Schrodinger Equation

Author

John Robin Inston

Published

October 15, 2026

1 Schrodinger Equation

References: Griffiths, Introduction to Quantum Mechanics, Chapters 1-2; Evans, Partial Differential Equations, Section 4.3 (dispersion).

The Schrodinger equation is the fundamental equation of non-relativistic quantum mechanics. It governs the evolution of the wave function \(\psi(t,x)\) of a particle, whose squared modulus is the probability density of finding the particle at position \(x\). Mathematically it is a linear dispersive PDE and is the diffusion equation with an imaginary diffusivity.

For a particle of mass \(m\) in a potential \(V(x)\) the Schrodinger equation is \[ i\hbar\,\psi_t=-\frac{\hbar^2}{2m}\Delta\psi+V\psi=\hat H\psi, \] where \(\hbar\) is the reduced Planck constant, \(\Delta\) is the Laplace Operator and \(\hat H=-\frac{\hbar^2}{2m}\Delta+V\) is the Hamiltonian operator. The wave function \(\psi(t,\cdot)\) lies in \(L^2(\mathbb{R}^n;\mathbb{C})\) and is normalised so that \[ \int_{\mathbb{R}^n}|\psi(t,x)|^2~dx=1. \]

In units with \(\hbar=m=1\) and \(V=0\) this becomes the free Schrodinger equation \(i\psi_t=-\frac12\Delta\psi\).

2 Interpretation

  • Born rule: \(|\psi(t,x)|^2\) is the probability density of the particle at \(x\), so \(\mathbb{P}(X_t\in A)=\int_A|\psi|^2~dx\).
  • Observables are self-adjoint operators on \(L^2\). Position is multiplication by \(x\), momentum is \(\hat p=-i\hbar\nabla\) and energy is \(\hat H\). The expected value of an observable \(\hat A\) is \(\langle\psi,\hat A\psi\rangle\).
  • Classical limit. Writing \(\psi=\sqrt\rho\,e^{iS/\hbar}\) and letting \(\hbar\to0\), the phase \(S\) solves the Hamilton-Jacobi Equation \(S_t+|\nabla S|^2/2m+V=0\), so classical mechanics is the geometric-optics limit of the wave equation.

Note: The Schrodinger equation is first order in time and complex, so \(\psi(0,\cdot)\) determines the whole evolution, in contrast to the real second-order Wave Equation.

3 Heuristic Derivation

The equation cannot be derived from classical mechanics, but is motivated by the de Broglie relations: a free particle of momentum \(p\) and energy \(E\) has the plane wave \[ \psi(t,x)=e^{i(kx-\omega t)},\qquad p=\hbar k,\quad E=\hbar\omega. \] Since \(E=p^2/2m\) classically, \(\hbar\omega=\hbar^2k^2/2m\). Now \[ i\hbar\psi_t=\hbar\omega\psi=E\psi,\qquad-\frac{\hbar^2}{2m}\psi_{xx}=\frac{\hbar^2k^2}{2m}\psi, \] so the plane wave solves \(i\hbar\psi_t=-\frac{\hbar^2}{2m}\psi_{xx}\). Adding the potential energy \(V\) to \(E\) gives the full equation, and linearity allows us to superpose plane waves.

4 Properties

Let \(\psi\) solve \(i\hbar\psi_t=\hat H\psi\) with real \(V\).

  1. Linearity (superposition): if \(\psi_1,\psi_2\) are solutions then so is \(c_1\psi_1+c_2\psi_2\) for any \(c_1,c_2\in\mathbb{C}\).
  2. Conservation of probability: \(\|\psi(t)\|_{L^2}=\|\psi(0)\|_{L^2}\) for all \(t\).
  3. Conservation of energy: \(\langle\psi,\hat H\psi\rangle\) is constant in time.
  4. Time reversibility: if \(\psi(t,x)\) is a solution then so is \(\overline{\psi(-t,x)}\).
  5. Unitary evolution: \(\psi(t)=e^{-it\hat H/\hbar}\psi(0)\) where \(e^{-it\hat H/\hbar}\) is a unitary group on \(L^2\).

Proof. (2) Let \(\rho=|\psi|^2=\psi\bar\psi\). Using \(\psi_t=\frac{1}{i\hbar}\hat H\psi\) and its conjugate, \[ \rho_t=\psi_t\bar\psi+\psi\bar\psi_t=\frac{1}{i\hbar}\left(\bar\psi\hat H\psi-\psi\overline{\hat H\psi}\right) =\frac{-\hbar}{2mi}\left(\bar\psi\Delta\psi-\psi\Delta\bar\psi\right), \] since the real potential terms \(V|\psi|^2\) cancel. The right side is \(-\nabla\cdot\vec J\) with the probability current \[ \vec J=\frac{\hbar}{2mi}\left(\bar\psi\nabla\psi-\psi\nabla\bar\psi\right)=\frac{\hbar}{m}\operatorname{Im}(\bar\psi\nabla\psi). \] This is a continuity equation \(\rho_t+\nabla\cdot\vec J=0\) exactly as in the Diffusion Equation. Integrating over \(\mathbb{R}^n\) and using decay of \(\psi\) at infinity, \(\frac{d}{dt}\int\rho~dx=0\).

(3) Differentiating \(\langle\psi,\hat H\psi\rangle\) and using that \(\hat H\) is self-adjoint, \(\frac{d}{dt}\langle\psi,\hat H\psi\rangle=2\operatorname{Re}\langle\hat H\psi,\psi_t\rangle=2\operatorname{Re}\langle\hat H\psi,\tfrac{1}{i\hbar}\hat H\psi\rangle=0\), as \(\langle\hat H\psi,\hat H\psi\rangle\) is real and is multiplied by \(1/(i\hbar)\).

Note: Compare with the heat equation. There \(\int u^2\) decays and the evolution is irreversible and smoothing. Here the factor \(i\) turns decay into rotation, so the evolution is unitary and information is never lost. Mathematically \(e^{t\Delta}\) is a contraction semigroup and \(e^{it\Delta}\) is a unitary group.

5 Stationary States

5.1 Time-Independent Equation

When \(V\) does not depend on \(t\), separate variables as \(\psi=T(t)\varphi(x)\): \[ \frac{i\hbar T'}{T}=\frac{\hat H\varphi}{\varphi}=E. \] Both sides must equal a constant \(E\) (the energy), so \(T(t)=e^{-iEt/\hbar}\) and \[ \hat H\varphi=E\varphi,\qquad\text{i.e.}\qquad -\frac{\hbar^2}{2m}\Delta\varphi+V\varphi=E\varphi. \] This is the time-independent Schrodinger equation, an eigenvalue problem for \(\hat H\). The solutions \(\psi=e^{-iEt/\hbar}\varphi(x)\) are stationary states, because \(|\psi|^2=|\varphi|^2\) does not depend on \(t\). The general solution is the superposition \[ \psi(t,x)=\sum_nc_ne^{-iE_nt/\hbar}\varphi_n(x),\qquad c_n=\langle\varphi_n,\psi(0)\rangle, \] with \(\sum|c_n|^2=1\). This is the analogue of the Fourier series solution of the Heat Equation, where the decay factors \(e^{-k\lambda_nt}\) are replaced by the phases \(e^{-iE_nt/\hbar}\).

5.2 Particle in a Box

Let \(V=0\) on \((0,L)\) and \(V=\infty\) outside, so that \(\varphi(0)=\varphi(L)=0\). The problem \(-\frac{\hbar^2}{2m}\varphi''=E\varphi\) is solved exactly as for the heat equation on a bar: \[ \varphi_n(x)=\sqrt{\frac2L}\sin\frac{n\pi x}{L},\qquad E_n=\frac{\hbar^2\pi^2n^2}{2mL^2},\qquad n=1,2,\dots \] The energy is quantised and the lowest energy \(E_1>0\) is not zero (the zero-point energy). The boundary conditions are what force the discreteness, just as they force the eigenvalues \(\lambda_n=(n\pi/L)^2\) in the diffusion problem.

5.3 Harmonic Oscillator

For \(V(x)=\frac12m\omega^2x^2\) the eigenvalues are \[ E_n=\hbar\omega\left(n+\tfrac12\right),\qquad n=0,1,2,\dots \] with eigenfunctions \(\varphi_n\propto H_n(\xi)e^{-\xi^2/2}\) where \(\xi=\sqrt{m\omega/\hbar}\,x\) and \(H_n\) are the Hermite polynomials. The ground state is the Gaussian \(\varphi_0\propto e^{-m\omega x^2/2\hbar}\), which has the form of the Normal Distribution density.

6 The Free Particle

6.1 Plane Wave Solutions and Dispersion

For \(V=0\) the plane waves \(e^{i(kx-\omega t)}\) solve the equation if \[ \omega(k)=\frac{\hbar k^2}{2m}. \] This dispersion relation is non-linear in \(k\), so waves of different wavelengths travel at different speeds. The phase velocity is \(\omega/k=\hbar k/2m\) and the group velocity, the speed at which a wave packet moves, is \[ \frac{d\omega}{dk}=\frac{\hbar k}{m}=\frac pm, \] which is the classical velocity. By contrast the Wave Equation has \(\omega=ck\) with no dispersion.

6.2 Fourier Solution

Take the Fourier transform in \(x\), \(\hat\psi(t,k)=\int e^{-ikx}\psi(t,x)~dx\). Since \(\partial_{xx}\to-k^2\), the equation becomes \(i\hbar\hat\psi_t=\frac{\hbar^2k^2}{2m}\hat\psi\) with solution \[ \hat\psi(t,k)=e^{-i\hbar k^2t/2m}\hat\psi_0(k), \] so that \[ \psi(t,x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{i(kx-\hbar k^2t/2m)}\hat\psi_0(k)~dk. \] Each mode only rotates in phase, so \(|\hat\psi|\) is unchanged and the \(L^2\) norm is conserved. Compare with \(\hat u=e^{-Dk^2t}\hat u_0\) for the Diffusion Equation.

6.3 Free Propagator

For \(V=0\) in one dimension, \[ \psi(t,x)=\int_{-\infty}^\infty K(t,x-y)\psi_0(y)~dy,\qquad K(t,x)=\sqrt{\frac{m}{2\pi i\hbar t}}\exp\left(\frac{imx^2}{2\hbar t}\right). \]

Proof. (Sketch) The heat kernel for \(u_t=Du_{xx}\) is \((4\pi Dt)^{-1/2}e^{-x^2/4Dt}\). The free Schrodinger equation is \(\psi_t=\frac{i\hbar}{2m}\psi_{xx}\), which is the heat equation with the formal diffusivity \(D=i\hbar/2m\). Substituting gives \[ (4\pi Dt)^{-1/2}=\left(\frac{2\pi i\hbar t}{m}\right)^{-1/2},\qquad \frac{x^2}{4Dt}=\frac{x^2m}{2i\hbar t}=-\frac{imx^2}{2\hbar t}, \] so \(e^{-x^2/4Dt}=e^{imx^2/2\hbar t}\). The result can be justified rigorously by computing the inverse Fourier transform of \(e^{-i\hbar k^2t/2m}\) as an oscillatory Gaussian integral (see Gaussian Integral) with \(t\) replaced by \(t+i\epsilon\) and letting \(\epsilon\to0\).

Note: Unlike the real heat kernel, \(|K|=\sqrt{m/2\pi\hbar t}\) does not decay in \(x\), so \(K\) is not a probability density and the evolution does not smooth the data. Instead, the solution disperses: the amplitude at a fixed point decays like \(t^{-1/2}\) in one dimension because the wave spreads out, while the norm is conserved.

Take \(\psi_0(x)=(2\pi\sigma^2)^{-1/4}e^{-x^2/4\sigma^2}\) (so \(|\psi_0|^2\) is the \(\mathcal{N}(0,\sigma^2)\) density). Then \(|\psi(t,x)|^2\) is again a Gaussian, centred at the origin with variance \[ \sigma(t)^2=\sigma^2\left(1+\frac{\hbar^2t^2}{4m^2\sigma^4}\right). \] The packet spreads linearly in \(t\) for large \(t\), compared with \(\sqrt t\) for diffusion. A narrower initial packet spreads faster, consistent with the Heisenberg uncertainty principle \(\sigma_x\sigma_p\ge\hbar/2\).

7 Relation to Other Equations

Equation Form Behaviour
Heat Equation \(u_t=D\Delta u\) Smoothing, irreversible
Schrodinger \(i\psi_t=-\frac12\Delta\psi\) Dispersive, reversible, \(L^2\)-conserving
Wave Equation \(u_{tt}=c^2\Delta u\) Non-dispersive waves, finite speed
  • Imaginary time. Substituting \(t=-i\tau\) in the Schrodinger equation gives \(\psi_\tau=\frac{\hbar}{2m}\Delta\psi-\frac1\hbar V\psi\), a diffusion equation with a potential term. Under this Wick rotation the unitary group \(e^{-it\hat H/\hbar}\) becomes \(e^{-\tau\hat H/\hbar}\), which projects onto the ground state as \(\tau\to\infty\).
  • Feynman path integral. The kernel of \(e^{-\tau\hat H/\hbar}\) is given by the Feynman-Kac Formula as an expectation over Brownian paths. The real-time kernel is the same formula as a formal oscillatory integral over paths weighted by \(e^{iS/\hbar}\).
  • Schrodinger bridges. The problem of finding the most likely evolution of a diffusion between two marginal distributions, which is used in Deep Generative Modeling (DGM), takes its name from Schrodinger’s coupled forward and backward diffusion equations.
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