Today: continue Matter Waves!
deBroglie thought electrons were waves with
λ
=h/p
el
Experiments demonstrated electrons have wave behavior,
with wavelength matching deBroglie’s wavelength
Revisit interpretation of what these waves are:
• Describing electrons (or any particle) with wave functions
• Interpreting (wave amplitude)
2
as probability density
• Making localized particles with wave packets
• Uncertainty principle … position and momentum
+
Three deBroglie waves are shown for particles of equal mass.
I
II
III
x
The highest speed and lowest speed are:
A, 2f
2A, 2f
A, f
x
x
Warmup question
Q1
b. I and II same and highest, III is lowest
c. all three have same speed
d. cannot tell from figures above
A: Amplitude. f: frequency
Last class we introduced the “wave function”
Ψ
(x,y,z,t). We discussed, that we can observe
its amplitude square (a ‘probability density’).
Probability density
= P(x,t) = 
Ψ

2
=
Ψ
*
Ψ
The likelihood of a particle being
detected at specific locations (and times).
Matter waves:
Probability density
= P(x,t) = 
Ψ

2
=
Ψ
*
Ψ
P(x,t=0)
Ψ
(x,t=0)
x
L
L
Wave function =
Ψ
(x,t)
L
L
x
Probability of electron
being in interval dx = P(x)·dx
dx
More general: Probability of finding
electron between x
1
and x
2
at time t:
P(x,t)dx =

Ψ
(x,t)
2
dx
x
1
x
2
x
1
x
2
This requires ‘normalization’ of
ψ
(x)
Normalization
Probability density
= P(x,t) = 
Ψ

2
=
Ψ
*
Ψ
P(x,t=0)
L
L
x
The probability of finding the particle
“Normalized wave function”
The probability of finding the particle
anywhere in space (i.e. between –
∞
and +
∞
)
must be 100%!
P(x,t)dx =
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 Spring '08
 staff
 Physics, Uncertainty Principle, Wavelength, wave function, probability density

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