Physics 249 Lecture 7, Sep 17th 2012
Reading: Chapter 5
HW 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 =
ℎ𝑐
𝜆
Then assume you can solve this for a wavelength of a massive particle.
𝜆=
ℎ
ℎ
=
𝑝 𝑚𝑣
𝑓=
𝐸
ℎ
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.
1
𝐸 = 𝑚𝑣 2 ,
2
𝐽
1.60𝑥10−19 𝑒𝑉
2𝐸 √
𝑚
𝑣 = √ = 2 ∙ 10 ∙
= 1.874𝑥106
−31
𝑚
9.11𝑥10
𝑠
ℎ
6.63𝑥10−34
𝜆=
=
= 0.388𝑛𝑚
𝑚𝑣 9.11𝑥10−31 ∙ 1.874𝑥106
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:
𝑝2
𝐸 = 2𝑚 = 𝑒𝑉0 ,
𝑝 = √2𝑚𝑒𝑉0
ℎ ℎ𝑐
ℎ𝑐
1.24𝑥103 𝑒𝑉 ∙ 𝑛𝑚
1.226
𝜆= =
=
=
=
𝑛𝑚
𝑝 𝑝𝑐 √2𝑚𝑐 2 𝑒𝑉0 √2 ∙ 0.511𝑥106 𝑒𝑉 ∙ (𝑒𝑉0 )𝑒𝑉 √𝑒𝑉0
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 = 2𝐷𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛼 = 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=
ℎ
𝑝
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.
𝐸 2 = (𝑝𝑐)2 + (𝑚𝑐 2 )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
𝜕 2𝑦
1 𝜕 2𝑦
=
𝜕𝑥 2 𝑣 2 𝜕𝑡 2
with solution
𝑦(𝑥, 𝑡) = 𝑦0 cos(𝑘𝑥 − 𝜔𝑡)
with
angular frequency in terms of frequency or period: 𝜔 = 2𝜋𝑓 =
wave number in terms of the wavelength: 𝑘 =
2𝜋
𝑇
2𝜋
𝜆
and the wave or phase (how a specific point of phase moves) velocity 𝑣𝑝 = 𝑓𝜆 =
𝜔
𝑘
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 − 𝜔1 𝑡) + 𝑦0 cos(𝑘2 𝑥 − 𝜔2 𝑡)
1
1
1
1
𝑦(𝑥, 𝑡) = 2𝑦0 cos ( (𝑘1 − 𝑘2 )𝑥 − (𝜔1 − 𝜔2 )𝑡) cos ( (𝑘1 + 𝑘2 )𝑥 − (𝜔1 + 𝜔2 )𝑡)
2
2
2
2
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
Δ𝜔
group velocity set by 𝑣𝑔 = Δ𝑘
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 =
𝑑𝜔
𝑑𝑘
where
𝑣𝑝 = 𝑓𝜆 =
𝑣𝑔 =
𝜔
,
𝑘
𝜔 = 𝑘𝑣𝑝
𝑑𝑣𝑝
𝑑𝜔
= 𝑣𝑝 + 𝑘
= 𝑣𝑝
𝑑𝑘
𝑑𝑘
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.
Δ𝑘Δ𝑥~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|2 𝑑𝑥
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) = 𝐴𝑒 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 =
𝑑𝜔 𝑑𝐸/ℏ 𝑑𝐸 𝑝
=
=
= =𝑣
𝑑𝑘 𝑑𝑝/ℏ 𝑑𝑝 𝑚
𝑝2
using 𝐸 = 2𝑚
𝜔
𝐸 = ℎ𝑓 = ℎ
= ℏ𝜔
2𝜋
ℎ
ℎ
𝑝= =
= ℏ𝑘
𝜆 2𝜋/𝑘
The wave packet propagates at the same velocity as the classical particle.
6) The Uncertainty principle.
Consider the classical uncertainty relationships:
Δ𝑘Δ𝑥~1
Δ𝜔Δ𝑡~1
Using 𝐸 = ℏ𝜔
Δ𝑝Δ𝑥~ℏ
𝑝 = ℏ𝑘
Δ𝐸Δ𝑡~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.