Physics 249 Lecture 7, Sep 17th 2012Reading: Chapter 5HW 2: Due Friday

1) de Broglie Wavelength

The de Broglie hypothesis: Hypothesized that just as light has wave like and particle like properties matter particles should also have particle like and wave like properties

Consider the relativistic relationships for a massless particle.

E=pc=hf=hcλ

Then assume you can solve this for a wavelength of a massive particle.

λ= hp= hmv

f= Eh

These equations led to a physical interpretation of the Bohr angular momentum quantization as a standing wave condition.

Typical particle wavelengths would be extremely small. Since wavelike properties are best demonstrated by interference or diffraction experiments dimensions involved would have to be very small. The smallest spacing’s available ware those between planes of atoms in crystalline solids which were about 0.1nm. The wavelength involved could be maximized by using electrons (low mass) and low momentums.

2) Example: nonrelativistic 10eV electrons.

E=12mv2 , v=√ 2 E

m=√2∙10 ∙

1.60 x10−19 JeV

9.11x 10−31 =1.874 x 106 ms

λ= hmv

= 6.63 x 10−34

9.11 x10−31 ∙1.874 x 106 =0.388nm

There is an easier solution that demonstrates the utility of energy units in eV

Given non relativistic velocity for an electron accelerated by a potential, often useful in this case since the velocities have to be nonrelativistic to give a wavelength large enough to exhibit wavelike phenomena:

E= p2

2m=eV 0 , p=√2meV 0

λ= hp= hc

pc= hc

√2mc2 eV 0

= 1.24 x103 eV ∙nm

√2 ∙0.511 x106eV ∙ (eV 0 )eV=1.226

√eV 0

nm

λ=1.226√10

nm=0.388nm

3) Davisson-Germer Experiment

Using Bragg diffraction

Consider scattering off the two layers of a crystal. The path to scatter off the lower plane is 2dsinθ longer.

The Bragg condition for constructive interference is nλ=2dsinθ

However, you have to be careful about what planes of atoms your electrons see:

Here: d=Dsinα

and: nλ=2dsinθ=2dcosα=2 Dsinαcosα=Dsin (2α )=Dsin(φ)

In the first experiment of this type they saw one instance of constructive interference for a spacing of 0.215nm at 50degrees indicating a wavelength of 0.165nm. 54 eV electrons have a wavelength of 0.167nm.

The diffraction of other particle including neutral particles can be demonstrated using lower energies. For neutral particles the energies are thermal since there was no way to accelerate them electrostatically.

4) Relativistic de Broglie Wavelength

Using

λ= hp

If the particle is relativistic we cannot simply relate the momentum to the kinetic energy. Instead we have to use p as in the relativistic energy relationship.

E2= (pc )2+(mc2)2

(could present more here or just refer to text book)

5) Wave packets

We have demonstrated that light and matter particles have both wave and particle like properties. If light and matter particles are waves that fact indicates that they are the solutions to wave equations.

Consider the classical wave equation

∂2 y∂ x2 =

1v2

∂2 y∂ t 2

with solution

y ( x , t )= y0 cos (kx−ωt )

with

angular frequency in terms of frequency or period: ω=2πf =2πT

wave number in terms of the wavelength: k=2πλ

and the wave or phase (how a specific point of phase moves) velocity vp=fλ=ωk

and maximum amplitude: y0

Light is an electromagnetic wave with amplitude set by the size of the Electro/Magnetic fields, which are what is oscillating.

For matter particles the wave equation (squared) will describe the probabilities of finding the particle as a given place and time. The probability field is what is oscillating. The maximum amplitude will be normalized by setting the probability of finding the particle anywhere at a given time to 1, or 100%.

This description is not so different from the classical wave equation for light where at any give time and place the electric or magnetic fields may be zero.

Consider adding two waves together with the same velocity but different wave numbers or frequencies.

y ( x , t )= y0 cos (k1 x−ω1t )+ y0 cos (k 2 x−ω2t )

y ( x , t )=2 y0cos ( 12 (k 1−k2 ) x−1

2 (ω1−ω2) t)cos ( 12 (k1+k 2) x−1

2 (ω1+ω2 ) t)

You get a slow frequency wave with the frequency set by the difference and a fast frequency wave with the frequency set by the average. The envelope moves with the

group velocity set by vg=ΔωΔk

If you add an infinite number of waves with infinitesimal separation in a narrow band around a given wave number you can localize the wave to a narrow position and make it zero everywhere else. This is known as a wave packet

The group velocity (the velocity that the packet of waves moves at) will be:

vg=dωdk

where

vp=fλ=ωk,ω=k v p

vg=dωdk

=v p+kd v p

dk=v p

Assuming the velocity does not change as a function of wave number or wavelength (like light in a vacuum).

Note that out of vacuum the group velocity can change, light can move at a different speed than c in a material, and the different parts of the wave move at different speeds causing a phenomenon called dispersion.

5) Uncertainty relationships for classical wave packets.

Given the wave above 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 k Δ x 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 viceverso. This will become a more interesting relationship when considering particle waves.

You can write down and analog relationship

ΔωΔt 1

6) Particle waves and wave packets

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:

P(x )dx=|ψ|2dx

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.

Example wave function solutions cos or sin function or

ψ ( x , t )=A e i(kx−ωt)

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).

vg=dωdk

=dE /ℏdp /ℏ=dE

dp= pm

=v

using E= p2

2m

E=hf=h ω2π

= ωℏ

p=hλ= h

2π /k= kℏ

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

6) The Uncertainty principle.

Consider the classical uncertainty relationships:

Δ k Δ x 1 ΔωΔt 1

Using E= ωℏ p= kℏ

Δ p Δ x ℏ ΔE Δt 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 on 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. After that the momentum will be uncertain. Examples and consequences will be discussed next time.