4.1 Schrödinger Equation and Wave Functions

Schrödinger Equation and Wave Functions

The Schrödinger equation is the central equation of quantum mechanics. It governs how quantum systems evolve and determines the allowed energy states of atoms and molecules. Solving it yields wave functions, which encode everything measurable about a quantum system.

This topic covers the structure of the Schrödinger equation, how to solve it for simple systems, the mathematical requirements wave functions must satisfy, and how to extract physical predictions using the Born interpretation.

Schrödinger Equation and Its Meaning

Physical Interpretation

The Schrödinger equation plays the same role in quantum mechanics that Newton's second law plays in classical mechanics: it's the equation of motion. Given a system (an electron in an atom, a vibrating bond), the Schrödinger equation tells you what quantum states are possible and how those states change with time.

There are two forms to know:

• Time-dependent Schrödinger equation: Describes how a quantum state evolves over time. The full form is:

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

• Time-independent Schrödinger equation: Applies when the potential energy doesn't change with time. It yields the stationary states of the system, which are states with definite energy. This is the form you'll solve most often in this course:

$\hat{H}\Psi(\mathbf{r}) = E\Psi(\mathbf{r})$

The time-independent equation is an eigenvalue equation. Solving it means finding which functions $\Psi$ and which energies $E$ satisfy it simultaneously.

Components of the Schrödinger Equation

The Hamiltonian operator $\hat{H}$ represents the total energy of the system. For a single particle of mass $m$ in a potential $V(x)$, it takes the form:

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

The first term is the kinetic energy operator and the second is the potential energy. The specific form of $V(x)$ defines the system you're studying.

The wave function $\Psi$ is a complex-valued function of position (and, in the time-dependent case, time). It contains all the information about the quantum state. You can't measure $\Psi$ directly, but you can extract every measurable quantity from it.

Eigenvalues and eigenfunctions: When you solve $\hat{H}\Psi_n = E_n\Psi_n$, the allowed energies $E_n$ are the eigenvalues and the corresponding functions $\Psi_n$ are the eigenfunctions (also called stationary states or orbitals, depending on context). The subscript $n$ is a quantum number that labels each solution.

Solving the Time-Independent Schrödinger Equation

Simple Quantum Systems

Particle in a one-dimensional box (infinite square well): A particle of mass $m$ is confined between $x = 0$ and $x = L$, with the potential energy infinite outside the box and zero inside.

The boundary conditions require $\Psi(0) = \Psi(L) = 0$, which restricts the solutions to standing waves:

$\Psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right), \quad n = 1, 2, 3, \ldots$

The corresponding quantized energies are:

$E_n = \frac{n^2 h^2}{8mL^2}$

A few things to notice: energy is quantized (only discrete values are allowed), the lowest energy $E_1$ is not zero (this is the zero-point energy), and the spacing between energy levels increases with $n$.

Quantum harmonic oscillator: Models a particle in a parabolic potential $V(x) = \frac{1}{2}kx^2$, which is a good approximation for molecular vibrations near equilibrium. The energy levels are:

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

where $\omega = \sqrt{k/m}$. Unlike the particle in a box, the energy levels here are evenly spaced by $\hbar\omega$. The wave functions involve Hermite polynomials multiplied by a Gaussian envelope, and they extend into the classically forbidden region (quantum tunneling).

Hydrogen Atom

The hydrogen atom consists of one proton and one electron interacting through the Coulomb potential $V(r) = -\frac{e^2}{4\pi\epsilon_0 r}$. Because this potential has spherical symmetry, you solve the Schrödinger equation in spherical coordinates $(r, \theta, \phi)$, and the wave function separates into radial and angular parts:

$\Psi_{nlm}(r, \theta, \phi) = R_{nl}(r) \cdot Y_l^m(\theta, \phi)$

• The radial functions $R_{nl}(r)$ involve associated Laguerre polynomials and depend on quantum numbers $n$ and $l$.
• The angular functions $Y_l^m(\theta, \phi)$ are spherical harmonics and depend on quantum numbers $l$ and $m_l$.

Three quantum numbers emerge from the solution:

• $n = 1, 2, 3, \ldots$ (principal quantum number, determines energy)
• $l = 0, 1, \ldots, n-1$ (angular momentum quantum number, determines orbital shape)
• $m_l = -l, \ldots, +l$ (magnetic quantum number, determines spatial orientation)

The energy levels depend only on $n$:

$E_n = -\frac{13.6 \text{ eV}}{n^2}$

This means states with the same $n$ but different $l$ are degenerate (same energy) in hydrogen. That degeneracy breaks in multi-electron atoms.

Properties of Wave Functions

Physically Acceptable Solutions

Not every mathematical solution to the Schrödinger equation is physically meaningful. A valid wave function must be:

• Continuous: No sudden jumps in value.
• Single-valued: Only one value of $\Psi$ at each point in space.
• Square-integrable: The integral $\int_{-\infty}^{\infty}|\Psi|^2\,dx$ must be finite, so you can normalize it. This rules out functions that blow up at infinity.

These constraints are what force energy quantization. Only certain values of $E$ produce solutions that satisfy all three conditions.

Normalization

The wave function must be scaled so that the total probability of finding the particle somewhere in all of space equals 1:

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

In three dimensions, this becomes an integral over all space. If you solve the Schrödinger equation and get an unnormalized solution $\Psi$, you find the normalization constant $N$ by requiring:

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

Orthogonality

Wave functions corresponding to different eigenvalues of the same Hermitian operator are orthogonal. For two stationary states $\Psi_m$ and $\Psi_n$, this means their overlap integral vanishes:

$\int_{-\infty}^{\infty}\Psi_m^*(x)\,\Psi_n(x)\,dx = 0 \quad \text{when } m \neq n$

Combined with normalization, this is written compactly using the Kronecker delta:

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

where $\delta_{mn} = 1$ if $m = n$ and $\delta_{mn} = 0$ if $m \neq n$. A set of functions satisfying this condition is called orthonormal. Orthonormality is essential when expanding an arbitrary state as a linear combination of eigenfunctions.

Born Interpretation for Probabilities

Probability Density

The wave function itself is not directly observable. The Born interpretation connects $\Psi$ to measurement: the squared modulus $|\Psi(x,t)|^2$ is the probability density. It tells you the probability per unit length (in 1D) of finding the particle at position $x$ at time $t$.

Because $\Psi$ is generally complex-valued, you compute the squared modulus as:

$|\Psi|^2 = \Psi^*\Psi$

where $\Psi^*$ is the complex conjugate of $\Psi$.

Calculating Probabilities

To find the probability of locating a particle between positions $a$ and $b$:

$P(a \leq x \leq b) = \int_a^b |\Psi(x,t)|^2\,dx$

For a normalized wave function, integrating over all space gives 1, consistent with certainty that the particle exists somewhere.

Expectation values give the average result you'd obtain from many measurements of an observable on identically prepared systems. For an observable with operator $\hat{O}$:

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

Common examples:

• Position: $\langle x \rangle = \int \Psi^* x\, \Psi\, dx$
• Momentum: $\langle p \rangle = \int \Psi^* \left(-i\hbar\frac{d}{dx}\right)\Psi\, dx$
• Energy: $\langle E \rangle = \int \Psi^* \hat{H}\, \Psi\, dx$

The expectation value is not necessarily a value the system can actually return upon measurement. It's the statistical average over many identical experiments.