1. Matter Waves
Chapter 6
§6.1–6.7

2. Homework
Chapter 6 problems 4, 5, 8, 16, 19, 20, 37, 43
Due March 10

3. de Broglie’s Wild Idea Maybe electrons act as waves!
After all, light can act like a particle.
Momentum of a photon:
p = h/l
p = momentum
h = Planck constant =  J s
l = wavelength

4. de Broglie’s Wild Idea What is an electron’s wavelength?
p = h/l, so l = h/p

5. Explains Bohr’s Quantization
standing waves nl = 2pr
Source: Griffith

6. Electrons: de Broglie and Bohr
nl = 2pr
nh/p = 2pr
nh/(2p) = rp
This is the Bohr condition that angular momentum is quantized in increments of h/(2p).

7. Experimental Verification
Electrons and neutrons diffract.
Diffraction patterns match l = h/p.

8. Waves and Quantization
Boundary conditions allow only certain wavelengths to be sustained

9. Born Interpretation What is a matter wave?
Square of amplitude |Y|2 is probability density
if Y is normalized to |Y|2dV = 1
Probability of being in region dV is ∫|Y|2dV
Still detected as particles

10. What is Y? “Wave function” Everything we “know” about a particle
Function of space and time Y(r,t)

11. Why? Because QM generalizes it
Fourier Analysis
Why? Because QM generalizes it

12. Basic Idea
Model a bounded or periodic function as a series of sines and cosines
The functions form a complete and orthogonal basis set
complete: series will converge to nearly any function
orthogonal: zero overlap

13. Fourier Series a0 + S an cos(nx) + S bn sin(nx) f(x) = 2
f(x) =
Finding an’s and bn’s is the job of Fourier analysis.

14. Example
“Sawtooth” wave

15. Analogy/Special Case Expressing a vector in terms of a basis set
Easiest if basis is orthonormal
E = {e1, e2, e3, ···}
A complete basis set of n vectors spans an n-dimensional vector space
X = x1e1 + x2e2 + x3e3 + ···
xn = X·en

16. Fourier series sin(nx) and cos(nx) form an basis set -dimensional
orthogonal
normalizable
basis set

17. Relation to Y Wavefunctions of bound particles are orthogonal
normalizable
a complete basis set