1. Schrödinger Wave Equation 1 Reading: Course packet Ch 5.7 SCHROEDINGER WAVE EQUATION (both images from Wikipedia.com)

2. Schrödinger Wave Equation 2 Schrödinger Wave Equation: More-or-less "god-given", just like Newton's law F(x) = dp/dt. It works, and if we can show that it fails, we'll refine or discard. You learned in Spins that the solution to the time-dependent SE is, in terms of the eigenstates of the Hamiltonian: where Notice the parallels to the rope problem we solved last week?

3. Schrödinger Wave Equation 3 How did this solution to the SWE come about? How do we solve this for  (x,t)? The operator H, the Hamiltonian or energy operator, is clearly special. So representing the kets or the wave function in the energy basis for expanding a general wave function is important! Note that the energy eigenstates are time INdependent if H is time independent. Write down the eigenvalue equation for the the Hamiltonian: We’ve already solved this equation for the energy eigenstates and energy eigenvalues for the case of the infinite and finite quare wells, so think of those values of E and those functions  (x) to be concrete.

4. Schrödinger Wave Equation 4 What follows is cast in terms of wave functions to give you practice, but in fact the ket notation can be used just as easily. The ket version of this discussion is in your text (Ch. 3) and was discussed in Spins. Expand the wave function in the energy basis. The time dependence comes from coefficients only: Substitute into Eigenfunctions of H

5. Schrödinger Wave Equation 5 What follows is cast in terms of wave functions to give you practice, but in fact the ket notation can be used just as easily.

6. Schrödinger Wave Equation 6 Keep working … So finally …

7. Schrödinger Wave Equation 7 So we have solved Schrödinger's time dependent equation! We know the time dependence, because we already know the eigenstates and energies, which we obtained from solving the eigenvalue equation.

8. Schrödinger Wave Equation 8 There are usually many functions that solve the eigenvalue problem, each function with its own associated eigenvalue. Label them with index n. In general, the state of particle is NOT an eigenstate of the Hamiltonian, but a superposition of eigenstates … Very important that the energy in the exponent matches the corresponding eigenstate.

9. Time dependence 9 n =1,2,3,4,5… for x a

10. Time dependence 10 n =1,2,3,4,5… for x a

11. Time dependence 11 At t = 0: Time dependent state: In abstract bra-ket notation: Notice that the constant b n in the previous slide has changed to c n. I used b n before instead of c n to avoid confusion with a time dependent c n (t), but it's so conventional to use c n, that I slipped back to it!

12. Time dependence 12 In position wave function representation: At t = 0: Time dependent state: For specific case of the particle in the infinite square well:

13. Time dependence 13 Class activity: Write short Maple program to animate a general state probability density. How do probability densities and expectation values (qualitative) change with time? First pick a single eigenstate (any one will do) Pick a superposition that is a sum of an even and and odd state. Pick a superposition that is a sum of two even or two odd states.

14. Time dependence 14 Discussion: Probability density always positive? What is a stationary state? Discuss time evolution. Discuss expectation value of x. Discuss simple model of atomic transitions

15. Schrödinger Wave Equation 15 Example: V(x) = 0 (free particle) 1st term is traveling wave (to right) and 2nd is traveling to left. Same eigenvalue E - “degenerate”

16. Schrödinger Wave Equation 16 Note the relationship between  and k for a free particle DISPERSION RELATION: Phase velocity Group velocity Not the same!!!

17. Schrödinger Wave Equation 17 Time dependent Schrödinger equation Energy eigenvalue equation (time independent SE) Eigenstates Time dependence (Connection to separation of variables) Mathematical representations of the above SCHROEDINGER WAVE EQUATION REVIEW

18. Schrödinger Wave Equation 18 The following treatment parallels the discussion of the 1-D wave equation more closely, but I like the approach earlier better. Schrödinger wave equation: As in 1-D non-dispersive eqn, use technique of separation of variables. is function of x only

19. Schrödinger Wave Equation 19 What shall we call the separation constant?

20. Schrödinger Wave Equation 20 Eigenvalue problem! Solutions (and there are many in general) depend on particular V(x). E represents the energy of a particle in that particular eigenstate, and there are many E values. Most general solution is: