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3. Wave Functions

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The wave function is the fundamental object in quantum mechanics, encoding all information about a system's state.

Position Representation

In position space, the state vector becomes a function:

$$\psi(\vec{r},t) = \langle \vec{r}|\psi(t)\rangle$$

Probability Interpretation

$$P(\vec{r} \in dV, t) = |\psi(\vec{r},t)|^2 dV$$

Normalization: $\int|\psi|^2d^3r = 1$

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Visualizing Wave Function Time Evolution

Infinite Square Well

Quantum Number (n = 1)

Animation Speed: 1.0x

Show Probability Density |ψ|²

Note: Time evolution shows the oscillating phase of the quantum state.

Wave function ψ(x,t) oscillates with frequency ω = E/ℏ
Probability density |ψ|² remains constant in time (stationary state)
Higher quantum numbers have higher energies and faster oscillations

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Probability Density |ψ(x)|²

1D Probability Distribution

Quantum Number n: 1

Resolution: 200 points

Show Classical Prediction (Uniform)

Interpretation:

Brighter regions = higher probability of finding the particle
Total integrated probability = 1 (normalization)
Quantum predictions differ dramatically from classical (uniform) distribution

Normalizing a Wave Function

BASIC

Problem: A particle in a 1D box has wave function ψ(x) = A sin(πx/L) for 0 < x < L. Find the normalization constant A.

Given:

ψ(x) = A sin(πx/L) for 0 < x < L
ψ(x) = 0 elsewhere
Normalization condition: ∫|ψ|² dx = 1

Find: Normalization constant A

Momentum Representation

Fourier transform relates position and momentum:

$$\tilde{\psi}(\vec{p},t) = \frac{1}{(2\pi\hbar)^{3/2}}\int e^{-i\vec{p}\cdot\vec{r}/\hbar}\psi(\vec{r},t)d^3r$$

Gaussian Wave Packet in Momentum Space

INTERMEDIATE

Problem: A particle has position-space wave function ψ(x) = A exp(-x²/2σ²). Find the momentum-space wave function φ(p).

Given:

ψ(x) = A exp(-x²/2σ²)
Fourier transform: φ(p) = (1/√(2πℏ)) ∫ exp(-ipx/ℏ)ψ(x)dx
Gaussian integral: ∫ exp(-ax² + bx)dx = √(π/a)exp(b²/4a)

Find: φ(p) in momentum space

Schrödinger Equation in Position Space

$$i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi + V(\vec{r})\psi$$

Probability Current & Continuity

Probability density: $\rho = |\psi|^2$

Probability current:

$$\vec{j} = \frac{\hbar}{2mi}(\psi^*\nabla\psi - \psi\nabla\psi^*) = \frac{\hbar}{m}\text{Im}(\psi^*\nabla\psi)$$

Continuity equation:

$$\frac{\partial\rho}{\partial t} + \nabla\cdot\vec{j} = 0$$

Properties of Wave Functions

✓ Normalized: $\int|\psi|^2d^3r = 1$
✓ Square-integrable: $\psi \in L^2(\mathbb{R}^3)$
✓ Continuous (usually)
✓ Single-valued
✓ Global phase arbitrary: $\psi$ and $e^{i\theta}\psi$ physically equivalent

Self-Check Question

What is the physical interpretation of |ψ(x,t)|²?

Self-Check Question

Why must wave functions be square-integrable (∫|ψ|²dx < ∞)?

Self-Check Question

If ψ(x) is narrow in position space, what happens to φ(p) in momentum space?

Self-Check Question

What does the continuity equation ∂ρ/∂t + ∇·j = 0 represent physically?

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Wave Functions in Modern Technology

1. Electron Microscopy

Materials Science

Transmission Electron Microscopes (TEM) use the wave nature of electrons (ψ(r) for electrons) to achieve sub-angstrom resolution. The electron wave function's wavelength λ = h/p is much smaller than visible light, enabling atomic-resolution imaging.

Examples:

STEM - scanning transmission electron microscopy (0.5 Å resolution)
Protein structure determination for drug design
Semiconductor defect analysis (Intel, TSMC chip development)
Graphene and 2D material characterization

Impact: Essential for nanotechnology, materials science, and structural biology

2. Bose-Einstein Condensates

Atomic Physics

Ultracold atoms (<1 μK) occupy the same macroscopic wave function ψ(r,t), creating a 'matter wave' visible to the naked eye. The collective wave function exhibits quantum interference on macroscopic scales.

Examples:

MIT BEC experiments (2001 Nobel Prize - Ketterle, Cornell, Wieman)
Atom interferometry for precision measurements
Quantum simulation of condensed matter systems
Tests of quantum mechanics at macroscopic scales

Impact: New state of matter enabling quantum technologies and fundamental tests

3. Molecular Wavefunctions in Chemistry

Computational Chemistry

Electronic wave functions ψ(r₁,r₂,...,rₙ) determine molecular structure, bonding, and reactivity. Quantum chemistry software solves Schrödinger equations for multi-electron systems to predict chemical properties.

Examples:

Drug discovery - predicting protein-ligand binding (Schrödinger Inc.)
Materials design - catalysts, batteries, solar cells
Reaction pathway calculations - transition state theory
Spectroscopy prediction - IR, Raman, UV-Vis spectra

Impact: Multi-billion dollar computational chemistry industry for pharmaceuticals and materials

💡 Understanding real-world applications helps connect abstract quantum concepts to tangible technology and motivates further study.

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📝 Chapter Summary

Key Equations

Position rep: ψ(r,t) = ⟨r|ψ(t)⟩

Probability: P(r)dV = |ψ(r,t)|²dV

Normalization: ∫|ψ|²d³r = 1

Fourier: φ(p) = (2πℏ)⁻³/²∫e⁻ⁱᵖ·ʳ/ℏψ(r)d³r

Current: j = (ℏ/m)Im(ψ*∇ψ)

Continuity: ∂ρ/∂t + ∇·j = 0

Key Concepts

ψ encodes complete information about quantum state
|ψ|² is probability density (Born interpretation)
ψ must be normalized, square-integrable, continuous
Position ↔ momentum via Fourier transform
Narrow in x → broad in p (uncertainty principle)
Probability is conserved (continuity equation)
Global phase e^(iθ) is unobservable

The wave function is the heart of quantum mechanics. It contains all knowable information about a system, evolves deterministically via Schrödinger's equation, but yields only probabilistic predictions upon measurement. Mastering wave functions is essential for understanding all quantum phenomena.

Practice Problems

Problem 1:Normalize the wave function $\psi(x) = A e^{-\alpha|x|}$ where $\alpha > 0$. Find $A$.

Solution:

Step 1: Impose normalization: $\int_{-\infty}^{\infty}|\psi|^2\,dx = 1$.

Step 2: By symmetry of $|x|$: $|A|^2 \cdot 2\int_0^{\infty} e^{-2\alpha x}\,dx = 1$.

Step 3: Evaluate the integral: $\int_0^{\infty} e^{-2\alpha x}\,dx = \frac{1}{2\alpha}$.

Step 4: Solve: $|A|^2 \cdot \frac{2}{2\alpha} = |A|^2/\alpha = 1$, so $|A|^2 = \alpha$.

Answer: $A = \sqrt{\alpha}$ (choosing the positive real root).

Problem 2:For the infinite square well ($V=0$ for $0 < x < L$, $V=\infty$ elsewhere), find the energy eigenvalues and normalized eigenfunctions.

Solution:

Step 1: Inside the well, the Schrodinger equation gives $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$, so $\psi(x) = A\sin(kx) + B\cos(kx)$ with $k = \sqrt{2mE}/\hbar$.

Step 2: Boundary condition $\psi(0) = 0$ requires $B = 0$. Boundary condition $\psi(L) = 0$ requires $\sin(kL) = 0$, so $kL = n\pi$ with $n = 1, 2, 3, \ldots$

Step 3: Energy eigenvalues: $E_n = \frac{\hbar^2 k^2}{2m} = \frac{n^2\pi^2\hbar^2}{2mL^2}$.

Step 4: Normalize: $\int_0^L |A|^2 \sin^2(n\pi x/L)\,dx = |A|^2 \cdot L/2 = 1$, giving $A = \sqrt{2/L}$.

Answer: $\psi_n(x) = \sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right)$, $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$ for $n = 1, 2, 3, \ldots$

Problem 3:Compute $\langle x \rangle$ and $\langle x^2 \rangle$ for the ground state of the harmonic oscillator, $\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-m\omega x^2/2\hbar}$.

Solution:

Step 1: $\langle x \rangle = \int_{-\infty}^{\infty} x|\psi_0|^2\,dx = 0$ by symmetry (the integrand is odd).

Step 2: For $\langle x^2 \rangle$, use the Gaussian integral $\int_{-\infty}^{\infty} x^2 e^{-\beta x^2}dx = \frac{\sqrt{\pi}}{2\beta^{3/2}}$ with $\beta = m\omega/\hbar$.

Step 3: $\langle x^2 \rangle = \sqrt{\frac{m\omega}{\pi\hbar}} \cdot \frac{\sqrt{\pi}}{2(m\omega/\hbar)^{3/2}} = \frac{\hbar}{2m\omega}$.

Answer: $\langle x \rangle = 0$ and $\langle x^2 \rangle = \frac{\hbar}{2m\omega}$, giving $\Delta x = \sqrt{\hbar/(2m\omega)}$.

Problem 4:Show that the probability current for a plane wave $\psi = Ae^{i(kx - \omega t)}$ equals $j = |A|^2 \hbar k / m$.

Solution:

Step 1: The probability current is $j = \frac{\hbar}{2mi}\left(\psi^*\frac{\partial\psi}{\partial x} - \psi\frac{\partial\psi^*}{\partial x}\right)$.

Step 2: Compute derivatives: $\frac{\partial\psi}{\partial x} = ik\psi$ and $\frac{\partial\psi^*}{\partial x} = -ik\psi^*$.

Step 3: Substitute: $j = \frac{\hbar}{2mi}\left(\psi^* \cdot ik\psi - \psi \cdot (-ik)\psi^*\right) = \frac{\hbar}{2mi}(2ik|\psi|^2)$.

Step 4: Simplify: $j = \frac{\hbar}{2mi} \cdot 2ik \cdot |A|^2 = \frac{\hbar k}{m}|A|^2$.

Answer: $j = |A|^2\hbar k/m = |A|^2 v$, where $v = p/m = \hbar k/m$ is the classical velocity.

Problem 5:A Gaussian wave packet has $\psi(x, 0) = \left(\frac{1}{2\pi\sigma_0^2}\right)^{1/4} e^{-x^2/(4\sigma_0^2)}e^{ik_0 x}$. Find $\Delta x$ and $\Delta p$ at $t = 0$ and verify the uncertainty principle.

Solution:

Step 1: The probability density is $|\psi|^2 \propto e^{-x^2/(2\sigma_0^2)}$. This is a Gaussian with standard deviation $\sigma_0$, so $\Delta x = \sigma_0$.

Step 2: The momentum-space wave function is $\phi(k) \propto e^{-(k - k_0)^2 \sigma_0^2}$ (Fourier transform of a Gaussian is a Gaussian).

Step 3: The momentum probability density $|\phi(k)|^2 \propto e^{-2(k-k_0)^2\sigma_0^2}$ has width $\Delta k = 1/(2\sigma_0)$, giving $\Delta p = \hbar\Delta k = \hbar/(2\sigma_0)$.

Step 4: Check: $\Delta x \cdot \Delta p = \sigma_0 \cdot \frac{\hbar}{2\sigma_0} = \frac{\hbar}{2}$.

Answer: $\Delta x\,\Delta p = \hbar/2$, exactly saturating the Heisenberg bound. The Gaussian wave packet is a minimum-uncertainty state at $t = 0$.

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Runnable Simulations

Gaussian Wave Packet: Normalization and Expectation Values

Python

script.py49 lines

import numpy as np

# Gaussian wave packet: psi(x) = (2*pi*sigma^2)^(-1/4) * exp(-(x-x0)^2/(4*sigma^2)) * exp(i*k0*x)
N = 1000
L = 20.0
x = np.linspace(-L/2, L/2, N)
dx = x[1] - x[0]

sigmas = [0.5, 1.0, 2.0, 4.0]
x0 = 0.0
k0 = 5.0

print("Gaussian Wave Packet Properties")
print("psi(x) = (2*pi*sigma^2)^(-1/4) * exp(-(x-x0)^2/(4*sigma^2)) * exp(i*k0*x)")
print("=" * 60)

for sigma in sigmas:
    psi = (2*np.pi*sigma**2)**(-0.25) * np.exp(-(x-x0)**2/(4*sigma**2)) * np.exp(1j*k0*x)

# Normalization check
    norm = np.trapezoid(np.abs(psi)**2, x)

# Position expectation value
    x_avg = np.trapezoid(x * np.abs(psi)**2, x)

# Position uncertainty
    x2_avg = np.trapezoid(x**2 * np.abs(psi)**2, x)
    delta_x = np.sqrt(x2_avg - x_avg**2)

# Momentum expectation value (via -i*hbar*d/dx)
    dpsi = np.gradient(psi, dx)
    p_avg = np.trapezoid(np.conj(psi) * (-1j) * dpsi, x).real

# Momentum uncertainty
    d2psi = np.gradient(dpsi, dx)
    p2_avg = np.trapezoid(np.conj(psi) * (-1) * d2psi, x).real
    delta_p = np.sqrt(abs(p2_avg - p_avg**2))

# Uncertainty product
    product = delta_x * delta_p

print(f"\nsigma = {sigma:.1f}:")
    print(f"  Normalization: {norm:.6f}")
    print(f"  <x> = {x_avg:.4f}, Delta_x = {delta_x:.4f} (theory: {sigma:.4f})")
    print(f"  <p> = {p_avg:.4f} (theory: {k0:.4f})")
    print(f"  Delta_p = {delta_p:.4f} (theory: {1/(2*sigma):.4f})")
    print(f"  Delta_x * Delta_p = {product:.4f} (min = 0.5000)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Probability Current and Conservation

Fortran

program.f9063 lines

program probability_current
  implicit none
  integer, parameter :: N = 200
  double precision, parameter :: pi = 3.14159265358979d0
  double precision :: L, dx, x, sigma, k0, x0
  double precision :: psi_r, psi_i, psi_r_next, psi_i_next
  double precision :: rho, j_curr, total_prob
  double precision :: dpsi_r, dpsi_i
  integer :: i

L = 20.0d0
  dx = L / dble(N)
  sigma = 1.0d0
  k0 = 3.0d0
  x0 = 0.0d0

print *, "Probability Current for Gaussian Wave Packet"
  print *, "j(x) = (hbar/m) * Im(psi* * dpsi/dx)"
  print *, "Continuity: d(rho)/dt + dj/dx = 0"
  print *, "============================================="
  print *, ""

print '(A10, A14, A14, A14)', "x", "rho(x)", "j(x)", "j/rho"
  print *, "----------------------------------------------------"

total_prob = 0.0d0

do i = 0, N
    x = -L/2.0d0 + dble(i) * dx

! psi = norm * exp(-(x-x0)^2/(4*sigma^2)) * exp(i*k0*x)
    psi_r = (2.0d0*pi*sigma**2)**(-0.25d0) * &
            exp(-(x-x0)**2 / (4.0d0*sigma**2)) * cos(k0*x)
    psi_i = (2.0d0*pi*sigma**2)**(-0.25d0) * &
            exp(-(x-x0)**2 / (4.0d0*sigma**2)) * sin(k0*x)

rho = psi_r**2 + psi_i**2

! dpsi/dx (analytical)
    dpsi_r = psi_r * (-(x-x0)/(2.0d0*sigma**2)) - psi_i * k0
    dpsi_i = psi_i * (-(x-x0)/(2.0d0*sigma**2)) + psi_r * k0

! j = Im(psi* dpsi/dx) = psi_r * dpsi_i - psi_i * dpsi_r
    j_curr = psi_r * dpsi_i - psi_i * dpsi_r

total_prob = total_prob + rho * dx

if (mod(i, 20) == 0) then
      if (rho > 1.0d-10) then
        print '(F10.3, ES14.4, ES14.4, F14.4)', x, rho, j_curr, j_curr/rho
      else
        print '(F10.3, ES14.4, ES14.4, A14)', x, rho, j_curr, "---"
      end if
    end if
  end do

print *, ""
  print '(A, F10.6)', " Total probability: ", total_prob
  print '(A, F10.4)', " Expected velocity j/rho = k0 = ", k0

end program probability_current

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

3. Wave Functions

Your Progress

Position Representation

Probability Interpretation

Visualizing Wave Function Time Evolution

Probability Density |ψ(x)|²

Normalizing a Wave Function

Momentum Representation

Gaussian Wave Packet in Momentum Space

Schrödinger Equation in Position Space

Probability Current & Continuity

Properties of Wave Functions

Self-Check Question

Self-Check Question

Self-Check Question

Self-Check Question

Wave Functions in Modern Technology

1. Electron Microscopy

2. Bose-Einstein Condensates

3. Molecular Wavefunctions in Chemistry

Related Topics & Learning Path

Postulates of QM

Hilbert Spaces

Measurement Theory

Uncertainty Principle

Infinite Square Well

Free Particle

Density Matrix

📝 Chapter Summary

Key Equations

Key Concepts

Practice Problems

Solution:

Solution:

Solution:

Solution:

Solution:

Runnable Simulations

Gaussian Wave Packet: Normalization and Expectation Values

Probability Current and Conservation