Chapter 40. Wave Functions and Uncertainty
The wave function characterizes particles in p
terms of the probability of finding them at various points in space. This scanning tunneling microscope image of graphite shows the most
of graphite shows the most probable place to find electrons.
Chapter Goal: To introduce the wave‐function description p
of matter and learn how it is interpreted.
Chapter 40. Wave Functions and Chapter . Wave Functions and Uncertainty
Topics:
•Waves, Particles, and the Double‐Slit Experiment •Connecting the Wave and Photon Views i
h
d h
i
•The Wave Function •Normalization •Wave Packets •The Heisenberg Uncertainty Principle
Review double slit
Review double slit
Connecting the Wave and Photon Views
The intensity of the light wave is correlated with the probability of detecting photons That is photons are more
probability of detecting photons. That is, photons are more likely to be detected at those points where the wave intensity is high and less likely to be detected at those points where the wave intensity is low.
The probability of detecting a photon at a particular point is directly proportional to the square of the light wave
is directly proportional to the square of the light‐wave amplitude function at that point:
Probability Density
We can define the probability density P(x) such that
–1. In one dimension, probability density has SI units of m
p
y
y
Thus the probability density multiplied by a length yields a dimensionless probability.
NOTE P(x) itself is not a probability. You must multiply the NOTE: P( ) itself is not a probabilit Yo m st m ltipl the
probability density by a length to find an actual probability. The photon probability density is directly proportional to p
p
y
y
yp p
the square of the light‐wave amplitude:
EXAMPLE 40.1 Calculating the probability density
QUESTION:
EXAMPLE 40.1 Calculating the probability density
Normalization
• A photon or electron has to land somewhere on the detector after passing through an experimental
detector after passing through an experimental apparatus. • Consequently, the probability that it will be detected at Consequently, the probability that it will be detected at
some position is 100%.
• The statement that the photon or electron has to land p
somewhere on the x‐axis is expressed mathematically as
• Any wave function must satisfy this normalization condition.
condition
Wave Packets
Wave Packets
Wave Packets
Suppose a single nonrepeating wave packet of duration Δt
is created by the superposition of many waves that span a y
p p
y
p
range of frequencies Δf. Fourier analysis shows that for any wave packet
We have not given a precise definition of Δt and Δf for a general wave packet.
The quantity Δt is The quantity Δt
is “about
about how long the wave packet lasts,
how long the wave packet lasts ” while Δf is “about the range of frequencies needing to be superimposed to produce this wave packet.”
EXAMPLE 40.4 Creating radio‐
EXAMPLE 40.4 Creating radio‐
frequency pulses
QUESTION:
EXAMPLE 40.4 Creating radio‐
EXAMPLE 40.4 Creating radio‐
frequency pulses
EXAMPLE 40.4 Creating radio‐
EXAMPLE 40.4 Creating radio‐
frequency pulses
The Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle
•The quantity Δx is the length or spatial extent of a wave p
packet.
•Δpx is a small range of momenta corresponding to the small range of frequencies within the wave packet.
•Any matter wave must obey the condition
This statement about the relationship between the position and momentum of a particle was proposed by Heisenberg in 1926 Physicists often just call it the
Heisenberg in 1926. Physicists often just call it the uncertainty principle.
The Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle
•If we want to know where a particle is located, we measure its position x with uncertainty Δx.
•If we want to know how fast the particle is going, we need to measure its velocity v
d
l
l l
x or, equivalently, its momentum px. This measurement also has some uncertainty Δpx.
uncertainty Δp
•You cannot measure both x and px simultaneously with arbitrarily good precision.
yg
p
•Any measurements you make are limited by the condition that ΔxΔpx ≥ h/2.
•Our knowledge about a particle is inherently uncertain.
General Principles
General Principles
General Principles
General Principles
Important Concepts
Important Concepts
Important Concepts
Important Concepts
The figure shows the detection of photons in an optical e periment Rank in order from lar est to
optical experiment. Rank in order, from largest to smallest, the square of the amplitude function of the electromagnetic wave at positions A, B, C, g
p
, , ,
and D.
A.
B
B.
C.
D.
D > C > B > A A>B>C>D
A > B > C > D A > B = D > C C > B = D > A This is the wave This
is the wave
function of a neutron. At what
neutron. At what value of x is the neutron most likely to be found?
A.
A
B.
C.
D.
x = 0
0
x = xA
x = xB
x = xC
The value of the constant a is
The value of the constant a
A.
B.
C.
D.
E.
a = 0.5 mm–1/2.
a = 1.0 mm–1/2.
a = 2.0 mm–1/2.
a = 1.0 mm–1.
a = 2.0 mm–1.
What minimum bandwidth must a medium have to transmit a 100‐ns‐long pulse?
A.
B.
C.
D.
E.
100 MHz
0.1 MHz
1 MHz
10 MHz
1000 MHz