Exam Notes - Matter Waves

Experimental evidence for matter waves using low-energy electrons.
Setup:
- Filament generates electrons by heating.
- Electrons are accelerated through a voltage in a vacuum chamber.
- Detector measures the current (number of electrons) at varying angles.
Constructive interference equation:
- $\text{extra path length} = m \lambda = d \sin(\alpha)$ , where:
  - $m$ is an integer.
  - $\lambda$ is the wavelength.
  - $d$ is the spacing between molecules.
  - $\alpha$ is the angle of detection.
Intensity is measured as a function of $\alpha$ (polar plot).
Maximum intensity is observed at a specific angle.
If the kinetic energy of electrons is high enough to interact with lower crystal levels, the Bragg equation is used.
Bragg equation:
- $n \lambda = 2d \sin(\theta)$
- Constructive interference occurs layer by layer within the crystal.
At low kinetic energies, electrons primarily interact on the surface.
High enough energy levels allow for Bragg diffraction, traditionally done with non-normal incidence.

For regular waves, the wave function represents a physical disturbance (e.g., electric field for electromagnetic waves).
Particles have properties like mass, energy, momentum, and angular momentum.
Question: What does the amplitude of a matter wave represent?
Particles are localized, while waves are extended in space.
In electromagnetic waves, disturbances are physical and measurable with mass and energy, representing photons.
In water waves and sound waves, displacement of particles or air molecules can be measured.
Davisson-Germer experiment demonstrates the wave nature of particles.
Increasing the energy (and thus momentum) of electrons changes the wavelength, allowing for prediction of the maximum angle.
This analysis suggests the wave nature isn't accidental.

Example: Electron accelerated through 1000 volts.
Check if non-relativistic analysis is valid:
- Energy gained by electron: $q \Delta V = (1.6 \times 10^{-19} \text{ C}) (1000 \text{ V}) = 1.6 \times 10^{-16} \text{ J}$
- Rest mass energy of electron: $m_0 c^2 = (9.1 \times 10^{-31} \text{ kg}) (3 \times 10^8 \text{ m/s})^2 = 9 \times 10^{-14} \text{ J}$
- Since 1.6 \times 10^{-16} \text{ J} << 9 \times 10^{-14} \text{ J}, non-relativistic analysis is valid.
Kinetic energy:
- $q \Delta V = \frac{1}{2} m v^2$
- $1.6 \times 10^{-16} \text{ J} = \frac{1}{2} (9 \times 10^{-31} \text{ kg}) v^2$
- $v \approx 1.7 \times 10^7 \text{ m/s}$
Momentum:
- $p = mv = (9 \times 10^{-31} \text{ kg}) (1.7 \times 10^7 \text{ m/s}) \approx 1.53 \times 10^{-23} \text{ kg m/s}$
Wavelength:
- $\lambda = \frac{h}{p} = \frac{6.626 \times 10^{-34} \text{ J s}}{1.53 \times 10^{-23} \text{ kg m/s}} \approx 4.5 \times 10^{-11} \text{ m}$
This wavelength is smaller than visible light wavelengths (on the order of microns).

Uses matter waves of electrons for higher resolution.
Electrons have much smaller wavelengths than visible light.
Increasing voltage increases electron speed and decreases wavelength, improving resolution.
At very high voltages, relativistic effects must be considered.
Practical limitations exist at very high voltages due to other issues.
Enables visualization of structures at atomic levels (10^-10 m).
Working principle:
- Similar to visible light microscope, but uses electrons instead of light.
- Electron source illuminates the object.
- Objective lens (using magnetic and electric fields) magnifies the image.
- A detector sensitive to electrons captures the image.
Uses electric and magnetic fields to control electron beam paths (geometric optics).
High-power lens (electric/magnetic fields) dramatically changes electron path for magnification.
Multiplication factor is about 40,000, much better than a physical light microscope.
Allows studying surface structures at the atomic level (e.g., crystal surfaces).

Question: Do matter waves exhibit superposition and diffraction?
Superposition: Waves pass through each other without changing properties.
- Linear differential equations allow wave solutions to be added.
If two spring waves moving in opposite directions meet, they momentarily cancel out before continuing.
Question: Can electrons pass through each other without interacting?
Diffraction: Waves scatter in all directions after passing through an opening.
- Essential for two-slit experiment interference patterns.
Question: Do matter waves diffract?
Question: What happens when a single electron undergoes diffraction?
If electrons diffract, their energy is distributed in multiple directions, but each electron lands in only one place on the screen.
Similar to the problem with the photon theory.

Combines many waves to describe the localization of a particle.
Electron is considered a collection of waves with different wavelengths.
Superposition of many waves at different strengths creates a localized pulse of wave energy.
Adding two disturbances (waves) with different wavelengths results in constructive and destructive interference, creating a localized structure.
Adding many matter waves of different wavelengths can create a localized structure.
Gaussian wave packet: A specific wave packet with a shape described by a Gaussian function.
- $y(x) = y_{\text{max}} e^{-\frac{(x - vt)^2}{2a^2}}$ , where:
  - $y_{\text{max}}$ is the maximum amplitude.
  - $v$ is the velocity.
  - $a$ is the width of the Gaussian.
The velocity $v$ represents the movement of a group of matter waves and is called the group velocity.
The Gaussian wave packet is made up of combining matter waves at different wavelengths, where: $\lambda = \frac{h}{\gamma m v}$

$\Delta \lambda$ : Standard deviation of wavelengths in the wave packet.
$\Delta x$ : Standard deviation of x values.
Combining wave fronts at different strengths results in a localized structure with some spread.
The localization of the particle is related to the distribution of wavelengths that make the wave packet.
$\Delta \lambda \times \Delta x = constant$
Broadening the range of wavelengths narrows the spread in the position, and vice versa.
The Fourier transform is a mathematical tool to easily add the waves.