving the Schro Eqn • Analytically • Numerically • Creation-Annihilation Ops

4. Postulates of Quantum Mechanics • The state of a physical system is completely described by a wavefunctionY. • All information is contained in the wavefunction Y • Probabilities are determined by the overlap of wavefunctions

5. Postulates of QM • Every measurable physical quantity has a corresponding operator. • The results of any individ measurement yields one of the eigenvalues ln of the corresponding operator. • Given a Hermetian Op with eigenvalues ln and eigenvectors Fn , the probability of measuring the eigenvalue ln is

6. Postulates of QM • If measurement of an observable gives a result ln , then immediately afterward the system is in state fn . • The time evolution of a system is given by • . corresponds to classical Hamiltonian

8. Picking Information out of Wavefunctions Expectation Values Eigenvalue Problems

9. Common Operators • Position • Momentum • Total Energy • Angular Momentum r = ( x, y, z ) - Cartesian repn L = r x p - work it out

10. Using Operators: A • Usual situation: Expectation Values • Special situations: Eigenvalue Problems the original wavefn aconstant (as far as A is concerned)

11. Expectation Values • Probability Density at r • Prob of finding the system in a region d3r about r • Prob of finding the system anywhere

12. Average value of position r • Average value of momentum p • Expectation value of total energy

13. Eigenvalue Problems Sometimes a function fn has a special property eigenfn eigenvalue Since this is simpler than doing integrals, we usually label QM systems by their list of eigenvalues (aka quantum numbers).

14. Eigenfns: 1-D Plane Wave moving in +x directionY(x,t) = A sin(kx-wt) or A cos(kx-wt) or A ei(kx-wt) • Y is an eigenfunction of Px • Y is an eigenfunction of Tot E • Y is not an eigenfunction of position X

15. Eigenfns: Hydrogenic atom Ynlm(r,q,f) • Y is an eigenfunction of Tot E • Y is an eigenfunction of L2 and Lz • Y is an eigenfunction of parity units eV

16. Eigenfns: Hydrogenic atom Ynlm(r,q,f) • Y is not an eigenfn of positionX, Y, Z • Y is not an eigenfn of the momentum vectorPx , Py , Pz • Y is not an eigenfn ofLx and Ly

18. Where Wavefunctions come from

19. Where do we get the wavefunctions from? • Physics tools • Newton’s equation of motion • Conservation of Energy • Cons of Momentum & Ang Momentum The most powerful and easy to use technique is Cons NRG.

20. Schrödinger Wave Equation Use non-relativistic formula for Total Energy Ops and http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians

21. Klein-Gordon Wave Equation Start with the relativistic constraint for free particle: Etot2 – p2c2 = m2c4 . p2 = px2 + py2 + pz2 [ Etot2 – p2c2 ] Y(r,t) = m2c4Y(r,t). See SP425  a Monster to solve

22. See SP425 Dirac Wave Equation Wanted a linear relativistic equation Etot2 – p2c2 = m2c4 p = ( px , py , pz) [ Etot2 – p2c2- m2c4 ] Y(r,t) = 0 Change notation slightly P4 = ( po , ipx , ipy , ipz) ~ [P42c2- m2c4 ] Y(r,t) = 0 difference of squares can be factored ~ ( P4c + mc2) (P4c-mc2) and there are two options for how to do overall +/- signs  4 coupled equations to solve.

23. Back to Time Dependent Schro Eqn Where H = KE + Potl E

24. Back to ER 5-5 Time Dependent Schro Eqn Where H = KE + Potl E

25. Time Independent Schro Eqn KE involves spatial derivatives only If Pot’l E not time dependent, then Schro Eqn separable ref: Griffiths 2.1

26. Drop to 1-D for ease

28. ER 5-6 What Good Wavefunctions Look Like

29. Sketching Pictures of Wavefunctions Prob ~ Y* Y KE + V = Etot

30. Bad Wavefunctions

31. Sketching Pictures of Wavefunctions To examine general behavior of wave fns, look for soln of the form where k is not necessarily a constant (but let’s pretend it is for a sec) KE

32. KE - KE + If Etot > V, then k Re Y ~ kinda free particle If Etot < V, then k Im Y ~ decaying exponential 2p/k ~ l ~ wavelength 1/k ~ 1/e distance

33. Sample Y(x) Sketches • Free Particles • Step Potentials • Barriers • Wells

34. Free Particle Energy axis V(x)=0 everywhere

35. 1-D Step Potential

37. 1-D Finite Square Well

38. 1-D Harmonic Oscillator

39. 1-D Infinite Square Well

40. 1-D Barrier

41. NH3 Molecule

42. E&R Ch 5 Prob 23 Discrete or Continuous Excitation Spectrum ?

43. E&R Ch 5, Prob 30 Which well goes with wfn ?

46. Techniques for solving the Schro Eqn. • Analytically • Solve the DiffyQ to obtain solns • Numerically • Do the DiffyQ integrations with code • Creation-Annihilation Operators • Pattern matching techniques derived from 1D SHO.

47. Analytic Techniques • Simple Cases • Free particle (ER 6.2) • Infinite square well (ER 6.8) • Continuous Potentials • 1-D Simple Harmonic Oscillator (ER 6.9, Table 6.1, and App I) • 3-D Attractive Coulomb (ER 7.2-6, Table 7.2) • 3-D Simple Harmonic Oscillator • Discontinuous Potentials • Step Functions (ER 6.3-7) • Barriers (ER6.3-7) • Finite Square Well (ER App H)

50. Eigenfns: Bare Coulomb - stationary statesYnlm(r,q,f) or Rnl(r) Ylm(q,f) Simple/Bare Coulomb