3. Wavefunctions¶

In this note describes the double-slit experiment, which directs us to the wavefunction. We will analyze consequences of the wavefunction framework.

Double-Slit Experiment¶

The physical world can be more accurately described using a wave description. Waves can interfere and be superposed. In Young's double-slit experiment, we can determine one plane wave can be represented by

\[ \vec{E} = \vec{E}_0e^{i(\vec{k}\cdot\vec{r} - \omega t + \phi)} \]

and we can add two waves together by addition.

\[ \vec{E}_{tot} = \vec{E}_{o, 1}e^{i\delta_1} \vec{E}_{o, 2}e^{i\delta_2} \]

We can also detect power or intensity by squaring the plane wave superposition. For two plane waves, the intensity is:

\[ \left|E\right|^2 = E_{o1}^2 + E_{o2}^2 + \vec{E}_{o1} \cdot \vec{E}_{o2}\left(e^{i(\delta_2 - \delta_1)} + e^{-i(\delta_2 - \delta_1)}\right) \]

Note that we must add amplitudes not intensities. Also, notee observable pattern only depends on phase difference and not absolute phase.

Quantum Wavefunction¶

\[ \Psi \]

Represents a wave amplitude
Not physical! Cannot picture it as something wiggling
Cannot observe it
Contains all the informatio nabout the system

How to use it?

\(|\Psi|^2\) is the probability denbsity to find the system at \(\vec{r}, t\).
Think of a large number of identical systems each with one particle
Repeated measurements generates probability

\[ P(\vec{r}, t) = |\Psi(\vec{r}, t)|^2 \]

To find the particle within a chunk \(d\vec{r}\), about \(\vec{r}\) at time \(t\) is

\[ P(\vec{r}, t)d\vec{r} = |\Psi(\vec{r}, t)|^2d\vec{r} \]

Wave Superposition¶

If \(\Psi_1\) and \(\Psi_2\) are two allowed states, then any linear combination is also allowed.

\[ \Psi = c_1\Psi_1 + c_2\Psi_2 \]

The scalars can be complex numbers.

Wavefunction for a Particle with Definite Momentum¶

We need to encode this information in waves. Just like a particle can be encoded with \(m, \vec{r}, \vec{v}, E, \vec{p}\), a wave should be characterized by some quantity.

What we want to encode is a wave that extends outwards in out directions. It will be a function of position and time. We have two useful quantities

\[ \begin{align*} E = hf = \frac{h}{2\pi}\omega = \hbar\omega\\ p = \frac{h}{\lambda} = \frac{hk}{2\pi} = \hbar k \end{align*} \]

where \(\hbar = h/2\pi\) and \(k = 2\pi/\lambda\).

Let's look at a 1D particle in \(\hat{x}\).

\[ \begin{align*} \vec{P} & = p_x\hat{x}\\ \Psi &= A\exp\left[i(kx - \omega k t)\right]\\ & = A\exp\left[\frac{i(p_xx - E(p_x)t)}{\hbar}\right] \end{align*} \]

I wish to take out the \(p_x\) from this expression. Notice what happens if I take a position derivative of \(\Psi\).

\[ \frac{\partial\Psi}{\partial x} = \frac{Ap_xi}{\hbar}\exp\left[\frac{i(p_xx - E(p_x)t)}{\hbar}\right] \]

Now rearranging the terms leads us to the following result:

\[ -i\hbar \frac{\partial}{\partial x}\Psi = p_x\Psi \]

which is known as th momentum operator. If, I wanted to extract An \(E\), I would want to take a time derivative and will get to the following result:

\[ i\hbar \frac{\partial}{\partial t} \Psi = E\Psi \]

In 3D, we would get something very similar with the momentum and time operators. We find the wave function to be

\[ \begin{align*} \Psi(\vec{r}, t) &= A\exp(\vec{k}\cdot\vec{r} - \omega kt)\\ & = A\exp\left(\frac{i(\vec{p}\cdot\vec{r} - E(\vec{p})t)}{\hbar}\right) \end{align*} \]

And then the time and position derivatives would lead to

\[ i\hbar \frac{\partial}{\partial t} \Psi = E\Psi \]

\[ -i\hbar \nabla\Psi = \vec{p}\Psi \]

Probability and Normalization¶

Another thing we need to note is that the wave function represents a density Recall, Max Born found that the probability density to find the system at \(\vec{r}, t\) is

\[ |\Psi(\vec{r}, t)|^2 \]

Since this is a density, we should normalize it:

\[ \int_{\text{all space}} |\Psi(\vec{r}, t)|^2d\vec{r} = 1 \]

However, for a plane wave, the integral would diverge. This is because a plane wave has amplitude everywhere in space. This tells us that a plane wave in not really a good thing to analyze in quantum mechanics. In other words, there must be something else that better accurately describes physics at this level. This is where the wavepacket comes in

3.1 wavepacket.png

Which will be discussed in [[5. Wavepackets]]. However, in order to understand the physics with wavepackets, some math will need to be covered first in 4. Fourier Stuff