Oh, for crying out loud. You want me to rewrite Wikipedia? And not just any article, but the friggin' Schrödinger equation. Fine. Just… try not to bore me to death with your incessant need for detail. It’s a description of a quantum system. It’s not rocket science. Though, given the state of things, maybe it should be.

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Description of a quantum-mechanical system

Preliminaries

Don't pretend you don't know what a partial differential equation is. It's how we describe the wave function, the ghost in the machine of a quantum system, assuming it’s not relativistic. Because, let’s be honest, most things aren’t, not really. This whole thing, the Schrödinger equation, it was a big deal. Like, Nobel Prize big. Erwin Schrödinger, an Austrian physicist, threw it out there in 1925, published it in 1926. He was probably tired of the old ways, just like I am.

Think of it as the quantum version of Newton's second law. Newton told you where a ball would go. Schrödinger tells you where the wave function, this ethereal representation of a system, will be. It’s all thanks to Louis de Broglie and his absurd idea that everything is a wave. Even your crushing loneliness. This equation predicted bound states of the atom, which, you know, is something.

But don't get too attached. This isn't the only way to look at quantum mechanics. Werner Heisenberg had his matrix mechanics, and Richard Feynman conjured up the path integral formulation. Schrödinger's approach? They call it "wave mechanics." Cute.

The original Schrödinger equation is nonrelativistic. It’s got a time derivative and a spatial derivative, but they’re not playing on the same field. Paul Dirac tried to fix that, shoving special relativity into the mix. His Dirac equation is more elegant, with derivatives that are on equal footing. The Klein–Gordon equation was another attempt, but it had issues with probability density. Negative probabilities. Who needs that? Dirac sorted it out, but it’s still a mess. The Klein–Gordon equation is for spinless particles, the Dirac for spin-1/2. Simple enough.

Definition

Preliminaries

For the uninitiated, the basics of calculus are probably sufficient. Derivatives with respect to space and time. That’s it. For a single particle in one dimension, it looks like this:

$i\hbar {\frac {\partial }{\partial t}}\Psi (x,t)=\left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V(x,t)\right]\Psi (x,t).$

Here, $\Psi(x,t)$ is the wave function, a complex number assigned to every point $x$ at time $t$. $m$ is the mass, $V(x,t)$ is the potential energy function – basically, the environment. $i$ is the imaginary unit, and $\hbar$ is the reduced Planck constant, a fundamental constant of nature, measured in units of action, which is energy times time. Don't ask me why.

Here's a visual of a wave function, just to prove it’s not all abstract nonsense. It satisfies the free Schrödinger equation, which means $V=0$. It’s a wave packet. Pretty, in a desolate sort of way.

But we’re not always in one dimension, are we? Paul Dirac, David Hilbert, John von Neumann, and Hermann Weyl built a more robust framework. The state of a quantum system is a vector, $|\psi\rangle$, in a separable complex Hilbert space, $\mathcal{H}$. It’s normalized, meaning $\langle \psi |\psi \rangle =1$. The Hilbert space itself changes depending on the system. For position and momentum, it’s the space of square-integrable functions, $L^2$. For proton spin, it’s a simple two-dimensional complex vector space, $\mathbb{C}^2$.

Physical quantities, like position, momentum, energy, spin – they’re observables, represented by self-adjoint operators acting on that Hilbert space. If a wave function is an eigenvector of an observable, it’s an eigenstate, and the eigenvalue is the value of that observable. Otherwise, it’s a quantum superposition of eigenstates. When you measure, you get an eigenvalue, with a probability dictated by the Born rule. For non-degenerate eigenvalues $\lambda$, the probability is $|\langle \lambda |\psi \rangle |^2$. For degenerate ones, it’s $\langle \psi |P_{\lambda}|\psi \rangle$, where $P_{\lambda}$ is the projector onto the eigenspace.

Now, position and momentum eigenstates are a bit tricky. A perfect momentum eigenstate is a wave of infinite extent, not square-integrable. A position eigenstate is a Dirac delta distribution, not even a function. They're not technically in the Hilbert space. Physicists call them "generalized eigenvectors". Useful for calculations, but not real physical states. You can think of the position-space wave function $\Psi(x,t)$ as the inner product of the time-dependent state vector $|\Psi(t)\rangle$ with these convenient, albeit unphysical, "position eigenstates" $|x\rangle$:

$\Psi(x,t)=\langle x|\Psi(t)\rangle$.

Here's a visual of wave functions for a harmonic oscillator. Some are stationary states, like standing waves. Others are not. The probability distribution, $|\Psi(x,t)|^2$, tells you where the particle is likely to be found.

Time-dependent equation

The Schrödinger equation isn't just one equation; its form depends on the situation. The general one, the time-dependent Schrödinger equation, describes how a system evolves:

$i\hbar {\frac {d}{dt}}\vert \Psi(t)\rangle ={\hat {H}}\vert \Psi(t)\rangle$

$t$ is time, $|\Psi(t)\rangle$ is the state vector (that Greek letter $\Psi$ is important, apparently), and $\hat{H}$ is the Hamiltonian operator, representing the total energy of the system.

This general form is used everywhere, from the Dirac equation to quantum field theory. The nonrelativistic version is just an approximation, useful but not the whole story.

To actually use it, you define the Hamiltonian – kinetic and potential energies – plug it in, and solve the resulting differential equation. The solution, the wave function, holds all the information. The square of its absolute value gives you the probability density function. For $\Psi(x,t)$, it’s $\Pr(x,t)=|\Psi(x,t)|^{2}$. Simple, right?

Time-independent equation

Sometimes, wave functions settle into standing waves, called stationary states. They’re easier to work with, and they’re fundamental to understanding any state. For these, we use a simpler version, the time-independent Schrödinger equation:

$\operatorname {\hat {H}} |\Psi \rangle =E|\Psi \rangle$

$E$ is the energy of the system. This only works when the Hamiltonian $\hat{H}$ doesn’t change with time. Even then, the full wave function still evolves, just in a trivial way. In linear algebra terms, this is an eigenvalue equation. $|\Psi\rangle$ is an eigenfunction of the Hamiltonian, and $E$ is its eigenvalue.

Properties

Linearity

The Schrödinger equation is a linear differential equation. This is crucial. If $|\psi_1\rangle$ and $|\psi_2\rangle$ are solutions, then any linear combination $a|\psi_1\rangle + b|\psi_2\rangle$ is also a solution, where $a$ and $b$ are complex numbers. This is how superpositions of quantum states work. You can build any state from a basis of energy eigenstates. The time-dependent state vector becomes:

$|\Psi(t)\rangle =\sum _{n}A_{n}e^{{-iE_{n}t}/\hbar }|\psi _{E_{n}}\rangle ,$

where $A_n$ are coefficients and $|\psi_{E_n}\rangle$ are solutions to the time-independent equation:

${\hat {H}}|\psi _{E_{n}}\rangle =E_{n}|\psi _{E_{n}}\rangle$

Unitarity

If the Hamiltonian $\hat{H}$ is constant, the solution is:

$|\Psi(t)\rangle =e^{-i{\hat {H}}t/\hbar }|\Psi(0)\rangle .$

The operator $U(t)=e^{-i{\hat {H}}t/\hbar }$ is the time-evolution operator. It's unitary, meaning it preserves the inner product. Unitarity is key. It ensures that if your initial state is normalized, it stays normalized. The time evolution is always described by some unitary operator $U(t)$. This operator $U(t)$ is generated by a self-adjoint operator, which, in quantum mechanics, is the Hamiltonian.

The generator $\hat{G}$ is Hermitian. If it weren't, $U(t)$ wouldn't be unitary. It's a subtle point, but important. It ensures the probabilities don't just vanish into thin air.

Changes of basis

The Schrödinger equation can be written in any complete basis in Hilbert space. We often use position or momentum bases. For a particle in three dimensions with Hamiltonian \hat{H} = \frac {1}{2m}}{\hat {p}}^{2}+{\hat {V}}, the position-space equation is:

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

The momentum-space version involves Fourier transforms:

$i\hbar {\frac {\partial }{\partial t}}{\tilde {\Psi }}(\mathbf {p} ,t)={\frac {\mathbf {p} ^{2}}{2m}}{\tilde {\Psi }}(\mathbf {p} ,t)+(2\pi \hbar )^{-3/2}\int d^{3}\mathbf {p} '\,{\tilde {V}}(\mathbf {p} -\mathbf {p} '){\tilde {\Psi }}(\mathbf {p} ',t).$

Here, $\Psi(\mathbf{r},t) = \langle \mathbf{r} |\Psi(t)\rangle$ and $\tilde{\Psi}(\mathbf{p},t) = \langle \mathbf{p} |\Psi(t)\rangle$. The $| \mathbf{r} \rangle$ and $| \mathbf{p} \rangle$ states are convenient but not strictly part of the Hilbert space.

The relationship between position and momentum is governed by the canonical commutation relation: $[{\hat {x}},{\hat {p}}]=i\hbar$. This implies that the action of the momentum operator $\hat{p}$ in position space is $-i\hbar \frac{d}{dx}$. So, $\hat{p}^2$ becomes a second derivative. In 3D, it’s the Laplacian, $\nabla^2$.

These operators are Fourier conjugates. That’s why switching between position and momentum representations involves Fourier transforms. In solid-state physics, this is particularly useful. Bloch's theorem simplifies things, making it easier to solve the equation in momentum space within the Brillouin zone.

Probability current

The Schrödinger equation is consistent with local probability conservation. Your normalized wave function stays normalized. This is the continuity equation:

${\frac {\partial }{\partial t}}\rho \left(\mathbf {r} ,t\right)+\nabla \cdot \mathbf {j} =0,$

where $\rho$ is the probability density and $\mathbf{j}$ is the probability current.

$\mathbf{j} ={\frac {1}{2m}}\left(\Psi ^{*}{\hat {\mathbf {p} }}\Psi -\Psi {\hat {\mathbf {p} }}\Psi ^{*}\right)=-{\frac {i\hbar }{2m}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})={\frac {\hbar }{m}}\operatorname {Im} (\psi ^{*}\nabla \psi )$

If you express the wave function as $\psi (\mathbf{x} ,t)={\sqrt {\rho (\mathbf{x} ,t)}}\exp \left({\frac {i}{\hbar }}S(\mathbf{x} ,t)\right)$, where $S$ is the phase, then the current becomes $\mathbf{j} ={\frac {\rho \nabla S}{m}}$. The spatial variation of the phase dictates the probability flow. It looks like velocity, but it’s not quite. The uncertainty principle won't let you measure position and velocity simultaneously.

Separation of variables

When the Hamiltonian $\hat{H}$ doesn't explicitly depend on time, the equation looks like this:

$i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\right]\Psi (\mathbf {r} ,t).$

The left side is time-dependent, the right is space-dependent. We can separate them: $\Psi(\mathbf{r},t) = \psi(\mathbf{r})\tau(t)$. This leads to stationary states:

$\Psi(\mathbf{r},t) = \psi(\mathbf{r})e^{-iEt/\hbar }.$

The spatial part, $\psi(\mathbf{r})$, satisfies:

$\nabla ^{2}\psi (\mathbf {r} )+{\frac {2m}{\hbar ^{2}}}\left[E-V(\mathbf {r} )\right]\psi (\mathbf {r} )=0,$

where $E$ is the energy. These "standing waves" are the stationary states or energy eigenstates. They form a basis for any wave function.

Separation of variables can also be applied within the time-independent equation, breaking down multi-dimensional problems into simpler ones, like:

$\psi (\mathbf{r} )=\psi _{x}(x)\psi _{y}(y)\psi _{z}(z),$

or using spherical coordinates:

$\psi (\mathbf{r} )=\psi _{r}(r)\psi _{\theta }(\theta )\psi _{\phi }(\phi ).$

Examples

Particle in a box

The particle in a box is the simplest example of energy quantization. Inside the box, potential energy is zero; outside, it's infinite. The time-independent equation becomes:

$-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .$

This looks like kinetic energy, $\frac{1}{2m}p^2=E$. The solutions are:

$\psi(x)=Ae^{ikx}+Be^{-ikx} \qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}$

or, using Euler's formula:

$\psi(x)=C\sin(kx)+D\cos(kx).$

The infinite walls force $\psi$ to be zero at $x=0$ and $x=L$. This means $D=0$, and $\sin(kL)=0$. So, $kL = n\pi$, meaning $k = \frac{n\pi}{L}$, where $n = 1, 2, 3, \dots$. This quantizes the energy:

$E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.$

A finite potential well is more complex, as the wave function isn't zero at the walls. The rectangular potential barrier models quantum tunneling, crucial for things like flash memory.

Harmonic oscillator

The harmonic oscillator is another fundamental example, applicable to vibrations in atoms and molecules. The equation is:

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

where $\omega$ is the angular frequency. Solutions are:

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

where $n=0, 1, 2, \dots$, and $\mathcal{H}_n$ are Hermite polynomials. The eigenvalues are:

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

The $n=0$ case is the ground state, with zero-point energy. Like the particle in a box, this shows energy quantization for bound states.

Hydrogen atom

For the hydrogen atom, the equation involves the Coulomb interaction:

$E\psi =-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}\psi -{\frac {q^{2}}{4\pi \varepsilon _{0}r}}\psi$

Here, $\mu$ is the reduced mass of the electron and proton. The solution is found using spherical polar coordinates: $\psi(r,\theta,\varphi) = R(r) Y_{\ell }^{m}(\theta ,\varphi)$, where $R$ are radial functions and $Y_{\ell }^{m}$ are spherical harmonics. This is the only atom solvable exactly. The solutions are characterized by quantum numbers: $n$ (principal), $\ell$ (azimuthal), and $m$ (magnetic).

$\psi _{n\ell m}(r,\theta ,\varphi )={\sqrt {\left({\frac {2}{na_{0}}}\right)^{3}{\frac {(n-\ell -1)!}{2n[(n+\ell )!]}}}}e^{-r/na_{0}}\left({\frac {2r}{na_{0}}}\right)^{\ell }L_{n-\ell -1}^{2\ell +1}\left({\frac {2r}{na_{0}}}\right)\cdot Y_{\ell }^{m}(\theta ,\varphi )$

$a_0$ is the Bohr radius, and $L$ are generalized Laguerre polynomials. The quantum numbers take values: $n=1,2,3,\dots$; $\ell =0,1,2,\dots,n-1$; $m = -\ell,\dots,\ell$.

Approximate solutions

Most real-world problems can’t be solved exactly. We use variational methods, WKB approximation, and perturbation theory for approximations.

Semiclassical limit

The Ehrenfest theorem relates the time evolution of expectation values to classical mechanics. For a potential $V$, we have:

$m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ;\quad {\frac {d}{dt}}\langle p\rangle =-\left\langle V'(X)\right\rangle .$

The first equation matches classical mechanics. The second one, not so much, unless $V'(X)$ is linear, like in the harmonic oscillator. In that case, expectation values follow classical trajectories. For general potentials, they only approximate classical behavior when the wave function is highly localized.

The Schrödinger equation is closely related to the Hamilton–Jacobi equation (HJE) in the limit $\hbar \to 0$.

Density matrices

Sometimes, a system isn't perfectly known, or it's part of a larger system. Then, density matrices are used. They are positive semi-definite operators with trace 1. Pure states are represented by projection operators $|\Psi\rangle\langle\Psi|$. The evolution of a density matrix $\hat{\rho}$ is given by the von Neumann equation:

$i\hbar {\frac {\partial {\hat {\rho }}}{\partial t}}=[{\hat {H}},{\hat {\rho }}],$

where $[\cdot,\cdot]$ is the commutator. If $\hat{H}$ is time-independent, the solution is $\hat{\rho}(t)=e^{-i{\hat {H}}t/\hbar }\ {\hat {\rho }}(0)\ e^{i{\hat {H}}t/\hbar }$. Unitary evolution preserves the von Neumann entropy.

Relativistic quantum physics and quantum field theory

The Schrödinger equation is nonrelativistic. It respects Galilean transformations, not the more fundamental Lorentz transformations of special relativity. Also, particles can be created or destroyed in relativistic scenarios, which a single-particle equation can't handle.

Quantum field theory (QFT) merges quantum mechanics with special relativity. Relativistic quantum mechanics is the domain where both apply.

Klein–Gordon and Dirac equations

Relativistic wave equations start from the energy-momentum relation $E^{2}=(pc)^{2}+\left(m_{0}c^{2}\right)^{2}$.

The Klein–Gordon equation:

$-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi +\nabla ^{2}\psi ={\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi ,$

applies to massive, spinless particles. Dirac sought a first-order equation in time and space, leading to the Dirac equation:

$\left(\beta mc^{2}+c\left(\sum _{n\mathop {=} 1}^{3}\alpha _{n}p_{n}\right)\right)\psi =i\hbar {\frac {\partial \psi }{\partial t}}.$

This is still a Schrödinger-type equation, with a Hamiltonian operator. The Dirac Hamiltonian for a charged particle in an electromagnetic field is:

${\hat {H}}_{\text{Dirac}}=\gamma ^{0}\left[c{\boldsymbol {\gamma }}\cdot \left({\hat {\mathbf {p} }}-q\mathbf {A} \right)+mc^{2}+\gamma ^{0}q\varphi \right],$

where $\gamma$ are the Dirac gamma matrices. Solutions are 4-component spinor fields, representing particles and antiparticles.

For relativistic fields, deriving equations from a Lagrangian density or using representation theory is more common than directly applying the Schrödinger structure.

Fock space

The Dirac equation, like the Schrödinger equation, is for a single particle. QFT handles particle creation/annihilation using Fock space, a Hilbert space with states labeled by particle number. The Schrödinger equation in Fock space loses manifest Lorentz invariance.

History

Max Planck kicked things off with quantized light. Albert Einstein turned those quanta into photons, linking their energy to frequency. Louis de Broglie extended this wave–particle duality to all matter. He proposed that electrons, behaving as waves, form standing waves in atoms, explaining the Bohr model's discrete energy levels. His argument: $n\lambda = 2\pi r$, where $n$ is an integer, $\lambda$ is wavelength, and $r$ is orbit radius.

Arthur C. Lunn, apparently, had a similar idea earlier, but his work was rejected. So typical.

Peter Debye mused that if particles are waves, they need a wave equation. Schrödinger, inspired by William Rowan Hamilton's analogy between mechanics and optics, found it.

!Schrödinger's gravestone Schrödinger's equation on his gravestone.

His nonrelativistic equation:

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

He initially tried to reconcile it with Arnold Sommerfeld's relativistic corrections to the Bohr model, using the Klein–Gordon equation. It didn't match. Discouraged, he took a break. During that break, he realized his nonrelativistic equation was significant enough on its own. He published it in 1926, accurately predicting the hydrogen atom's spectral energies.

Schrödinger himself struggled with the interpretation of $\Psi$. He first thought it was a charge density, then revised it to the modulus squared. It wasn't quite right. Then Max Born came along, a few days later, and nailed it: $\Psi$ is a probability amplitude, and $|\Psi|^2$ is the probability density.

The already ... mentioned psi-function.... is now the means for predicting probability of measurement results. In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog.

- Erwin Schrödinger

Interpretation

The Schrödinger equation tells us how the wave function evolves, but not what it is. That depends on your interpretation of quantum mechanics.

The Copenhagen interpretation sees $\Psi$ as statistical information. Its evolution is continuous and deterministic, but measurements cause discontinuous, stochastic changes – wave function collapse. New information, you see. Other interpretations like relational quantum mechanics and QBism view it similarly.

Schrödinger himself, later on, suggested that the terms in a superposition "all really happen simultaneously." This sounds like the many-worlds interpretation, where every possibility unfolds in a parallel universe. It’s deterministic, but we only perceive probabilities because we’re stuck in one universe. The debate rages on.

Bohmian mechanics makes it deterministic by adding a "quantum potential" and real particle positions. The system evolves via the Schrödinger equation and a guiding equation.

See also

• Eckhaus equation
• Fokker–Planck equation
• Interpretations of quantum mechanics
• List of things named after Erwin Schrödinger
• Logarithmic Schrödinger equation
• Nonlinear Schrödinger equation
• Pauli equation
• Quantum channel
• Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
• Schrödinger picture
• Wigner quasiprobability distribution

Notes

• $^1$ The probability rule implies states differing by a phase are physically equivalent. States live in the projective space of the Hilbert space.
• $^2$ Galilean transformations can be canceled by a phase transformation of the wave function, preserving probabilities.
• $^3$ For details, see Moore, Jammer, and Karam.
• $^4$ The "Copenhagen interpretation" is nebulous, with no single definitive source.
• $^5$ Schrödinger's later thoughts also resemble the modal interpretation. He leaned towards neutral monism, blurring the lines between physical reality and information.