Physics 249 Lecture 9, Sep 21st 2012 Reading: Chapter 5, Next week Chapter 6 HW 2: Due Today, HW3: Posted on the course homepage 1) Uncertainty relationships for classical wave packets. Given the wave packets described in the previous lecture we find there is a relationship between the range of wave numbers or wavelengths and the localization in x. Integrating the series of waves we can derive the relationship. Δ!Δ!~1 For the wave packet to exist as a discreet entity if it is highly localized in space it must have a broader range of frequencies or vice versa. This will become a more interesting relationship when considering particle waves. You can write down and analog relationship Δ!Δ!~1 The exact value of the relationship depends on the shape of the wave packet. 2) Particle waves and wave packets and the probability interpretation The wave properties of particles indicate that they are the solution to a wave equation. That wave equation solution needs an interpretation that is consistent with the results that you observe interference and diffraction phenomena for particles. A consistent 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: !(!)!" = ! !!"   In an interference experiment if there is compete destructive interference of waves at a point that just 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 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 are coherent. In practice to produce coherent waves of light or 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 matter particles.

  Example wave function solutions cos or sin functions as above or ! !, ! = !!!(!"!!") The wave packet will be a superposition of these wave functions just as in the classical case. Consider the group velocity for a superposition of particles waves (non relativistic).

!! =!"!" =

!"/ℏ!"/ℏ =

!"!" =

!! = !

using ! = !!

!!

! = ℎ! = ℎ!2! = ℏ!

! =ℎ! =

ℎ2!/! = ℏ!

The wave packet propagates at the same velocity as the classical particle.

3) The Uncertainty principle. Consider the classical uncertainty relationships: Δ!Δ!~1 Δ!Δ!~1 Using ! = ℏ! ! = ℏ! Δ!Δ!~ℏ Δ!Δ!~ℏ To be a physical wave packet the distributions of p and k 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 delta px. For a Gaussian wave packet if we interpret these equations as the standard deviation on the on Gaussian distributions of position and momentum or energy and time 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 Δ!Δ! ≥ !

!ℏ Δ!Δ! ≥ !

!ℏ

We will look at how this is a real effect seen in measurement below. 4) Review of diffraction. Diffraction can be used to explore some important aspects of the uncertainty principle. First lets review the concept of diffraction. A propagating wave front behaves as if at any point the wave front emits waves spherically in all directions. Consider this idea for a small slit and calculate the wave interference on a far wall. See diagram: Consider the electric field magnitude of a light wave. zsin(theta) is the path difference between all the other wave front sources along the slit and a reference wave front source at the center of the slit and phi is the phase difference.

Ep = Ei!

= ("zDE0 )sin(!t + 2"

#zsin$ )!

= E0

D#D/2

D/2

$ sin(!t + 2"#zsin$ )dz

= 2E0

D 2"#

sin$sin(2"

#D2

sin$ )sin(!t)

For a small slit D, comparable to the wavelength, you see a maximum at theta equals zero (evaluating the limit) and minimum at: 2!"D2sin# = n!, sin# = "

Dn

which describes the diffraction pattern. For large very large D you see a maximum at theta equals zero, straight ahead, with a

width to the minimums of!! = 2 "D

, which is very small. However, also the nth

minimums, lets consider n=100, would have a small width of!! = 200 "D ,

for very large

D, and the magnitude would be down by 100 squared since the intensity of the light goes down as the square of the electric field. The only place you see non zero in the interference pattern is straight ahead.

A wave/wave front in free space, equivalent to having an arbitrarily large slit and the wave is only seen straight ahead in the direction of travel. For small D any obstacle of size of order the wavelength that limit the range of the type of integral above and will cause diffraction effects 5) Uncertainty principle and diffraction. Diffraction 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. 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:

sin! = "D

or you can tell positions that are separated by a distance of at least.

!x > Lsin! = "DL

Where the distance delta(x) 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.

! = ℎ! =ℎ!!

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. However, the first photon will transfer momentum to the target changing its momentum by delta(p). Thus we don’t measure quite the right 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

Δ! =12! =

!2! =

ℎ!2! =

ℎ2!

In some unknown direction. Define that in the x direction the targets momentum was changed by Δ!!. However we can only observed the returning photon if it has an angle small enough to enter our optical device. Therefore the x momentum can be anything up to Δ!! = Δ!! =

!!!"#$!! =

!!Δ! !

!= !

!!!!

!!

So the measurement of the x position changed the x momentum an unknown amount that is possibly as big as delta(px), 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

ΔxΔ!! ≥2!"!

ℎ4!!! ,ΔxΔ!! ≥

12ℎ

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! 6) Application of the uncertainty principle. Jumping forward to particle physics. We know now that the electromagnetic force is propagated by virtual photons. The idea is that these temporary photon can exist and even transfer negative (unphysical) momentum as long as they are consistent with zero within the uncertainty principle. Let’s investigate a consequence of this. The virtual photons have energy maximum E and exist for a maximum time of t such that Et~hbar. Or you can say they have maximum momentum p and cross a maximum distance r such that pr~hbar where r=tc. Under the Heisenberg uncertainty principle their physics properties are totally uncertain and consistent with zero within uncertainty. If they transfer a smaller amount of momentum then they can cross a larger distance or vice-versa. Then for force is F= dp/dt = dp/dr dr/dt ~ hbar c/r2. The functional dependence is a constant governing the strength over r2. The strength of the interaction depends on the charges involved. F=CqQ/r2 . The photons are transferred when there are charges the number of photons is going to be proportional to qQ times some constant that governs the strength of individual electromagnetic interactions.