1. Modern Physics 6b Physical Systems, week 7, Thursday 22 Feb. 2007, EJZ Ch.6.4-5: Expectation values and operators Quantum harmonic oscillator → blackbody applications week 8, Ch.7.1-3: Schrödinger Eqn in 3D, Hydrogen atom week 9, Ch.7.4-8: Spin and angular momentum, applications

2. Review energy and momentum operators Apply to the Schrödinger eqn: E  (x,t) = T  (x,t) + V  (x,t) Find the wavefunction for a given potential V(x)

3. Expectation values Most likely outcome of a measurement of position, for a system (or particle) in state  x,t  : Order matters for operators like momentum – differentiate  (x,t):

4. Expectation values Exercise: Consider the infinite square well of width L. (a)What is ? (b) What is ? (c)What is ? (Guess first) (d)What is ? (Guess first)

6. This is one of the classic potentials for which we can analytically solve Sch.Eqn., and it approximates many physical situations. Harmonic oscillator

7. Simple Harmonic oscillator (SHO) What values of total Energy are possible? What is the zero-point energy for the simple harmonic oscillator? Compare this to the finite square well.

8. Solving the Quantum Harmonic oscillator 0. QHO Preview Substitution approach: Verify that y 0 =Ae -ax^2 is a solution 2. Analytic approach: rewrite SE diffeq and solve 3. Algebraic method: ladder operators a±

9. QHO preview: What values of total energy are possible? What is the zero-point energy for the Quantum Harmonic Oscillator? Compare this to the finite square well and SHO

10. QHO: 1. Substitution: Verify solution to SE:

12. 2. QHO 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 before: * Higher states can be constructed with ladder operators

13. 3. QHO algebraically: use a ± to get  n Ladder operators a ± generate higher-energy wave- functions from the ground state  0. Griffiths Quantum Section 2.3.1 Result:

14. Griffiths Prob.2.13 QHO Worksheet

16. 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:

17. Applications of Quantum mechanics Choose your Minilectures for Ch.7 Blackbody radiation: resolve ultraviolet catastrophe, measure star temperatures http://192.211.16.13/curricular/physys/0607/lectures/BB/BBKK.pdf Photoelectric effect: particle detectors and signal amplifiers Bohr atom: predict and understand H-like spectra and energies Structure and behavior of solids, including semiconductors STM (p.279),  -decay (280), NH 3 atomic clock (p.282) Zeeman effect: measure magnetic fields of stars from light Electron spin: Pauli exclusion principle Lasers, NMR, nuclear and particle physics, and much more...

18. Scanning Tunneling Microscope

19. Alpha Decay

20. Ammonia Atomic Clock