Wave Mechanics

Matter waves

Particles exhibit wave-like behavior (e.g., diffraction). The momentum/wave vector and energy/frequency relations (de Broglie) are:

\[ \vec{p} = \hbar \vec{k} \quad \text{and} \quad E = \hbar \omega \]

Time evolution

The particle wave function \(\psi\) follows the time-dependent Schrödinger Equation (SE):

\[ i \hbar \frac{\partial}{\partial t} \psi(\vec{r},t) = \hat{H} \psi(\vec{r},t), \quad \hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\vec{r},t) \]

Time-independent SE

For a potential without explicit time dependence \(V(\vec{r},t)=V(\vec{r})\) and a well-defined energy \(E = \hbar \omega\), the wave function can be written as \(\psi(\vec{r},t) = \phi(\vec{r}) e^{i\omega t}\) with \(\phi(\vec{r})\) following the time-independent SE:

\[ \hat{H}\phi(\vec{r}) = E \phi(\vec{r}) \]

Probability density (Born rule)

The wave function is related to the particle’s probability density \(P(\vec{r},t)\) (i.e., the probability of finding the particle at position \(\vec{r}\)) following the Born rule:

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

(Note: This requires the wave function to be normalized, i.e., \(\int |\psi(\vec{r},t)|^2 d^3r = 1\). However, not all wave functions can be normalized, e.g., ‘plane waves’.)

Expectation value of position

This is the average position of the particle when many identical measurements are performed, and it is given by:

\[ \langle \vec{r} \rangle = \int \psi^*(\vec{r},t) \, \vec{r} \, \psi(\vec{r},t) \, d^3r \]

Variance of position

The variance of the position \(\vec{r}\) quantifies the spread of the probability distribution, i.e., it expresses the uncertainty in the particle’s position:

\[ \mathrm{Var}(\vec{r}) = \langle \vec{r}^2 \rangle - \langle \vec{r} \rangle^2, \quad \text{with} \quad \langle \vec{r}^2 \rangle = \int \psi^*(\vec{r},t) \, \vec{r}^2 \, \psi(\vec{r},t) \, d^3r \]

Wave function in momentum space

A wave function in coordinate space \(\psi(\vec{r})\) has a corresponding wave function in momentum space \(\Psi(\vec{k})\), and they are related by a Fourier transform (for one particle in three-dimensional space):

\[ \Psi(\vec{k}) = \frac{1}{(2\pi)^{3/2}} \int d^3r \, \psi(\vec{r}) e^{-i\vec{k}\cdot \vec{r}}, \quad \psi(\vec{r}) = \frac{1}{(2\pi)^{3/2}} \int d^3k \, \Psi(\vec{k}) e^{i\vec{k}\cdot \vec{r}} \]

---

EXAMPLE 1: Measurement outcomes and probability distributions

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm

x = np.linspace(-6,6,400)
large_var = norm.pdf(x, 0, 2)
small_var = norm.pdf(x, 0, 0.6)
np.random.seed(0)
measurements_wide = np.random.normal(0, 2, 10)
measurements_narrow = np.random.normal(0, 0.6, 10)
plt.figure(figsize=(10,5))
plt.fill_between(x, 0, 2.2, where=(x>=-2) & (x<=2), color='red', alpha=0.1)
plt.fill_between(x, 0, 2.2, where=(x>=-0.6) & (x<=0.6), color='blue', alpha=0.1)
plt.plot(x, large_var*2.5, 'r--', label='Large variance distribution')
plt.plot(x, small_var*2.5, 'b-', label='Small variance distribution')
plt.scatter(measurements_wide, [0.3]*10, color='red', marker='^', s=50, alpha=0.7, label='Measurements (wide)')
plt.scatter(measurements_narrow, [0.6]*10, color='blue', s=20, alpha=0.7, label='Measurements (narrow)')
plt.axvline(0, color='black', linestyle='--', linewidth=2, label='Expectation value <x>')
plt.xlabel('Measurement outcome x')
plt.ylabel('Probability density / outcomes')
plt.legend(loc='upper left')
plt.grid(False)

Figure 1: Measurement outcomes and probability distributions for small (blue) and large (red) variance. Shaded bands show ±σ around the expectation value .

EXAMPLE 2: Free 1D Electron: Plane Wave Solution

The time-dependent Schrödinger equation for a free particle in 1D:

\[ i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} \]

Plane wave solution:

\[ \psi(x,t) = e^{i(kx - \omega t)}, \quad \omega = \frac{\hbar k^2}{2m} \]

• Solution: plane wave with constant amplitude
• Momentum: well-defined, \(p = \hbar k\)
• Normalizability: Not normalizable, so not physical by itself

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0, 10, 400)
y = np.cos(2*np.pi*0.8*x - 0.5)

plt.figure(figsize=(8,4))
plt.plot(x, y, color='blue', linewidth=2)
plt.xlabel('x')
plt.ylabel('Re[ψ(x,t)]')
plt.ylim(-2,2)
plt.grid(False)

Figure 2: Plane wave solution: real part of the wave function for a free 1D electron

EXAMPLE 3: Gaussian Wavepacket in 1D

Initial wavepacket:

\[ \psi(x,0) = e^{-x^2/(2\sigma^2)} e^{i k_0 x} \]

Time evolution:

\[ \psi(x,t) \sim e^{-(x - \hbar k_0 t / m)^2 / (2 \sigma_t^2)} e^{i(k_0 x - \omega t)}, \quad \sigma_t = \sigma \sqrt{1 + \left(\frac{\hbar t}{m \sigma^2}\right)^2} \]

• Localized wavepacket (superposition of plane waves)
• Moves with group velocity \(v_g = \hbar k_0/m\)
• Spreads in time due to dispersion
• Physical: normalizable and finite probability

x = np.linspace(-8, 8, 400)
psi0 = np.exp(-x**2/2)
psi_t = np.exp(-(x-2)**2/4)
plt.figure(figsize=(8,4))
plt.plot(x, psi0, color='red', linewidth=2, label='t=0')
plt.plot(x, psi_t, color='blue', linewidth=2, label='t>0')
plt.xlabel('x')
plt.ylabel('|ψ(x,t)|²')
plt.legend()
plt.grid(False)

Figure 3: Gaussian wavepacket evolution: initial (red) and later (blue) distributions

Additional notes:

• Analytic solutions exist only for very simple systems. More particles require approximations or numerical methods.
• Electromagnetic interactions are relativistic; a full description uses quantum electrodynamics (QED).
• Some quantum properties (e.g., spin) cannot be described by wave functions alone, but the general Schrödinger equation \(\hat{H}\psi=i\hbar \partial_t \psi\) still governs time evolution.