5.1 X-Ray Scattering
5.2 De Broglie Waves
5.3 Electron Scattering
5.4 Wave Motion
5.5 Waves or Particles?
5.6 Uncertainty Principle
5.7 Probability, Wave Functions, and the Copenhagen Interpretation
5.8 Particle in a Box
CHAPTER 5
Wave Properties of Matter and Quantum Mechanics I
I thus arrived at the overall concept which guided my studies: for both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time.
- Louis de Broglie, 1929
Many experimental observations of particles can only be understood by assigning them “wave-like” properties.
5.1: X-Ray Scattering
Max von Laue suggested that if x rays (~ 10
-11m) were a form of electromagnetic radiation, interference effects should be
observed.
Crystals act as three-dimensional gratings, scattering the waves
and producing observable interference effects.
Bragg’s Law
William Lawrence Bragg simplified the von Laue picture.
He interpreted the x-ray scattering as the reflection of the incident x-ray beam from a unique set of planes of atoms within the crystal.
Each Bragg plane has
different density of atoms
Bragg’s Law
What are the conditions for constructive interference?
1. θincidence= θreflection (can be shown to arise from interference of exiting waves).
2. The difference in path lengths must be an integral number of wavelengths (nλ).
Result Bragg’s Law:
2 sin d n
X-ray diffraction has become a powerful technique in materials science to study crystal structure.
A Bragg spectrometer scatters x rays from several crystals.
The intensity of the diffracted beam is determined as a function of scattering angle by rotating the crystal and the detector.
When a beam of x rays passes through the powdered crystal (powder diffraction), the dots become a series of rings.
The Bragg Spectrometer (X-ray crystallography)
Example
2 sin d n
X-rays scattered from NaCl are observed to have an intense maximum at an angle of 20°from the incident direction. Assuming n = 1, what must be the wavelength of the incident radiation?
5.2: De Broglie Waves
Prince Louis V. de Broglie began studying the problems of the Bohr model in 1920.
Particularly struck by the fact that photons had both wave and particle properties.
He suggested that particles with non-zero mass should have wave properties.
In Ph.D. thesis he used Einstein’s theory of special relativity along with Planck’s quantum hypothesis to propose these properties.
For a photon, special relativity
E = pc
quantum hypothesis
E = hf
Therefore, De Broglie’s fundamental prediction is encapsulated in the equation:
De Broglie wavelength of a “matter wave”
Matter waves = De Broglie Waves
Example: What is the de Broglie wavelength of a 1000 kg automobile traveling at 100 m/s?
This is so much smaller than the dimensions of the object we can’t hope to observe wave behavior.
Example: What is the de Broglie wavelength of an electron with K = 1 eV?
Therefore wave behavior might be important in atomic theory.
Atomic dimension
Bohr’s Quantization Condition
One of Bohr’s assumptions in his hydrogen atom model was the angular momentum of the electron-nucleus system in a stationary state is quantized in units of h/2π.
Can see how this assumption can be interpreted physically by considering the electron as a wave.
In particular we consider the electron as a standing wave in an orbit around the proton.
This standing wave will have nodes
and be an integral number of wavelengths.
Then, the angular momentum becomes:
L n
5.3: Electron Scattering
de Broglie’s ideas had no experimental proof.
proof provided by Davisson and Germer who experimentally observed that electrons were diffracted much like X-rays by nickel single crystal.
Once nickel crystal surface was properly cleaned, observed large increases
in the scattered intensity at certain angles.
Electrons are diffracted like waves
Aware of de Broglie’s ideas found electrons obeyed Braggs law for diffraction.
Example: original experiment d = 0.215 nm, K = 54 eV.
sin
d n
0.167 sin d 0.215
51
o(observed: 50
o)
Electron diffraction in transmission
George P. Thomson (1892–1975), son of J. J. Thomson, reported seeing the effects of electron diffraction in transmission experiments.
The first target was celluloid, and soon after that gold, aluminum, and platinum were used.
The randomly oriented polycrystalline sample of SnO2 produces rings as shown in the figure at right.
Particle diffraction
Subsequently also demonstrated that hydrogen and helium atoms exhibit wave properties.
Neutron diffraction, like X-ray diffraction is powerful probe of crystal and molecular structure.
Example: Determine the wavelength of neutron at 300K.
Given the validity of the de Broglie hypothesis, we want to develop a wave description of particle motion.
Simplest wave is a sinusoid. If we take a snapshot of such a wave at time t = 0.
Ψ(x,t) is the displacement:
for sound → pressure
for EM radiation → E, B fields
The wave Ψ(x,t) will propagate & at some later time:
5.4: Wave Motion
The expression for the displacement of a wave is
This is a solution to the wave equation:
Define the wave number k and the angular frequency ω as:
The wave function is now:
5.4: Wave Motion
and
Wave Properties
The phase velocity is the velocity of a point on the wave that has a given phase (for example, the crest) and is given by
A phase constant Φ shifts the wave:
.
How to describe a moving particle as a wave?
Our ultimate goal in quantum theory is to use waves to describe a moving particle.
To represent a particle which is localized in space we will need a wave representation which is localized in space rather than the
sinusoidal example we started with.
i.e. we need to construct a pulse, or wave packet.
Mathematically how do we localize the wave?
Answer: by combining many waves with different amplitudes and frequencies.
Superposition of waves
Principle of Superposition
One fundamental properties of waves as opposed to particles is the principle of superposition.
When two or more waves traverse the same region, they act
independently of each other and the resultant amplitude is the sum of the of the amplitudes of each individual wave.
the addition can lead to
constructive or destructive
interference depending on
phase of waves.
Superposition of waves wave packet
to demonstrate how localization of the wave is obtained, let us combine (superpose) two sinusoidal waves of the same amplitude but slightly different wave vectors k
1and k
2.
u
gr
k
Beat
Beat envelop moves at Group Velocity:
group
u
Wave packets
To further localize the wave within x by producing a wave packet, we need to combine more than two waves over a range of wave vectors ∆k.
Define ∆x to be ∆λ/2 from the previous diagram. The phase difference between these two point must be (half
wavelength of the packet).
Similarly at given location can find the time ∆t where wave is localized:
/ 2 x
x
1x
22 1
x x x
2
k x
2
t
Fourier Series to localize a wave
Mathematically to localize a wave in space we need to sum over many waves with different k, ω, and amplitudes.
The sum of many waves that form a wave packet is called a Fourier series:
Summing an infinite number of waves yields the Fourier integral:
Fourier analysis beyond the scope of the course but will look at one example, the Gaussian Wave Packet.
Gaussian Wave Packet is often used to describe particles and is
an example of a “minimum uncertainty wave packet”.
1 k x t 2
A Gaussian wave packet describes the envelope of a pulse wave.
When you move from a summation to an integral the definition of the group velocity becomes:
Note, Gaussian wave packet has relations:
Gaussian Function
gr
u d
k dk
minimum uncertainty wave packet
de Broglie Waves as Wave Packets
Considering the group velocity of a de Broglie wave packet yields:
Given E
2= (pc)
2+ (mc
2)
2this leads to:
Thus group velocity of a de Broglie wave is equivalent to the translational velocity of “particle”.
The relationship between the phase velocity and the group velocity is Given v
ph= ω/k
Hence the group velocity may be greater or less than the phase velocity.
A medium is called nondispersive when the phase velocity is the same for all frequencies and equal to the group velocity.
otherwise, a medium is dispersive.
5.5: Waves or Particles?
Results leave number of questions:
If particles have wave properties what is oscillating?
Can we represent matter as waves and particles simultaneously?
Young’s double-slit diffraction experiment demonstrates the wave property of light.
However, dimming the light results in
single flashes on the screen representative
of particles.
Electron Double-Slit Experiment
C. Jönsson of Tübingen,
Germany, succeeded in 1961 in showing double-slit interference effects for electrons by
constructing very narrow slits and using relatively large distances between the slits and the
observation screen.
This experiment demonstrated
that precisely the same behavior
occurs for both light (waves)
and electrons (particles).
Which slit?
From our experience the electron is a particle and a particle can only go through one of slits.
To determine which slit the electron went through:
We set up a light shining on the double slit and use a powerful microscope to look at the region.
After the electron passes through one of the slits, light bounces off the electron; we observe the reflected light, so we know which slit the electron came through.
The difficulty is that the momentum of the
photons used to determine which slit the electron went through is sufficiently great to strongly modify the momentum of the electron itself, thus changing the direction of the electron! The attempt to identify which slit the electron is passing through will in itself change the interference pattern.
Thus, any attempt to determine which slit the electron passes through destroys the
interference pattern.
In trying to determine which slit the electron passes through we are examining the particle-like behavior of the electron.
One the other hand, when we observe the interference pattern we are using the wave-like behavior of the electron.
Wave-particle duality solution
Bohr pointed out that particle-like and wave-like aspects are complimentary.
Bohr’s principle of complementarity: It is not possible to describe physical observables simultaneously in terms of both particles and waves.
where, Physical observables are those quantities such as position, velocity, momentum, and energy that can be experimentally measured.
In any given instance we must use either the particle description or the wave description.
We will see later on that the “amplitude” of the de Broglie wave is related to the probability of finding the object at x,y,z and t.
where magnitude large: more likely to find the object.
