PH 401

1
PH 401

• Dr. Cecilia Vogel
• Lecture 6

2
Review

• Free Particle
• Fourier Synthesis and Analysis
• group velocity

Outline

• Gaussian wavefunctions free particle
• Momentum by operator

3
Representation

• The wavefunction contains all the info available
  about the state of the particle.
• momentum info can be found by Fourier analysis.
• The momentum amplitude ALSO contains all the info
  available about the state of the particle.
• position info can be found by Fourier synthesis.
• Each is just a different representation of the
  particles state.

4
Gaussian Wavefunctions

• Often a Gaussian (aka normal, aka bell-curve)
  function is a good approximation for the
  wavefunction
• more mouthwatering mathematics.
• eqn 4.54, 4.55
• The expectation value of x
• ltxgt x0
• The uncertainty in x
• Dx L

5
Gaussian Wavefunctions

• If the wavefunction is Gaussian, the momentum
  amplitude will be, too
• more mouthwatering mathematics.
• eqn 4.58 and question 4-12
• The expectation value of p
• ltpgt ?k0 p0
• The uncertainty in p
• Dp ?/2L

6
Uncertainty Principle

• At t0,
• DxDp ?/2
• Gaussian is minimum uncertainty wavefunction
• look at graphs
• As we will soon see, for tgt0
• DxDp gt ?/2

7
x Operator

• Consider
• To find expectation value of position, we can
• multiply Y by x
• then multiply by Y and integrate
• For this reason
• position is said to be represented by the
  multiply by x operator

8
p Operator

• Consider

• For each partial wave
• bring down the momentum value

9
p Operator

• Consider
• To find expectation value of momentum, we can
• take deriv of Y with respect to x, and multiply
  by -i?
• then multiply by Y and integrate
• For this reason
• momentum is said to be represented by the -i?
  ?/?x operator

10
For Monday

• chapter