11.6 Schrödinger equation

Foundations of quantum mechanics

Quantum mechanics describes how physics works at the atomic and subatomic scale. It departs from classical physics in a fundamental way: instead of predicting exact outcomes, it deals in probabilities and wave-particle duality. This framework underpins modern technologies like lasers, transistors, and MRI machines.

Wave-particle duality

Matter and energy behave as both waves and particles, depending on how you observe them. The photoelectric effect shows light acting as particles (photons), while electron diffraction shows electrons acting as waves. Young's double-slit experiment, performed with electrons, is one of the clearest demonstrations of this duality.

The de Broglie wavelength connects a particle's momentum to its wave behavior:

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

• $h$ is Planck's constant ($6.626 \times 10^{-34}$ J·s)
• $p$ is the particle's momentum

A heavier or faster particle has a shorter wavelength, which is why wave behavior is negligible for everyday objects but dominant at atomic scales.

Uncertainty principle

Formulated by Werner Heisenberg in 1927, this principle sets a hard limit on how precisely you can simultaneously know certain pairs of quantities (called conjugate variables). For position and momentum:

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

• $\Delta x$ is the uncertainty in position
• $\Delta p$ is the uncertainty in momentum
• $\hbar = \frac{h}{2\pi}$ is the reduced Planck's constant

This isn't about imperfect instruments. It's a built-in feature of nature. The more precisely you pin down a particle's position, the less you can know about its momentum, and vice versa.

Probability in quantum systems

In classical mechanics, if you know the initial conditions perfectly, you can predict exactly what happens next. Quantum mechanics replaces that determinism with probability.

• The wave function describes a particle's quantum state and determines the probability of finding it in various locations.
• Before measurement, a particle can exist in a superposition of multiple states at once.
• Measurement collapses the wave function to a single definite outcome.
• Entanglement links two particles so that measuring one instantly constrains what you'll find when measuring the other, regardless of distance.

Schrödinger equation formulation

Developed by Erwin Schrödinger in 1925, this equation plays the same role in quantum mechanics that Newton's second law plays in classical mechanics: it tells you how a system evolves. Given a wave function at one moment, the Schrödinger equation predicts what it will look like at any future time.

Time-dependent vs time-independent

The time-dependent Schrödinger equation describes how a quantum system changes over time:

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

When the potential energy doesn't change with time, you can separate out the time dependence entirely. That gives you the time-independent Schrödinger equation, which applies to stationary states:

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

Stationary states have a fixed energy $E$, and their probability distributions don't change over time. Most textbook problems (particle in a box, hydrogen atom) start here because the math is more manageable.

Wave function concept

The wave function $\Psi(\mathbf{r},t)$ is a complex-valued function of position and time that encodes everything you can know about a quantum system. You can't measure $\Psi$ directly, but you extract physical predictions from it.

A valid wave function must be normalized, meaning the total probability of finding the particle somewhere in all of space equals 1:

$\int_{-\infty}^{\infty} |\Psi(\mathbf{r},t)|^2 \, d\mathbf{r} = 1$

If this integral doesn't equal 1, you multiply $\Psi$ by a constant to fix it.

Operators in quantum mechanics

In quantum mechanics, every measurable quantity (observable) has a corresponding operator that acts on the wave function. These operators are Hermitian, which guarantees that measurement outcomes are real numbers (not complex).

The three most common operators:

• Position: $\hat{x} = x$
• Momentum: $\hat{p} = -i\hbar\frac{\partial}{\partial x}$
• Hamiltonian (total energy): $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})$

The Hamiltonian has two parts: the first term represents kinetic energy, and $V(\mathbf{r})$ is the potential energy. This is the operator that appears in both forms of the Schrödinger equation.

Solutions to Schrödinger equation

Analytical (exact) solutions exist only for a handful of simple systems. These model problems are important because they build your intuition for how quantum systems behave, and more complex systems can often be understood as variations of them.

Particle in a box

This is the simplest quantum confinement problem. A particle is trapped between two rigid walls separated by distance $L$, with infinite potential energy outside the box (so the particle can never escape).

The allowed energy levels are:

$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$

• $n = 1, 2, 3, \ldots$ is the quantum number ($n = 0$ is not allowed)
• $m$ is the particle's mass
• $L$ is the width of the box

The corresponding wave functions are standing waves:

$\Psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$

Two things to notice: energy is quantized (only specific values are permitted), and the lowest energy ($n = 1$) is not zero. The particle always has some minimum kinetic energy, even in its ground state.

Quantum harmonic oscillator

This models any system where a restoring force is proportional to displacement, like vibrating molecules or phonons in a crystal lattice. The energy levels are equally spaced:

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

• $\omega$ is the angular frequency of the oscillator
• $n = 0, 1, 2, \ldots$

The $\frac{1}{2}\hbar\omega$ term is the zero-point energy: even in the ground state ($n = 0$), the oscillator still has energy. This is a direct consequence of the uncertainty principle, since a particle sitting perfectly still at the bottom of the well would have definite position and zero momentum simultaneously.

The wave functions involve Hermite polynomials, which you typically don't need to memorize for this course, but you should know that they spread out more as $n$ increases.

Hydrogen atom model

The hydrogen atom is the most physically important exactly solvable system. An electron moves in the Coulomb potential of a single proton, and the Schrödinger equation yields:

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

where $n = 1, 2, 3, \ldots$ is the principal quantum number. The negative sign means the electron is bound; $n = 1$ is the most tightly bound (ground state), and as $n \to \infty$, the energy approaches zero (ionization).

Three quantum numbers emerge from the solution:

• $n$ (principal): determines energy and overall size of the orbital
• $l$ (angular momentum): ranges from $0$ to $n-1$, determines orbital shape (s, p, d, f)
• $m_l$ (magnetic): ranges from $-l$ to $+l$, determines orbital orientation

The wave functions are products of radial functions and spherical harmonics, which give rise to the familiar orbital shapes (spherical for s, dumbbell for p, etc.). These solutions explain the discrete spectral lines observed in hydrogen emission and absorption.

Applications of Schrödinger equation

The Schrödinger equation isn't just a theoretical exercise. It provides the foundation for understanding atomic structure, predicting material properties, and designing quantum technologies.

Tunneling effect

Classically, a particle without enough energy to get over a barrier simply bounces back. In quantum mechanics, the wave function doesn't abruptly stop at the barrier; it decays exponentially inside it. If the barrier is thin enough, there's a nonzero probability the particle appears on the other side. This is quantum tunneling.

The tunneling probability depends on:

• The height of the barrier above the particle's energy
• The width of the barrier
• The particle's mass (lighter particles tunnel more easily)

Real-world examples include:

• Alpha decay: alpha particles tunnel out of the nucleus despite the strong nuclear potential barrier
• Scanning tunneling microscopy: uses tunneling current to image surfaces at atomic resolution
• Tunnel diodes and flash memory: electronic devices that exploit tunneling for their operation

Quantum wells and barriers

A quantum well is created by sandwiching a thin layer of one semiconductor between layers of a wider band-gap semiconductor. Electrons (or holes) get confined in the thin layer, producing discrete energy levels similar to the particle-in-a-box model.

Applications include:

• Quantum well lasers used in fiber-optic communications
• High-electron-mobility transistors (HEMTs) for high-frequency electronics

These structures are engineered at the nanometer scale using techniques like molecular beam epitaxy.

Molecular orbitals

When atoms bond to form molecules, their atomic orbitals combine to create molecular orbitals that spread across the entire molecule. The standard approach is the linear combination of atomic orbitals (LCAO) method.

Molecular orbitals explain why some combinations of atoms form stable bonds and others don't. They predict molecular spectra, reactivity, and geometry. Computational chemistry uses these principles extensively for applications like drug design and materials engineering.

Interpretation of wave function

The wave function is the central mathematical object in quantum mechanics, but what does it physically mean? This question connects the abstract formalism to actual measurements.

Born interpretation

Max Born proposed in 1926 that the wave function itself isn't directly observable. Instead, the square of its magnitude gives the probability density:

$P(x) = |\Psi(x)|^2$

This means $|\Psi(x)|^2 \, dx$ is the probability of finding the particle in a small interval $dx$ around position $x$. The Born interpretation reconciles the wave-like mathematics with the fact that detectors always register particles at definite locations. Upon measurement, the wave function "collapses" to the observed value.

Probability density

The probability density $\rho(x) = |\Psi(x)|^2$ tells you where the particle is most likely to be found. For a normalized wave function:

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

You can visualize probability density as the "probability clouds" shown in orbital diagrams. Denser regions of the cloud correspond to higher probability. For the particle in a box, the probability density has nodes (points of zero probability) at specific locations that depend on the quantum number $n$.

Expectation values

The expectation value of an observable is the average result you'd get if you measured that quantity on many identical copies of the system. For an operator $\hat{A}$:

$\langle A \rangle = \int \Psi^* \hat{A} \, \Psi \, dx$

For example, the expectation value of position $\langle x \rangle$ tells you the average position, and $\langle \hat{H} \rangle$ gives the average energy. These expectation values are what you compare to experimental measurements. In the classical limit (large quantum numbers), expectation values reproduce classical results, consistent with the correspondence principle.

Limitations and extensions

The Schrödinger equation works extremely well for non-relativistic quantum systems. But it has known limitations that require more advanced theories.

Relativistic considerations

When particles move at speeds approaching the speed of light, the Schrödinger equation breaks down because it's built on non-relativistic mechanics.

• The Dirac equation extends quantum mechanics to include special relativity for spin-½ particles (like electrons). It naturally predicts electron spin and the existence of antimatter.
• The Klein-Gordon equation handles spinless relativistic particles.
• Quantum electrodynamics (QED) is the full relativistic quantum theory of electromagnetic interactions, and it's one of the most precisely tested theories in all of physics.

Many-body systems

For systems with more than a few interacting particles, exact solutions to the Schrödinger equation are impossible. Approximation methods become essential:

• Hartree-Fock method: treats each electron as moving in the average field of all the others
• Density functional theory (DFT): reformulates the problem in terms of electron density rather than the full wave function, making it computationally tractable for solids and large molecules
• Quantum Monte Carlo: uses statistical sampling to simulate complex quantum systems

Approximation methods

Several general-purpose techniques handle systems that are close to, but not exactly, solvable:

• Perturbation theory: starts with an exactly solvable system and treats the difference as a small correction. Works well when the perturbation is genuinely small.
• Variational method: you guess a trial wave function with adjustable parameters, then minimize the energy. This always gives an upper bound on the true ground state energy.
• WKB approximation: useful for slowly varying potentials, often applied to tunneling problems.
• Tight-binding model: simplifies electronic structure in solids by focusing on nearest-neighbor interactions.

Experimental verification

Quantum mechanics has been tested to extraordinary precision, and the Schrödinger equation's predictions have been confirmed across a wide range of experiments.

Double-slit experiment

Originally performed by Thomas Young with light in 1801, this experiment has since been repeated with electrons, neutrons, atoms, and even large molecules. When particles pass through two slits, an interference pattern builds up on the detector, even when particles are sent through one at a time.

If you try to measure which slit each particle goes through, the interference pattern disappears. This is the complementarity principle: wave behavior and particle behavior are mutually exclusive observations. Quantum eraser experiments have further explored how the availability of "which-path" information affects the interference pattern.

Atomic spectra

The discrete emission and absorption lines of atoms provide direct evidence for quantized energy levels. For hydrogen, the Schrödinger equation predicts the spectral lines with high accuracy.

Finer details in the spectrum reveal additional physics:

• Fine structure comes from spin-orbit coupling (the interaction between the electron's spin and its orbital motion)
• Hyperfine structure arises from the interaction between the electron and the nuclear magnetic moment
• The Zeeman effect shows spectral lines splitting in an external magnetic field

High-precision spectroscopy of hydrogen has been used to test QED predictions to parts per trillion.

Scanning tunneling microscopy

Invented by Gerd Binnig and Heinrich Rohrer in 1981 (earning them the 1986 Nobel Prize), the scanning tunneling microscope (STM) is a direct application of quantum tunneling. A sharp conducting tip is brought within about 1 nm of a surface, and a voltage is applied. Electrons tunnel across the gap, producing a current that depends exponentially on the tip-sample distance.

By scanning the tip across the surface and measuring the tunneling current, you can map out the surface with atomic resolution. STMs can even be used to push individual atoms around on a surface, enabling nanotechnology at its most fundamental level.