7.6 Schrödinger Equation and Wave Functions

Schrödinger Equation

The Schrödinger equation is the central equation of quantum mechanics. It governs how quantum systems evolve and determines the allowed energies and states of particles. If classical mechanics has Newton's second law, quantum mechanics has the Schrödinger equation.

The wave function $\Psi(x,t)$ is the solution to this equation, and it encodes everything you can know about a quantum system. You can't measure the wave function directly, but its square gives you the probability of finding a particle at a given location.

Time-Dependent and Time-Independent Forms

There are two forms of the Schrödinger equation, and each serves a different purpose.

The time-dependent Schrödinger equation describes how a quantum state changes over time. It's the more general form and applies to any quantum system.

The time-independent Schrödinger equation is a simplified version used when the potential energy doesn't change with time. It gives you the stationary states of a system, which are states with definite energy. You get this form by separating the time and position parts of the full equation.

Both equations revolve around the Hamiltonian operator $\hat{H}$, which represents the total energy (kinetic + potential) of the system. Solving the time-independent equation means finding the energy eigenvalues $E$ (the allowed energies) and the corresponding eigenfunctions $\psi(x)$ (the spatial wave functions for those energies).

Both forms contain the reduced Planck's constant $\hbar$, which sets the scale at which quantum effects become significant.

Mathematical Formulation

Time-dependent form:

$i\hbar\frac{\partial}{\partial t}\Psi(x,t) = \hat{H}\Psi(x,t)$

• $i$ is the imaginary unit
• $\hbar$ is the reduced Planck's constant
• $\Psi(x,t)$ is the full wave function (depends on both position and time)
• $\hat{H}$ is the Hamiltonian operator

Time-independent form:

$\hat{H}\psi(x) = E\psi(x)$

• $E$ is the energy eigenvalue (a number, not an operator)
• $\psi(x)$ is the spatial part of the wave function

The Hamiltonian for a single particle in one dimension typically looks like:

$\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)$

The first term is the kinetic energy operator, and $V(x)$ is the potential energy. The mass of the particle is $m$.

Solutions to the Schrödinger equation yield wave functions and energy levels. For bound states (like a particle trapped in a box), the energy spectrum is discrete, meaning only certain energies are allowed. For unbound states (like a free particle), the energy spectrum is continuous.

Wave Function Interpretation

Physical Meaning and Probability Density

The wave function $\Psi(x,t)$ is a complex-valued function that contains all information about a quantum system's state. You can't observe it directly, but you can extract measurable predictions from it.

The Born interpretation provides the physical link: the quantity $|\Psi(x,t)|^2$ is the probability density for finding the particle at position $x$ at time $t$. So if you want the probability of finding a particle in some small region, you integrate $|\Psi|^2$ over that region.

For this probability interpretation to make sense, the wave function must be normalized. The total probability of finding the particle somewhere must equal one:

$\int_{-\infty}^{\infty} |\Psi(x,t)|^2 \, dx = 1$

If your wave function doesn't satisfy this, you multiply it by a constant to make it work.

Measurement and collapse: Before measurement, a particle can exist in a superposition of states. When you actually measure a property (like position), the wave function "collapses" to a definite value. This is one of the most debated aspects of quantum mechanics, with different interpretations (Copenhagen, Many-Worlds) offering different explanations for what collapse means physically.

Wave functions also give rise to distinctly quantum phenomena:

• Interference: Wave functions can add constructively or destructively, as seen in the double-slit experiment
• Superposition: A quantum system can exist in a combination of multiple states simultaneously
• Tunneling: A particle can pass through a potential barrier it classically shouldn't have enough energy to cross (this is how alpha decay works and how scanning tunneling microscopes operate)

Wave Function Properties and Implications

Because the wave function is complex-valued, it carries phase information. This phase is what makes quantum interference possible: two wave functions with the same amplitude but different phases can cancel each other out.

De Broglie wavelength connects a particle's momentum to its wavelike behavior:

$\lambda = \frac{h}{p}$

This explains why electrons produce diffraction patterns. Particles with larger momentum have shorter wavelengths, which is why quantum wave behavior is negligible for everyday objects.

Heisenberg uncertainty principle follows directly from the wave nature of quantum states. You cannot simultaneously know a particle's exact position and exact momentum:

$\Delta x \, \Delta p \geq \frac{\hbar}{2}$

This isn't a limitation of measurement equipment. It's a fundamental property of nature built into the mathematics of wave functions. A wave function that's sharply peaked in position (small $\Delta x$) must be spread out in momentum (large $\Delta p$), and vice versa.

Quantum entanglement occurs when two particles share a wave function that can't be written as a product of individual wave functions. Measuring one particle instantly constrains what you'll find when you measure the other, regardless of distance. This is the basis of the Einstein-Podolsky-Rosen (EPR) paradox.

Quantum superposition means a system can be in a linear combination of states. This principle underlies quantum computing, where qubits exploit superposition to exist in combinations of 0 and 1 simultaneously.

Solving the Schrödinger Equation

Particle in a Box Model

The particle in a box (also called the infinite square well) is the simplest system that demonstrates energy quantization. A particle of mass $m$ is confined to a region of length $L$ with impenetrable walls.

Setup:

• Inside the box ($0 < x < L$): $V(x) = 0$
• Outside the box: $V(x) = \infty$

Because the potential is infinite outside, the particle has zero probability of being found there, so $\psi(x) = 0$ outside the box.

Solving step by step:

1. Write the time-independent Schrödinger equation inside the box (where $V = 0$): $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi$

2. Apply boundary conditions: the wave function must vanish at both walls, so $\psi(0) = 0$ and $\psi(L) = 0$.

3. The general solution is a sine function (cosine won't satisfy $\psi(0) = 0$). Requiring $\psi(L) = 0$ forces the argument of the sine to be an integer multiple of $\pi$.

4. The allowed energy levels are: $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \quad n = 1, 2, 3, \ldots$

5. The corresponding normalized wave functions are: $\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$

Notice that $n$ starts at 1, not 0. The ground state ($n = 1$) already has nonzero energy, and the energy grows as $n^2$. Each wave function represents a standing wave with $n$ half-wavelengths fitting inside the box.

Applications: This model approximates electrons in a conducting wire, conjugated molecules, and particles in semiconductor quantum wells.

Quantum Harmonic Oscillator

The quantum harmonic oscillator describes a particle in a parabolic potential well. It's one of the most important models in physics because many systems behave approximately like harmonic oscillators near equilibrium: molecular vibrations, phonons in solids, and even quantized electromagnetic fields.

Setup:

• Potential energy: $V(x) = \frac{1}{2}kx^2$, where $k$ is the spring constant
• Angular frequency: $\omega = \sqrt{k/m}$

The Schrödinger equation becomes:

$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}kx^2\psi = E\psi$

Key results:

The allowed energy levels are evenly spaced:

$E_n = \left(n + \frac{1}{2}\right)\hbar\omega, \quad n = 0, 1, 2, \ldots$

Two things to notice here. First, $n$ starts at 0 (unlike the particle in a box). Second, the ground state energy is $E_0 = \frac{1}{2}\hbar\omega$, not zero. This zero-point energy means a quantum oscillator is never completely at rest.

The wave functions involve Hermite polynomials $H_n$ multiplied by a Gaussian envelope:

$\psi_n(x) = \frac{1}{\sqrt{2^n n!}}\left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-\frac{m\omega x^2}{2\hbar}} H_n\left(\sqrt{\frac{m\omega}{\hbar}}\,x\right)$

You probably won't need to memorize this full expression, but you should recognize that the Gaussian factor $e^{-m\omega x^2/2\hbar}$ ensures the wave function decays at large $|x|$, and the Hermite polynomial determines the number of nodes.

Applications: Vibrational spectra of diatomic molecules, phonons in solid-state physics, and the quantization of the electromagnetic field (photons).

Wave Function Properties

Normalization and Orthogonality

Normalization is the requirement that the total probability of finding the particle somewhere equals one:

$\int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = 1$

If you solve the Schrödinger equation and get a wave function that doesn't satisfy this, you multiply by a normalization constant to fix it.

Orthogonality means that different energy eigenfunctions are "perpendicular" in a mathematical sense. The overlap integral of two different eigenfunctions equals zero:

$\int_{-\infty}^{\infty} \psi_m^*(x)\,\psi_n(x) \, dx = \delta_{mn}$

Here $\delta_{mn}$ is the Kronecker delta: it equals 1 when $m = n$ and 0 when $m \neq n$. This combined condition (normalized and orthogonal) is called orthonormality.

Why does this matter? Orthogonality lets you expand any arbitrary quantum state as a sum of energy eigenfunctions, similar to how you can decompose any vector into components along perpendicular axes. The inner product (also written in Dirac notation as $\langle\psi_m|\psi_n\rangle$) is the tool for computing these overlaps and for calculating transition probabilities between states.

Expectation Values and Uncertainty

Expectation values give you the average result you'd get if you measured a quantity many times on identically prepared systems. For an observable with operator $\hat{A}$:

$\langle A \rangle = \int_{-\infty}^{\infty} \psi^*(x)\,\hat{A}\,\psi(x) \, dx$

For example, the expectation value of position uses $\hat{A} = x$, and the expectation value of momentum uses $\hat{A} = -i\hbar\frac{d}{dx}$.

The uncertainty principle connects the spread in measurements of complementary variables. For position and momentum:

$\Delta x \, \Delta p \geq \frac{\hbar}{2}$

Here $\Delta x$ and $\Delta p$ are standard deviations, not small changes. This inequality is a theorem that follows from the math of wave functions and operators, not just a philosophical statement.

Parity describes how a wave function behaves when you flip the sign of $x$:

• Even parity: $\psi(-x) = \psi(x)$ (symmetric about the origin)
• Odd parity: $\psi(-x) = -\psi(x)$ (antisymmetric about the origin)

Parity is useful because if the potential $V(x)$ is symmetric, every energy eigenfunction must have definite parity (either even or odd). This can simplify solving problems significantly.

Completeness means that the set of all energy eigenfunctions forms a complete basis. Any well-behaved wave function can be written as a sum (or integral) over these eigenfunctions:

$\sum_n |\psi_n\rangle\langle\psi_n| = \mathbf{1}$

This is the quantum analog of Fourier decomposition and is essential for solving more complex quantum systems by expanding unknown states in terms of known solutions.