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Modern Physics 6b Physical Systems, week 7, Thursday 22 Feb. 2007, EJZ

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

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Modern Physics 6b Physical Systems, week 7, Thursday 22 Feb. 2007, EJZ

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  1. Modern Physics 6bPhysical 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 • Choose for next quarter: EM, QM, Gravity? 2/3. Vote on Tuesday.

  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 y(x,t): Order matters for operators like momentum – differentiate y(x,t):

  4. Expectation values • Exercise: Consider the infinite square well of width L. • What is <x>? • (b) What is <x2>? • What is <p>? (Guess first) • What is <p2>? (Guess first)

  5. Expectation values • Exercise: Consider the infinite square well of width L. • What is <x>? • (b) What is <x2>? • What is <p>? (Guess first) • What is <p2>? (Guess first) A: L/2 C: <p>=0 D: <p2>=2mE

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

  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 y0=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:

  11. 2. QHO analytically: solve the diffeq directly: Rewrite SE using * At large x~x, has solutions * Guess series solution h(x) * Consider normalization and BC to find that hn=an Hn(x) where Hn(x) are Hermite polynomials * The ground state solution y0 is the same as before: * Higher states can be constructed with ladder operators

  12. 3. QHO algebraically: use a± to get yn Ladder operators a±generate higher-energy wave-functions from the ground state y0. Griffiths Quantum Section 2.3.1 Result:

  13. Griffiths Prob.2.13 QHO Worksheet

  14. 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 vg = 2 vp, consistent with classical particle:

  15. Applications of Quantum mechanics 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), a-decay (280), NH3 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... Choose your Minilectures for Ch.7

  16. Scanning Tunneling Microscope

  17. Alpha Decay

  18. Ammonia Atomic Clock

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