Physics 249 Lecture 9, Sep 24th 2012
Reading: Chapter 5, Next time Chapter 6
HW3: Posted on the course homepage
1) Particle waves and wave packets and the probability interpretation
The wave properties of particles indicate that they are described by the solutions to a
wave equation. That wave equation solution needs an interpretation that is useful for
predicting the results that you observe, for instance, interference and diffraction
phenomena for particles. The correct interpretation is that the wave function (actually the
wave function squared) represents the probability to observe a particle at a given position
and time. For instance to show the probability as a function of x you can write the
probability distribution as:
𝑃(𝑥)𝑑𝑥 = |𝜓|2 𝑑𝑥
In an interference experiment if there is compete destructive interference of waves at a
point that means the particle has zero probability of being observed at those coordinates
according to the wave function squared. This is a consistent interpretation with that of
light waves where if the electric and magnetic fields of the wave (or wave packet) are
always zero due to destructive inference you observe no light there. Finally note that the
intensity pattern of the light classically is a function of the field strength squared just as
the probability pattern of particle interference is due to the square of the particle wave
function.
A note on interference: Interference between two waves occurs when the sources of the
waves have the same wavelength and are coherent, are in phase or have a well defined
phase difference. In practice to produce coherent waves of light, light particles or matter
particles the source must be the same.
In fact in the particle interpretation a single photon or matter particle in an interference
experiment follows both paths and interferers with itself. This guarantees that the two
waves are coherent. This interpretation is confirmed by progressing from high intensity
light experiments, where you see the entire interference pattern immediately, to low
intensity light/particle experiments where you can observe the interference pattern build
up over time with the location of each particle hit governed by the expected probability
distribution but otherwise random. The same can be done with light or matter particles.
2) The Uncertainty principle.
Consider the classical uncertainty relationships:
Δ𝑘Δ𝑥~1
Δ𝜔Δ𝑡~1
Using 𝐸 = ℏ𝜔
Δ𝑝Δ𝑥~ℏ
𝑝 = ℏ𝑘
Δ𝐸Δ𝑡~ℏ
In a wave packet the distributions of p and x can’t be made arbitral narrow at once. In
essence if you limit the possible value of one parameter, for instance by measuring where
the particle is very precisely then the particle will have a distribution of possible
momentums.
For instance you can measure the particle very precisely by sending it through and
extremely narrow slit width delta x. After that the momentum will be uncertain. It can
leave the slit with a large range of momentum in the x direction, Δ𝑝𝑥 .
For a Gaussian wave packet if we interpret these equations as the standard deviation on
the on Gaussian distributions of position and momentum or time and energy then the
exact value for this relationship is ½h. For any other shape of wave packet we find that
the value is larger than ½h. Therefore
1
Δ𝑝Δ𝑥 ≥ ℏ
2
1
Δ𝐸Δ𝑡 ≥ ℏ
2
We will look at how this is a real effect seen in measurement below.
3) Uncertainty principle and diffraction.
Diffraction, a wave phenomena, places a practical limit on the ability to measure the
position of objects with light.
Consider trying to measure an object as being at positions d1 or d2 a distance L away.
You do this by seeing the light from the object or if they are not luminous by bouncing
light off them via Compton scattering.
𝜆𝑓 − 𝜆𝑖 =
ℎ
(1 − 𝑐𝑜𝑠𝜃𝑐 )
𝑚𝑐
If the angle between the two positions is theta then when the light that comes from them
will be seen at different angles by theta. You observe the light with your eye, or a
telescope or by microscope all of which have some diameter D. You can only distinguish
the angle and thus the position if the central diffractive peaks of the light are separated,
i.e. the second peak is at the minimum of the first. Approximating the optical device as a
slit then:
Using sin q =
l
D
n at n=1
sin q =
l
D
or you can tell positions that are separated by a distance of at least.
Dx = L sinq =
l
D
L
Where the distance Dx is a distance perpendicular to the direction you have your optical
device pointed.
To resolve smaller distances you can use shorter wavelength light. However, when the
light has shorter wavelength it will be higher energy.
𝐸 = ℎ𝑓 =
ℎ𝑐
𝜆
Now simultaneously try to measure the momentum. We can do that by repeating the
experiment a short time later and measuring the new position with the same accuracy and
seeing how far the particle moved in that time. We will measure the position again by
bounding photons off the object via Compton scattering.
However, the first photon will transfer momentum to the target changing its momentum
by delta(p). Thus we don’t measure quite the initial momentum.
In a reference frame where the total momentum is zero the photon will have equal and
opposite momentum to the target after the collision. Considering the magnitude of the
targets change in momentum the photon must have lost half of it’s momentum or energy.
The targets momentum is changed by
Δ𝑝 =
1
𝐸
ℎ𝑓
ℎ
𝑝=
=
=
2
2𝑐 2𝑐 2𝜆
In some unknown direction obeying the Compton formula. Define that in the x direction
the targets momentum was changed by Δ𝑝𝑥 . However we can only observed the returning
photon if it has an angle small enough to enter our optical device. Therefore the x
1
1
𝐷
1 ℎ 𝐷
momentum can be anything up to Δ𝑝𝑥 = Δ𝑝𝑥 = 2 𝑝𝑠𝑖𝑛𝜃𝑑 = 2 Δ𝑝 𝐿 = 2 2𝜆 𝐿
So the measurement of the x position changed the x momentum an unknown amount that
is possibly as big as Δ𝑝𝑥 , since all you know is you saw the photon. Note that also that
the scattered photon has a wavelength twice as large since it lost half its energy.
Putting these two numbers together
ΔxΔ𝑝𝑥 =
2𝜆𝐿 ℎ 𝐷
1
1
, ΔxΔ𝑝𝑥 = ℎ > ℏ
𝐷 4𝜆 𝐿
2
2
Measure the x position via diffraction (the limit on how well we can measure any object
optically) by using a shorter wavelength improves the precision for delta x. However,
then we find that gain in precision for delta x comes at the cost of introducing a larger
uncertainty in the momentum delta p.
We find that just from considering quantum the wave nature that there are limitations to
our ability to measure objects!