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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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)
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):
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
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
Griffiths Prob.2.13 QHO Worksheet
Free particle: V=0
Applications of Quantum mechanics
Blackbody radiation: resolve ultraviolet catastrophe, measure star temperatures http://220.127.116.11/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