1. Time-independent Schrodinger eqn QM Ch.2, Physical Systems, 12.Jan.2003 EJZ Assume the potential V(x) does not change in time. Use * separation of variables and * boundary conditions to solve for . Once you know , you can find any expectation value!

2. Outline: “Derive” Schroedinger Eqn (SE) Stationary states ML1 by Don and Jason R, Problem #2.2 Infinite square well Harmonic oscillator, Problem #2.13 ML2 by Jason Wall and Andy, Problem #2.14 Free particle and finite square well Summary

3. Schroedinger Equation

4. Stationary States - introduction If evolving wavefunction  (x,t) =  (x) f(t) can be separated, then the time-dependent term satisfies (ML1 will show - class solve for f) Separable solutions are stationary states...

5. Separable solutions: (1) are stationary states, because * probability density is independent of time [2.7] * therefore, expectation values do not change (2) have definite total energy, since the Hamiltonian is sharply localized: [2.13] (3)  i = eigenfunctions corresponding to each allowed energy eigenvalue E i. General solution to SE is [2.14]

6. ML1: Stationary states are separable Guess that SE has separable solutions  (x,t) =  (x) f(t) sub into SE=Schrodinger Eqn Divide by  f: LHS(t) = RHS(x) = constant=E. Now solve each side: You already found solution to LHS: f(t)=_________ RHS solution depends on the form of the potential V(x).

7. ML1: Problem 2.2, p.24

8. Now solve for  (x) for various V(x) Strategy: * draw a diagram * write down boundary conditions (BC) * think about what form of  (x) will fit the potential * find the wavenumbers k n =2  * find the allowed energies E n * sub k into  (x) and normalize to find the amplitude A * Now you know everything about a QM system in this potential, and you can calculate for any expectation value

9. Square well: V(0<x<a) = 0, V=  outside What is probability of finding particle outside? Inside: SE becomes * Solve this simple diffeq, using E=p 2 /2m, *  (x) =A sin kx + B cos kx: apply BC to find A and B * Draw wavefunctions, find wavenumbers: k n a=  n  * find the allowed energies: * sub k into  (x) and normalize: * Finally, the wavefunction is

10. Square well: homework 2.4: Repeat the process above, but center the infinite square well of width a about the point x=0. Preview: discuss similarities and differences

11. Ex: Harmonic oscillator: V(x) =1/2 kx 2 Tipler’s approach: Verify that  0 =A 0 e -ax^2 is a solution Analytic approach (2.3.2): rewrite SE diffeq and solve Algebraic method (2.3.1): ladder operators

12. HO: Tipler’s approach: Verify solution to SE:

13. HO: Tipler’s approach..

14. HO analytically: solve the diffeq directly Rewrite SE using * At large  ~x, has solutions * Guess series solution h(  ) * Consider normalization and BC to find that h n =a n H n (  ) where H n (  ) are Hermite polynomials * The ground state solution  0 is the same as Tipler’s * Higher states can be constructed with ladder operators

15. HO algebraically: use a ± to get  n Ladder operators a ± generate higher-energy wave- functions from the ground state  0. Work through Section 2.3.1 together Result: Practice on Problem 2.13

16. Harmonic oscillator: Prob.2.13 Worksheet

17. ML2: HO, Prob. 2.14 Worksheet

18. Ex: Free particle: V=0 Looks easy, but we need Fourier series If it has a definite energy, it isn’t normalizable! No stationary states for free particles Wave function’s v g = 2 v p, consistent with classical particle: check this.

19. Finite square well: V=0 outside, -V 0 inside BC:  NOT zero at edges, so wavefunction can spill out of potential Wide deep well has many, but finite, states Shallow, narrow well has at least one bound state

20. Summary: Time-independent Schrodinger equation has stationary states  (x) k,  (x), and E depend on V(x) (shape & BC) wavefunctions oscillate as e i  t wavefunctions can spill out of potential wells and tunnel through barriers