Lecture 9: Quantum Mechanics: Wave-Particle Duality and Uncertainty

Wave-Like Behavior of Particles

All particles exhibit wave-like behavior.
Wave amplitude is related to the probability of finding the particle.
Wavelength is inversely proportional to the particle's momentum.
The wave properties of matter are not apparent in everyday life because the wavelength is scaled by Planck's constant, which is very small.

The Electron Beam Experiment

To illustrate the wave nature of matter, experiments often use electrons because their wave properties are observable at small scales.
An electron beam is directed at a screen (like in an old TV tube).
If an object with two tiny holes is placed in the path of the beam, one might expect to see two spots on the screen. However, an interference pattern is observed instead.

Interference Pattern Explained

The initial hypothesis might be that the electrons are interfering with each other due to their electric fields.
However, the experiment is modified to send electrons through one at a time.

Single Electron Interference

Even when electrons are sent through the two holes one at a time, the interference pattern still emerges after a sufficient number of electrons have passed through.
This indicates that each electron is interfering with itself.
This experiment has been conducted and verified, demonstrating the strange quantum behavior.

The Effect of Blocking One Hole

If one of the holes is blocked, the interference pattern disappears.
This suggests that the electron somehow knows whether there is more than one option available.
The electron explores all available options simultaneously, even if it doesn't physically go through all of them.
This principle is being explored in the development of quantum computers, which aim to perform numerous calculations simultaneously to solve complex problems.

Wave-Particle Duality

Electrons, which are considered particles, also possess wave properties.
Our classical understanding of particles and waves as distinct entities is insufficient.
Particles come in clumps but also exhibit wave behavior.

The Wave Function

The presence or existence of a particle is what's waving.
Erwin Schrodinger described this using a wave function ($\psi$).
The wave function contains all information about the particle, including its position and momentum.
The square of the wave function ( $\psi^2$ ) gives the probability of finding the particle at a specific point in space.
If you sum (integrate) $\psi^2$ over all space, it equals 1, because the particle must be somewhere.
$\int_{-\infty}^{\infty} \psi^2 dx = 1$
This process is called normalization.
A normalized wave function means $\psi^2$ is exactly equal to the probability. If its not normalized then $\psi^2$ is proportional to the probability.
$\psi$ can be positive or negative, useful for describing waves.

Wave Function and Probability Example

Given a wave function, the probability of finding a particle at a point is determined by the square of the wave function at that point.
For example, you are given a graph of a wave function $\psi$ over distance x with points x, y, and z. The point y is the local minima, the point z is almost on the x-axis and the point x is in between the two.
$y$ is the most probable location because when you square $\psi$ at point y it produces the highest value, and z is the least probable because the point z when you square $\psi$ at point z it produces the lowest value.

Wavelength and Momentum

Wavelength ($\lambda$) is inversely related to the particle's momentum ($\p$):
$\lambda = \frac{h}{p}$
Where h is Planck's constant.

Wavelength and Energy

Momentum is related to kinetic energy ($\KE$) by:
\KE = \frac{p^2}{2m}
Where m is the mass of the particle, and $v$ is its velocity.
Therefore, kinetic energy can be expressed in terms of wavelength:
\KE = \frac{h^2}{2m\lambda^2}

Hypothetical Scenario

If Planck's constant were much larger, the quantum effects would be apparent in everyday life.
Example: Calculating the size of Planck's constant required for a ping pong ball (mass = 2.7 grams, velocity = 12 m/s) to have a wavelength of 1 cm.
$h = \lambda \cdot p$
$p = mv$
$h = (0.01 \text{ m}) \cdot (0.0027 \text{ kg}) \cdot (12 \text{ m/s}) = 3.2 \times 10^{-4} \text{ Js}$
This value is much larger than the actual Planck's constant ($\approx 6.626 \times 10^{-34} \text{ Js}$), illustrating why quantum effects are only noticeable at small scales.

Implications of Probability

The wave function only provides the probability of finding a particle.
Any experiment will yield a range of results due to this inherent uncertainty.
This is relevant for all behaviors involving electrons.

Heisenberg's Uncertainty Principle

There is a fundamental limit to how precisely both the position and momentum of a particle can be known simultaneously.
The better you know the position, the less you know about the momentum, and vice versa.
$\Delta p \cdot \Delta x \geq \frac{h}{2 \pi}$
Where $\Delta p$ is the uncertainty in momentum and $\Delta x$ is the uncertainty in position.

Einstein's Beliefs

Einstein was skeptical of the probabilistic nature of quantum mechanics, famously saying, "I can't believe the old guy plays dice with the universe."

Michelson Interferometer for Electrons

The Michelson interferometer experiment can be performed with electrons to demonstrate their wave-like behavior.
An electron beam is split, reflected, and recombined, creating an interference pattern.
If the detector sees no electrons, it implies destructive interference.

Path Length Difference and Interference

For destructive interference to occur, the path length difference must be half a wavelength.
Path difference: $2(L2 - L1)$
Destructive interference condition:
$2(L2 - L1) = \frac{\lambda}{2}$
$L2 - L1 = \frac{\lambda}{4} = \frac{h}{4mv}$

Diffraction in Crystals

Crystals can act as partial reflectors for radiation due to their atomic structure.
X-rays are commonly used because they can penetrate materials like aluminum.
When X-rays interact with a crystal: Some X-rays bounce off the surface and some slip through
The rays split and bounce off different planes.

Bragg's Law

William Bragg used X-ray diffraction to determine the spacing of atoms within crystals.
This technique is now used to determine the structures of proteins and other molecules.

Electron Diffraction

Electrons, like X-rays, can be diffracted by crystals, providing further evidence of their wave-like nature.
The experiment with X rays and aluminum foil are compared to electrons going through aluminum foil.
Electrons and X-rays with the same wavelength will produce the same diffraction patterns.
This is because they undergo the same constructive and deconstructive interference process.

Diffraction Patterns and Wavelength

Using protons instead of electrons with same velocity results in a smaller wave length and creates a different interference pattern.
The formula for wavelength is related to $mv$ , where m is mass and v is velocity.
Since the mass of protons is bigger then the mass of electrons the value for wavelength will be smaller.

Particle in a Box

Consider a particle bouncing back and forth in a one-dimensional box (or pipe).
Classically, the probability of finding the particle is the same everywhere in the box.
Quantum mechanically, the particle behaves as a standing wave.

Standing Waves

In a standing wave, there are nodes at either end with an antinode in the middle.
Since the particle is trapped in the box there is a zero likely hood of it breaching the ends.

Wave Function and Probability Distribution

The wave function is built using standard standing wave patterns. With all standing waves there is a node and an antinode with the addition that all patterns are bounded at the wall ends.
The probability distribution is the square of the wave function.
In contrast to the classical expectation the odds of finding the particle are the same through out. The quantum view is that the odds are not the same.
The lowest energy level produces a peak in the middle so you are likely to find it there, and if the energy is higher there are locations where you will and will not find it.

Energy Quantization

The kinetic energy of the particle is quantized.
$E = \frac{p^2}{2m} = \frac{h^2}{2m\lambda^2}$
For the simplest case, half a wave exists in the box:
$E = \frac{h^2}{8mL^2}$
More generally, the energy levels are given by:
$En = n^2 E0$
Where n is an integer (1, 2, 3, …) and $E_0$ is the lowest energy level.
For a tennis ball in a pipe since it has wave properties then it cant receive energy unless it is at a specific value.

Standing Waves on a Circle

Rutherford's model of the atom proposed that electrons orbit the nucleus like planets around the sun.
Due to their wave nature, electrons perceive the entire orbit at once, forming standing waves in a circle.
The ends of the wave must match up smoothly.

Angular Momentum Quantization

If the 2 ends match up smoothly the number of waves you will have is discreet, it can only be whole numbers.
If you take this two dimensional concept you discover a momentum relationship orbiting a point.
$L = r_{\perp} p$
Where L is angular momentum about a point, r is distance, and p is momentum perpendicular.
$P = \frac{h}{\lambda}$
$n\lambda = 2\pi r$
$\lambda = \frac{2\pi r}{n}$
The orbit has to contain a wave equal to discrete number.
There will always be at least 1 discreet number.
Therefore angular momentum comes in discreet numbers.
Angular momentum calculation:
$L = r \cdot \frac{h}{\lambda} = r \cdot \frac{h}{\frac{2\pi r}{n}} = \frac{nh}{2\pi}$
The formula above shows how angular momentum doesn't care about size.
The important take way is that the minimal spin any object can have is $\frac{h}{2\pi}$
Therefore particles that behave like waves dictate the universe.

Solid State Electronics

Electrons wave properties are used to construct electronic devices.
Even when the concept of quantum mechanics wasn't discovered the nineteen twenties still developed radio.