6.8 The Schrödinger equation

This lesson is a small detour outside acoustics, included because the Schrödinger equation is a striking example of how the same PDE machinery — separation of variables, eigenvalue problems, orthogonal mode expansions — underwrites a field of physics that looks superficially very different. The clamped-string analysis of 6.3 is, almost line for line, the analysis of a quantum particle confined in a box. Recognising that parallel is one of the simpler ways to grasp the structural unity of linear-PDE physics.

We will not develop quantum mechanics; this is a one-lesson tour of the wave-function-as-PDE picture, the time-independent equation that emerges from separation of variables, and the canonical worked example that every quantum textbook starts with. A reader who wants more should pick up Griffiths’ Introduction to Quantum Mechanics or Shankar’s Principles of Quantum Mechanics — this lesson is meant to make the first few chapters of either feel familiar.

The time-dependent Schrödinger equation

In non-relativistic quantum mechanics, the state of a single particle of mass $m$ moving in a potential $V(\mathbf{r})$ is described by a complex-valued wavefunction $\Psi(\mathbf{r}, t)$ that obeys the time-dependent Schrödinger equation

i \hbar\, \frac{\partial \Psi}{\partial t} \;=\; -\frac{\hbar^2}{2 m}\, \nabla^2 \Psi \;+\; V(\mathbf{r})\, \Psi.

Here $\hbar = h / (2 \pi) \approx 1.055 \times 10^{-34}\,\mathrm{J \cdot s}$ is the reduced Planck constant and $V$ is the same classical potential energy the particle would feel. The wavefunction itself has no direct physical interpretation, but its squared magnitude $|\Psi(\mathbf{r}, t)|^2$ is the probability density for finding the particle at $\mathbf{r}$ at time $t$ , with $\int |\Psi|^2 dV = 1$ for a normalised state.

Three structural facts about the equation are worth noting before any solving:

It is first order in time but with a factor of $i$ on the time derivative. Compare with the heat equation $u_t = D u_{xx}$ , which is also first order in time. The diffusion equation dissipates — modes decay exponentially with rate $D k^2$ . The Schrödinger equation, because of the $i$ , oscillates — modes pick up phase factors $e^{-i E t / \hbar}$ but their magnitudes stay constant. The factor of $i$ converts dissipation into unitary phase evolution, and it is what keeps probability conserved.
It is second order in space, like the wave equation and the heat equation. The Laplacian piece — the kinetic-energy operator $-\hbar^2 \nabla^2 / (2m)$ — gives the equation its PDE character.
It is linear in $\Psi$ . Any linear combination of solutions is also a solution. This is the superposition principle of quantum mechanics, and it is the same superposition principle that makes Fourier series work for the wave equation.

The combination of first order in time (one initial function: $\Psi(\mathbf{r}, 0)$ ) and second order in space (boundary conditions on $\Psi$ at the edges of the domain) is the order-counting from 6.1, exactly as it was for the heat equation.

Separation of variables

The standard first move on the time-dependent Schrödinger equation is to look for product solutions $\Psi(\mathbf{r}, t) = \psi(\mathbf{r})\, \phi(t)$ . The separation procedure proceeds line by line as before, with one twist: the separation constant is an energy rather than a frequency.

▶ Worked example: separating time from space, every step

Starting point. Substitute the product ansatz $\Psi(\mathbf{r}, t) = \psi(\mathbf{r})\, \phi(t)$ into

i \hbar\, \partial_t \Psi \;=\; -\frac{\hbar^2}{2 m}\, \nabla^2 \Psi \;+\; V(\mathbf{r})\, \Psi.

Step 1 — Substitute and compute the derivatives.

\partial_t \Psi \;=\; \psi(\mathbf{r})\, \phi'(t), \qquad \nabla^2 \Psi \;=\; \phi(t)\, \nabla^2 \psi(\mathbf{r}).

The equation becomes

i \hbar\, \psi(\mathbf{r})\, \phi'(t) \;=\; -\frac{\hbar^2}{2 m}\, \phi(t)\, \nabla^2 \psi(\mathbf{r}) \;+\; V(\mathbf{r})\, \psi(\mathbf{r})\, \phi(t).

Step 2 — Divide both sides by $\psi(\mathbf{r})\, \phi(t)$ .

i \hbar\, \frac{\phi'(t)}{\phi(t)} \;=\; \frac{1}{\psi(\mathbf{r})} \left[ -\frac{\hbar^2}{2 m}\, \nabla^2 \psi(\mathbf{r}) \;+\; V(\mathbf{r})\, \psi(\mathbf{r}) \right].

The left side depends only on $t$ ; the right depends only on $\mathbf{r}$ . Two functions of independent variables can be equal at all values only if both equal the same constant. Call that constant $E$ (the choice of letter anticipates that it will turn out to be the energy):

i \hbar\, \frac{\phi'(t)}{\phi(t)} \;=\; E, \qquad \frac{1}{\psi(\mathbf{r})} \left[ -\frac{\hbar^2}{2 m}\, \nabla^2 \psi(\mathbf{r}) \;+\; V(\mathbf{r})\, \psi(\mathbf{r}) \right] \;=\; E.

Step 3 — Solve the time ODE. Multiply through:

i \hbar\, \phi'(t) \;=\; E\, \phi(t).

A first-order linear ODE with constant coefficients; solve as in 5.2. The solution is

\phi(t) \;=\; \phi(0)\, e^{-i E t / \hbar}.

Note the minus sign — it comes from the $i$ on the left and the convention of writing the time dependence as $e^{-i E t / \hbar}$ . Compared with the wave equation, where each mode oscillates as $\cos(\omega t)$ at a real frequency, here each mode picks up a pure phase of magnitude 1. The magnitude $|\phi(t)|^2 = |\phi(0)|^2$ is constant in time — the time evolution is unitary, preserving normalisation.

Step 4 — The remaining spatial equation. Multiply through by $\psi$ :

-\frac{\hbar^2}{2 m}\, \nabla^2 \psi(\mathbf{r}) \;+\; V(\mathbf{r})\, \psi(\mathbf{r}) \;=\; E\, \psi(\mathbf{r}).

This is the time-independent Schrödinger equation. It is an eigenvalue problem (refresher →): the linear differential operator on the left,

\hat{H} \;\equiv\; -\frac{\hbar^2}{2 m}\, \nabla^2 \;+\; V(\mathbf{r}),

called the Hamiltonian, acts on the spatial wavefunction $\psi$ and yields a constant multiple $E\, \psi$ . The number $E$ is the energy eigenvalue; the function $\psi$ is the energy eigenstate (or “stationary state”). The solvable problem has been reduced from a partial differential equation in $\mathbf{r}$ and $t$ to an eigenvalue problem in $\mathbf{r}$ alone.

Step 5 — Reassemble. Each pair $(E_n, \psi_n)$ produces one stationary solution of the full time-dependent equation:

\Psi_n(\mathbf{r}, t) \;=\; \psi_n(\mathbf{r})\, e^{-i E_n t / \hbar}.

The general solution is a superposition over all allowed eigenstates:

\Psi(\mathbf{r}, t) \;=\; \sum_n c_n\, \psi_n(\mathbf{r})\, e^{-i E_n t / \hbar},

with coefficients $c_n$ set by the initial wavefunction via Fourier projection onto the eigenstate basis,

c_n \;=\; \int \psi_n^*(\mathbf{r})\, \Psi(\mathbf{r}, 0)\, dV.

This is exactly the modes-and-mode-sums structure of 6.5, with the eigenfunctions $\psi_n$ of $\hat H$ playing the role the eigenfunctions $X_n$ of $-\partial_x^2$ played there. Orthogonality, completeness, projection — all carry over.

Worked example: the infinite square well

Solving the time-independent Schrödinger equation for a particular potential $V(\mathbf{r})$ requires the eigenvalues and eigenfunctions of $\hat H$ on that potential. The simplest non-trivial case is a particle confined to a 1-D box.

The problem. A particle of mass $m$ moves in one dimension, free inside the interval $0 < x < L$ but unable to escape because the potential is infinite outside:

V(x) \;=\; \begin{cases} 0, & 0 < x < L, \\ \infty, & \text{otherwise.} \end{cases}

Outside the well the wavefunction must be zero (the particle cannot be where the energy is infinite); inside the well the time-independent Schrödinger equation reduces to

-\frac{\hbar^2}{2 m}\, \psi''(x) \;=\; E\, \psi(x),

with boundary conditions $\psi(0) = \psi(L) = 0$ (continuity of $\psi$ across the wall).

The recipe. Rearrange the equation:

\psi''(x) \;+\; k^2\, \psi(x) \;=\; 0, \qquad k^2 \;\equiv\; \frac{2 m E}{\hbar^2}.

This is exactly the clamped-string equation from 6.3. The boundary conditions are exactly the clamped-string boundary conditions. The allowed wavenumbers are exactly the clamped-string wavenumbers:

k_n \;=\; \frac{n \pi}{L}, \qquad n = 1, 2, 3, \ldots

with eigenfunctions $\psi_n(x) = \sqrt{2/L}\, \sin(n \pi x / L)$ , normalised so that $\int_0^L |\psi_n|^2\, dx = 1$ . Solving $k_n^2 = 2 m E_n / \hbar^2$ for the energy:

\boxed{\;E_n \;=\; \frac{n^2 \pi^2 \hbar^2}{2 m L^2}.\;}

The energies are quantised — only the discrete values $E_n$ are allowed — and they go as $n^2$ , so the spacing between levels grows with $n$ . The lowest level $E_1$ is non-zero (the particle cannot have zero kinetic energy, by Heisenberg’s uncertainty principle: confinement to a box of width $L$ forces a momentum uncertainty of order $\hbar / L$ , hence a kinetic energy of order $\hbar^2 / (m L^2)$ ).

quantum number n = 1

The wavefunction ψₙ(x) = √(2/L) sin(n π x / L) has n half-wavelengths inside the well, exactly like the clamped-string mode of the same n. The probability density |ψₙ|² has n peaks; the particle is found preferentially at the antinodes of ψₙ and never at the nodes. The energy levels scale as Eₙ ∝ n²: doubling the index quadruples the energy, an unequal spacing characteristic of confinement. (Quoted energies use the electron mass and a 1 nm well.)

The wavefunction $\psi_n(x)$ has $n$ half-wavelengths inside the well, just like the $n$ -th mode of the clamped string. The probability density $|\psi_n(x)|^2$ has $n$ peaks; the particle is found preferentially at the antinodes of $\psi_n$ and never at the nodes — an interference-pattern character that has no classical analogue. The energy ladder $E_n \propto n^2$ rises faster than linearly: doubling the index quadruples the energy.

The quoted energies use the electron mass and a 1 nm well, giving electronvolt-scale energies. Same equation, same eigenstates, but the energy scale changes radically with the mass and the box size:

An electron in a 1 nm box: $E_1 \approx 0.38\,\mathrm{eV}$ . Comparable to room-temperature thermal energies and to chemical bond energies — relevant for atomic-scale physics.
A nitrogen molecule in a 1 mm box: $E_1 \approx 10^{-25}\,\mathrm{eV}$ . Utterly negligible. Quantum confinement is irrelevant at macroscopic scales.
A proton in a 1 fm nucleus: $E_1 \approx 200\,\mathrm{MeV}$ . The nuclear-physics scale.

The same equation answers questions across forty orders of magnitude in energy by simply changing the inputs.

The parallel with acoustics

This is the structural unity worth pausing on. The infinite-square-well eigenfunctions $\sin(n \pi x / L)$ are the same functions as the clamped-string modes from 6.3 — not merely similar in spirit, literally the same sines. The boundary conditions ( $\psi = 0$ at the walls vs $u = 0$ at the fixed ends) are the same Dirichlet condition. The orthogonality and completeness theorems carry over verbatim. The Fourier-projection step that pulls out the coefficients is identical.

What differs is the physical interpretation of those functions and the time evolution attached to each:

Feature	Clamped string	Particle in a box
Spatial mode shape	$\sin(n\pi x / L)$	$\sin(n\pi x / L)$
Eigenvalue	$\omega_n^2 = (c k_n)^2$	$E_n = \hbar^2 k_n^2 / (2m)$
Time evolution of each mode	$\cos(\omega_n t)$	$e^{-i E_n t / \hbar}$
Physical meaning of $X_n$	displacement at time $t = 0$	probability amplitude at any $t$
What $	X_n	^2$ means

Two equations from different centuries, different fields, with different physical interpretations, share an eigenvalue problem. The unifying feature is that both Hamiltonians (spatial wave operator $-c^2 \partial_x^2$ for the wave equation, $-\hbar^2 \partial_x^2 / (2m) + V$ for Schrödinger) are self-adjoint linear operators on a function space, and the spectral theorem guarantees both have a complete orthogonal eigenbasis. Once you have that theorem, every linear PDE on a bounded domain becomes the same problem.

This is one of the central facts of mathematical physics. The fact that the same mathematical structure controls vibrating strings, electromagnetic cavities, electron states in atoms, and the energy levels of molecules is what makes a single course in linear-PDE methods so useful: learn separation of variables once, apply it across most of physics.

⏳ The history — Schrödinger 1926, and the two quantum mechanicses

Quantum mechanics was discovered twice in the same year. Werner Heisenberg’s 1925 paper introduced matrix mechanics: physical observables were represented by infinite matrices and the dynamics by matrix multiplication. The mathematics was unfamiliar to physicists — Born and Jordan had to teach Heisenberg what a matrix was — but it correctly predicted the spectral lines of the hydrogen atom and the spectra of more complicated atoms.

Erwin Schrödinger, working independently in early 1926, was guided by de Broglie’s 1924 hypothesis that matter has wave-like character. He wrote down the wave equation $i\hbar\, \partial_t \Psi = \hat H \Psi$ and showed that its eigenvalues for the hydrogen-atom potential gave the Bohr energy levels exactly. The mathematics was the separation-of-variables technique already familiar from acoustics — which is precisely the parallel this lesson develops.

The two formulations looked utterly different. Heisenberg’s was algebraic and discrete; Schrödinger’s was differential and continuous. Within months of publication (1926), Schrödinger himself proved that the two were mathematically equivalent — different representations of the same theory. Paul Dirac’s 1930 textbook The Principles of Quantum Mechanics and John von Neumann’s 1932 Mathematische Grundlagen der Quantenmechanik gave the unified abstract formulation in terms of operators on Hilbert space, which is the formulation modern physics uses. The same Hilbert space, complete with self-adjoint operators and the spectral theorem, that runs through the rest of Foundations 6.

Beyond the infinite well

A few standard generalisations, mentioned without development:

Finite square well. Replace the infinite walls with a finite step $V_0$ . Inside the well, the eigenfunctions remain sinusoidal; outside, they decay exponentially. Boundary conditions become continuity of $\psi$ and $\psi'$ at the walls, leading to a transcendental quantisation equation (no longer the clean $k_n L = n \pi$ ). The number of bound states is finite — a particle can escape if its energy exceeds $V_0$ . Same separation-of-variables logic, but with Robin-style matching at the walls and a continuum of unbound states above $V_0$ .
Harmonic oscillator. Set $V(x) = \tfrac12 m \omega^2 x^2$ . The eigenvalue equation becomes Hermite’s differential equation, with eigenfunctions $\psi_n(x) = H_n(\xi)\, e^{-\xi^2 / 2}$ for $\xi = x \sqrt{m \omega / \hbar}$ and eigenvalues $E_n = \hbar \omega (n + 1/2)$ . Evenly spaced energy levels with a non-zero ground state — the quantum analogue of a vibrating mode of a classical oscillator. The same operator algebra (raising and lowering operators, the harmonic-oscillator ladder) reappears in quantum optics, in molecular vibrations, in lattice phonons, and in quantum field theory.
Hydrogen atom. $V(\mathbf{r}) = -e^2 / (4 \pi \epsilon_0 r)$ . Separation of variables in spherical coordinates gives an angular ODE (whose solutions are spherical harmonics) and a radial ODE (whose solutions are associated Laguerre polynomials). The eigenvalues are the Bohr energy levels $E_n = -13.6\,\mathrm{eV} / n^2$ . Same playbook — separation of variables, eigenvalue ODEs, mode-sum solutions — applied to a problem of greater geometric complexity.

The full development of any of these is a textbook chapter on its own. What this lesson supplies is the structural understanding that they are all PDE-eigenvalue problems of the kind we built in this chapter.

Concluding the chapter

That closes Foundations 6. The chapter started with a question — what is a PDE? — and ended with quantum mechanics. The unifying thread is that linear PDEs on bounded domains are eigenvalue problems for self-adjoint operators, and their solutions are sums over orthogonal eigenmodes. The wave equation, the heat equation, Laplace’s equation, the Helmholtz equation, and the Schrödinger equation are all instances of that single structure — different physics, identical mathematics.

The next two chapters develop the continuum version of the eigenmode picture: Foundations 7 — Fourier series and the Fourier transform shows what happens to mode catalogues when the domain becomes unbounded and the discrete ladder $\{k_n\}$ becomes a continuous variable $k$ .