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

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

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

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- 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.

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)

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):

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)

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

Harmonic oscillator

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

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.

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±

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

QHO: 1. Substitution: Verify solution to SE:

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

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:

Griffiths Prob.2.13 QHO Worksheet

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:

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