1. Uncertainty Principle
Chapter 6
§6.8–6.9

2. Fourier Analysis Unbounded Function f(x) = A(k) cos(kx) dk
x = space, k = wavenumber = 2p/l
cos(kx) is a wave with wavelength 2p/k
A(k) is the Fourier transform of f(x)
f(x) = spatial pattern; A(k) = spectrum
x and k are conjugates

3. Fourier Analysis f(t) = A(w) cos(wt) dk
t = time, w = angular frequency= 2p/T
cos(wt) is a wave with period 2p/w
A(w) is the Fourier transform of f(t)
f(t) = time signal; A(w) = spectrum
t and w are conjugates

4. Meaning of FT time domain frequency domain FT space domain
wavenumber domain FT

5. Frequency and wavelength not determined, position known exactly
Waves and Uncertainty
Frequency and wavelength known exactly, position and time not determined
Frequency and wavelength less specific, position and time more specific
Frequency and wavelength not determined, position known exactly

6. Fourier Transforms
The narrower the distribution in space f(x), the more wavenumbers must be specified
The narrower the distribution in time f(t), the more frequencies must be specified

7. (Du)2 = <(u – <u>)2>
Uncertainty, defined
Root Mean Square (RMS)
(Du)2 = <(u – <u>)2>
<x> = expectation of x
It turns out
Dx Dk ≥ ½
and
Dt Dw ≥ ½

8. Position and Momentum Dx Dk ≥ ½
Recall p = h/l = h/(2p/k) = kh/(2p) = k h
So k = p/h, Dk = Dp/h
Dx Dp ≥ h/2

9. Time and Energy Dt Dw ≥ ½ Recall E = hf = h(w/2p) = wh
So w = E/h, Dw = DE/h
Dt DE ≥ h/2

10. Uncertainty and Confinement
Confine mass m to space of size a
Non-relativistic, K = p2/(2m)
Dx Dp = h/2
What is the minimum K?