= Schrödinger equation
{c}
{title2=$i\hbar\partial_t\psi=H\psi$}
{wiki}
The <Schrödinger equation> governs the evolution of a <quantum state> between instances of <measurement in quantum mechanics>. For a particle of <mass> $m>0$ moving in a prescribed real <potential energy> $V(\mathbf r,t)$, its position <wavefunction> satisfies
$$
i\hbar\frac{\partial\psi}{\partial t}=H(t)\psi,
\qquad H(t)=-\frac{\hbar^2}{2m}\nabla^2+V(\mathbf r,t),
\qquad \hbar=\frac{h}{2\pi}.
$$
Here $h$ is the <Planck constant>, $\nabla^2$ is the <Laplace operator>, and the <Hamiltonian operator> combines <kinetic energy> with <potential energy>. Specify an initial <wavefunction>, an <operator domain> and any <boundary conditions>: the differential expression alone does not completely specify a physical model. The <Born rule> then turns the evolving <probability amplitude> into position <probabilities> through $\rho=|\psi|^2$ and integration over a region.
This scalar <Schrödinger operator> describes <nonrelativistic quantum mechanics>, with relevant <momenta> much smaller than $mc$, where $c$ is the <speed of light>. It omits the constant rest <energy> $mc^2$, assumes fixed particle number, and suppresses active <spin> dynamics and coupling to a <magnetic vector potential>. A free <electron> with a fixed, uncoupled <spin> state can still use this scalar model; the <Pauli equation> supplies the additional <spin> components and magnetic coupling when needed. A prescribed real $V$ describes a closed evolution model or an approximation in which the source of that <potential energy> is treated as external. Dissipation, particle creation and a general <measurement in quantum mechanics> need further modelling.
For a time-independent <self-adjoint> <Hamiltonian operator>, the <spectral theorem> defines <unitary time evolution> by
$$
\psi(t)=e^{-iHt/\hbar}\psi(0),\qquad
\|\psi(t)\|^2=\|\psi(0)\|^2.
$$
Initial states in the <operator domain> satisfy the differential equation in the <Hilbert space> sense; the <unitary operator> also evolves all other states in that <Hilbert space>. Time-dependent <Hamiltonian operators> need suitable regularity and domain hypotheses; their evolution is not generally the exponential of $-itH(t)/\hbar$.
The local reason for conservation is especially tangible. Multiply the <Schrödinger equation> by the <complex conjugate> $\psi^*$, subtract its conjugate multiplied by $\psi$, and use the fact that the real <potential energy> terms cancel. For smooth <wavefunctions> this gives the <probability continuity equation>
$$
\partial_t\rho+\nabla\cdot\mathbf j=0,\qquad
\mathbf j=\frac{\hbar}{m}\operatorname{Im}(\psi^*\nabla\psi).
$$
The <divergence theorem> identifies the change in probability inside a region with minus the outward <probability current>. Decay at infinity, or appropriate probability-conserving <boundary conditions>, preserves <wavefunction normalization>. Conservation of total probability does not require a stationary <probability density>.
Its usefulness comes from being a <linear partial differential equation>: <quantum superpositions> of solutions are solutions, while different <quantum phases> can produce <interference>. For a fixed <Hamiltonian operator>, the <time-independent Schrödinger equation> finds the <energy eigenstates>; each then acquires just its own time-dependent <quantum phase>. One <eigenvalue problem> therefore organizes both allowed <energies> and the evolution of arbitrary <quantum superpositions>, with a <continuous spectrum> included when required. This is the attractive bridge between a spatial wave problem and a clock: <energy> differences determine the rate at which relative <quantum phases> change. It makes atomic <spectral lines>, <quantum tunnelling> and <quantum scattering> calculable from a specified model.
For an explicit example without confinement, take a <free particle> on the whole line, $V=0$, initially in a <Gaussian wave packet> with position <standard deviation> $\sigma_0$ and mean <momentum> $\hbar k_0$:
$$
\psi(x,0)=\frac{e^{-x^2/(4\sigma_0^2)+ik_0x}}{(2\pi\sigma_0^2)^{1/4}},\qquad
t_s=\frac{2m\sigma_0^2}{\hbar},\quad \tau=\frac{t}{t_s},\quad v_0=\frac{\hbar k_0}{m}.
$$
Unlike a single <plane wave>, this <wavefunction> satisfies <wavefunction normalization>. A <Fourier transform> writes it as a <quantum superposition> of <plane waves>. Their <dispersion relation> is $\omega(k)=\hbar k^2/(2m)$, so each Fourier component gains the factor $e^{-i\hbar k^2t/(2m)}$. Completing the square in the resulting <Gaussian integral> gives
$$
\psi(x,t)=\frac{1}{(2\pi\sigma_0^2)^{1/4}\sqrt{1+i\tau}}
\exp\!\left[-\frac{(x-v_0t)^2}{4\sigma_0^2(1+i\tau)}+ik_0x-\frac{i\hbar k_0^2t}{2m}\right],
$$
with the square root chosen continuously from $t=0$. Its position <probability density> and <standard deviation> are
$$
\rho(x,t)=\frac{\exp[-(x-v_0t)^2/(2\sigma_t^2)]}{\sqrt{2\pi}\,\sigma_t},\qquad
\sigma_t=\sigma_0\sqrt{1+\tau^2}.
$$
The centre moves with the <group velocity> $\omega'(k_0)=v_0$ and the <wave packet> spreads because different <wavenumbers> have different <group velocities>. This is unitary <wave dispersion>, without a random <force> or loss of probability. Independently, the initial <momentum> <standard deviation> is $\Delta p=\hbar/(2\sigma_0)$ and the initial position-momentum <covariance> vanishes. Free evolution gives $x(t)=x(0)+p(0)t/m$ in the <Heisenberg picture>, hence
$$
\sigma_t^2=\sigma_0^2+\frac{t^2}{m^2}(\Delta p)^2
=\sigma_0^2+\frac{\hbar^2t^2}{4m^2\sigma_0^2}.
$$
This independently checks the spreading law. The <Gaussian integral> of $\rho$ is one at every time; the decreasing peak is exactly offset by increasing width.
As hypothetical preparation parameters, choose a free <electron> with $\sigma_0=1\,\mathrm{nm}$ and $k_0=2\,\mathrm{nm}^{-1}$. Using $\hbar\simeq1.054571817\times10^{-34}\,\mathrm{J\,s}$ and $m\simeq9.1093837\times10^{-31}\,\mathrm{kg}$ gives
$$
t_s\simeq17.276\,\mathrm{fs},\qquad
v_0\simeq2.31535\times10^5\,\mathrm{m\,s^{-1}},\qquad
\Delta p\simeq5.27286\times10^{-26}\,\mathrm{kg\,m\,s^{-1}}.
$$
The rounded <electron> <mass> uses the 2022 adjustment.<CODATA Value: electron mass by National Institute of Standards and Technology>[Electron mass, 2022 CODATA adjustment; rounded here to eight significant figures]{ref} At $t=t_s$, the centre is $4\,\mathrm{nm}$ and $\sigma_t=\sqrt2\,\mathrm{nm}$; at $t=2t_s$, they are $8\,\mathrm{nm}$ and $\sqrt5\,\mathrm{nm}$. The mean <kinetic energy> stays
$$
\langle H\rangle=\frac{\hbar^2}{2m}\left(k_0^2+\frac{1}{4\sigma_0^2}\right)\simeq0.1619\,\mathrm{eV}.
$$
It includes both mean motion and the <momentum> spread. The mean speed is about $7.72\times10^{-4}c$ and the <momentum> spread corresponds to $1.93\times10^{-4}c$, making the nonrelativistic model accurate for the appreciably occupied part of this ideal <Gaussian distribution>. Its mathematically unbounded tails are an idealization.
\Image[/schrodinger-free-gaussian-spreading.png]
{title=Translation and spreading of a free <Gaussian wave packet>}
{description=Calculated <electron> position <probability densities> for the hypothetical $1\,\mathrm{nm}$ preparation above, together with the position <standard deviation>. The area under every curve is one. The dashed width curve illustrates pure translation without spreading for comparison; it is not the free solution. No curve represents measured data.}
{source=schrodinger-free-gaussian-spreading.py}
The complex width also explains where the probability goes. Write $\psi=\sqrt\rho\,e^{i\theta}$, where $\theta$ is the dimensionless <quantum phase>. Differentiating the exact solution gives
$$
\partial_x\theta=k_0+\frac{\tau(x-v_0t)}{2\sigma_0^2(1+\tau^2)},\qquad
\frac{j}{\rho}=\frac{\hbar}{m}\partial_x\theta
=v_0+\frac{t}{t_s^2+t^2}(x-v_0t).
$$
Relative to the moving centre, the <probability current> $j_{\mathrm{rel}}=j-v_0\rho$ points outward on both sides for $t>0$. This is a flux statement, not an additional assumption about individual particle trajectories. At $t=t_s$ and $x-v_0t=\pm\sigma_0$, the relative flow speeds $j_{\mathrm{rel}}/\rho$ are $\pm\sigma_0/(2t_s)$, about $\pm2.8942\times10^4\,\mathrm{m\,s^{-1}}$. The probability within a moving interval of half-width $\sigma_0$ falls from about $0.6827$ initially to $0.5205$ at $t_s$, exactly because of this outward <probability current>.
\Image[/schrodinger-gaussian-phase-current.png]
{title=Curved <quantum phase> drives spreading <probability current>}
{description=The same free <Gaussian wave packet> at $t=t_s$. With $y=(x-v_0t)/\sigma_0$, removing the carrier $k_0x$ and a spatially constant <quantum phase> leaves $y^2/8$. The right panel plots $t_sj_{\mathrm{rel}}=(\sigma_0\rho)y/2$: negative to the left, positive to the right. Flux through the marked moving interval endpoints explains its decreasing probability.}
{source=schrodinger-gaussian-phase-current.py}
Historically, the 1926 <Quantisierung als Eigenwertproblem (Erste Mitteilung) by Erwin Schrödinger> began with the stationary <hydrogen atom> problem. It distinguishes discrete negative <energies> from a positive <continuous spectrum>; it is not yet the general evolution equation displayed above.<Quantisierung als Eigenwertproblem (Erste Mitteilung) by Erwin Schrödinger>[Printed pp. 361–363, especially p. 362: discrete and continuous spectrum]{ref} The opening of <quantum mechanics> discusses its original vibrating-string analogy.
In the 12 December 1933 lecture <Nobel Lectures, Physics 1922–1941/The fundamental idea of wave mechanics by Erwin Schrödinger>, <Erwin Schrödinger> explains the move from independent rays to waves through <diffraction>, then applies that analogy to <atomic physics>. His closing pages explicitly acknowledge the difficulty of reconciling wave language with observed particle tracks. This retrospective physical argument motivates the model; it does not derive the modern <Born rule> from <classical mechanics>.<Nobel Lectures, Physics 1922–1941/The fundamental idea of wave mechanics by Erwin Schrödinger>[Nobel Lectures, Physics 1922–1941, printed pp. 313–316; inspected in the Columbia University mirror]{ref}
The dates separate proposal from measurement: the <de Broglie wavelength> belongs to the 1924 thesis, the <Schrödinger equation> to 1926, and the <Davisson–Germer experiment>'s <electron diffraction> beams to 1927.<The discovery of electron waves by Clinton J. Davisson>[Printed pp. 389–390: 1924 proposal, deliberate 1926 search and early-1927 diffraction discovery]{ref} That linked experiment describes the prepared <nickel> crystal, evacuated <electron> gun, angular collector scans, retarding-potential selection and heating controls. Its measured collector <electric current> tests wave-dependent scattering probabilities; it neither measures a <wavefunction> directly nor establishes this equation uniquely.
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