Schrödinger Equation

Formula

In the Schrödinger picture, a closed-system pure state obeys

$$i\hbar \frac{d}{dt} \lvert\psi(t)\rangle = H(t)\lvert\psi(t)\rangle.$$

Given

$$\lvert\psi(t_0)\rangle = \lvert\psi_0\rangle,$$

the solution is

$$\lvert\psi(t)\rangle = U(t,t_0) \lvert\psi_0\rangle,$$

where

$$i\hbar \frac{\partial}{\partial t} U(t,t_0) = H(t)U(t,t_0), \qquad U(t_0,t_0)=I.$$

For a time-independent Hamiltonian,

$$U(t,t_0) = \exp\left[ -\frac{i}{\hbar} H(t-t_0) \right].$$

For a general time-dependent Hamiltonian,

$$U(t,t_0) = \mathcal T \exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t} H(s)\,ds \right].$$

At a Glance

Setting	Equation
Abstract pure state	$i\hbar\,d\lvert\psi\rangle/dt=H(t)\lvert\psi\rangle$
Fixed-basis coefficient vector	$i\hbar\,\dot{\mathbf c}(t)=H(t)\mathbf c(t)$
Position-space wavefunction	$i\hbar\,\partial_t\psi(\mathbf r,t)=(H\psi)(\mathbf r,t)$
Spinless particle in a scalar potential	$i\hbar\,\partial_t\psi=[-\hbar^2\nabla^2/(2m)+V]\psi$
Density operator	$i\hbar\,\dot\rho=[H,\rho]$
Stationary eigenvalue problem	$H\phi_n=E_n\phi_n$
Stationary-state solution	$\lvert\psi_n(t)\rangle=e^{-iE_n(t-t_0)/\hbar}\lvert E_n\rangle$

The time-dependent equation is the dynamical law. The time-independent equation is an eigenvalue problem used when $H$ has no explicit time dependence.

Meaning

The Schrödinger equation determines the state curve once the Hamiltonian, initial time, and initial state are specified. It is:

• linear in the state;
• first order in time;
• norm preserving when the evolution is generated by a suitable self-adjoint Hamiltonian;
• representation independent in its abstract form.

The equation does not construct the Hamiltonian, choose a measurement, assign an outcome, or model general irreversible open-system dynamics. Those require additional physical input.

Initial-Value Character

Because the equation is first order in time, one initial state is specified:

$$\lvert\psi(t_0)\rangle = \lvert\psi_0\rangle.$$

An independent value of $d\lvert\psi\rangle/dt$ is not supplied; the Hamiltonian determines it. Coordinate-space equations can be second order in space, so spatial boundary conditions remain necessary even though only one time datum is required.

For a time-independent self-adjoint $H$, the propagator satisfies

$$U(t_2,t_1)U(t_1,t_0) = U(t_2,t_0),$$ $$U^\dagger(t,t_0) = U^{-1}(t,t_0) = U(t_0,t).$$

Coordinate-Space Form

The position-space wavefunction is

$$\psi(\mathbf r,t) = \langle\mathbf r\rvert \psi(t)\rangle.$$

For a spinless particle of mass $m$ in a scalar potential,

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

and

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

In one dimension,

$$i\hbar \frac{\partial\psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2\psi(x,t)}{\partial x^2} +V(x,t)\psi(x,t).$$

This familiar local PDE assumes a nonrelativistic spinless particle, Cartesian coordinates, the standard kinetic term, and no vector potential. It must be accompanied by a coordinate domain, spatial boundary conditions, and an initial wavefunction.

For a nonlocal Hamiltonian kernel,

$$H(\mathbf r,\mathbf r';t) = \langle\mathbf r\rvert H(t) \lvert\mathbf r'\rangle,$$

the equation becomes

$$i\hbar \frac{\partial\psi(\mathbf r,t)}{\partial t} = \int H(\mathbf r,\mathbf r';t) \psi(\mathbf r',t) \,d^dr'.$$

The local differential equation is therefore a special representation, not the definition of Schrödinger evolution.

Electromagnetic Coupling

For a spinless particle of charge $q$ in electromagnetic potentials,

$$H(t) = \frac{1}{2m} \left[ -i\hbar\nabla -q\mathbf A(\mathbf r,t) \right]^2 +q\phi(\mathbf r,t).$$

The wavefunction equation is

$$i\hbar\partial_t\psi = \left\lbrace \frac{1}{2m} \left[ -i\hbar\nabla -q\mathbf A \right]^2 +q\phi \right\rbrace \psi.$$

The canonical momentum $-i\hbar\nabla$ and kinetic momentum $-i\hbar\nabla-q\mathbf A$ are different. Gauge transformations change the potentials and wavefunction phase together while leaving physical predictions invariant.

Spin adds internal components and spin-dependent terms to the Hamiltonian. The abstract Schrödinger equation remains valid for the enlarged state space.

Matrix Form

In a fixed orthonormal basis,

$$\lvert\psi(t)\rangle = \sum_n c_n(t)\lvert e_n\rangle,$$

the coefficients satisfy

$$i\hbar \dot c_m(t) = \sum_n H_{mn}(t)c_n(t),$$

with

$$H_{mn}(t) = \langle e_m\rvert H(t) \lvert e_n\rangle.$$

In vector notation,

$$i\hbar \dot{\mathbf c}(t) = H(t)\mathbf c(t).$$

If the basis depends on time,

$$\lvert\psi(t)\rangle = \sum_n c_n(t)\lvert e_n(t)\rangle,$$

then

$$i\hbar\dot c_m = \sum_n \left[ H_{mn} -i\hbar \langle e_m\rvert \dot e_n\rangle \right] c_n.$$

The connection term from the moving basis is essential. Omitting it changes the physical evolution.

Time-Independent Hamiltonians

If

$$H\lvert E_n\rangle = E_n\lvert E_n\rangle,$$

then

$$\lvert\psi_n(t)\rangle = e^{-iE_n(t-t_0)/\hbar} \lvert E_n\rangle$$

solves the time-dependent equation. For

$$\lvert\psi(t_0)\rangle = \sum_n c_n\lvert E_n\rangle,$$

the solution is

$$\lvert\psi(t)\rangle = \sum_n c_n e^{-iE_n(t-t_0)/\hbar} \lvert E_n\rangle.$$

Continuous spectral sectors require corresponding integrals and normalization measures.

The energy probabilities $\lvert c_n\rvert^2$ remain constant, while relative phases evolve. Those relative phases can make observables and probability densities time dependent.

Stationary Equation

For a time-independent Hamiltonian, separated solutions have

$$\psi_n(\mathbf r,t) = \phi_n(\mathbf r) e^{-iE_n(t-t_0)/\hbar},$$

where

$$H\phi_n = E_n\phi_n.$$

The equation

$$H\phi=E\phi$$

is the time-independent Schrödinger equation. It determines spectral values and eigenfunctions together with the Hamiltonian domain and boundary conditions. It is not a replacement for the time-dependent law.

A superposition confined to one degenerate energy eigenspace is stationary because every component acquires the same phase. A superposition of distinct energies generally is not stationary.

Density-Operator Form

For a pure-state projector

$$\rho(t) = \lvert\psi(t)\rangle \langle\psi(t)\rvert,$$

or for a general closed-system density operator, unitary evolution gives

$$i\hbar \frac{d\rho}{dt} = [H,\rho].$$

For time-independent $H$,

$$\rho(t) = U(t,t_0) \rho(t_0) U^\dagger(t,t_0).$$

This von Neumann equation is the density-operator counterpart of the Schrödinger equation. General open-system evolution usually requires a quantum channel or master equation with additional terms.

Time Ordering

If

$$[H(t),H(t')] \neq0$$

for some times, then

$$\exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t}H(s)\,ds \right]$$

without time ordering is generally incorrect. The symbol $\mathcal T$ orders later-time Hamiltonians to the left.

If

$$[H(t),H(t')]=0$$

for all relevant times, the ordinary exponential is valid. Piecewise-constant Hamiltonians must be multiplied in chronological order even when each interval has a simple exponential.

Conservation Laws

For a suitable self-adjoint $H(t)$,

$$\frac{d}{dt} \langle\psi(t)\rvert \psi(t)\rangle = 0.$$

Thus normalization is preserved. In coordinate space, the same statement appears as a continuity equation with boundary flux.

Energy expectation obeys

$$\frac{d}{dt} \langle H\rangle = \left\langle \frac{\partial H}{\partial t} \right\rangle.$$

Norm conservation therefore does not imply energy conservation. A self-adjoint time-dependent Hamiltonian generates unitary evolution while exchanging energy with an external drive.

For a time-independent Hamiltonian,

$$\frac{d}{dt} \langle H\rangle =0.$$

Additive Energy Shifts

Replacing

$$H(t) \longmapsto H(t)+C(t)I$$

multiplies every closed-system state by

$$\exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t}C(s)\,ds \right].$$

This common phase does not change closed-system probabilities or expectation values. Energy differences and relative phases between differently shifted sectors remain physically relevant.

Symbols

Symbol	Mathematical type	Meaning
$\lvert\psi(t)\rangle$	State vector	Schrödinger-picture pure state
$\psi(\mathbf r,t)$	Complex function	Position representation of the state
$H(t)$	Self-adjoint operator	Hamiltonian
$U(t,t_0)$	Unitary operator	Time-evolution operator
$\mathcal T$	Ordering operation	Chronological time ordering
$E_n$	Real scalar	Energy eigenvalue
$\lvert E_n\rangle$	Eigenvector	Energy eigenstate
$\rho(t)$	Positive trace-one operator	General quantum state
$m$	Positive scalar	Particle mass
$V$, $\phi$, $\mathbf A$	Functions or multiplication operators	Scalar, electric, and vector potentials

Hats on $H$, $X$, and $P$ are optional typography conventions. They should be used consistently within one calculation.

Units and Dimensions

Every term in the Schrödinger equation has units of energy times the state:

$$\left[ i\hbar \frac{\partial\psi}{\partial t} \right] = [H\psi] = \text{energy}\times[\psi].$$

The ratio

$$\frac{H}{\hbar}$$

has angular-frequency units. In SI,

$$[\hbar]=\mathrm{J\,s}, \qquad [H]=\mathrm J.$$

In one-dimensional coordinate normalization,

$$[\psi(x,t)] = L^{-1/2}.$$

Natural-unit conventions can hide $\hbar$ or express $H$ directly in frequency units. Restore the convention before numerical substitution.

Domain and Regularity Conditions

For a time-independent unbounded self-adjoint $H$, the unitary operator

$$e^{-iH(t-t_0)/\hbar}$$

acts on every Hilbert-space vector. The strong differential equation

$$i\hbar \frac{d}{dt} \lvert\psi(t)\rangle = H\lvert\psi(t)\rangle$$

requires the initial state to lie in $\mathcal D(H)$. Less regular states can still have well-defined unitary evolution while requiring a weak or integral interpretation of the differential equation.

For $H(t)$, well-posed evolution requires suitable regularity and compatibility of the time-dependent self-adjoint domains. Finite-dimensional matrix equations hide these issues.

In coordinate space, the differential expression, spatial domain, interface rules, and boundary conditions together define the Hamiltonian. Square integrability alone does not make arbitrary initial data admissible.

Assumptions

• The state belongs to a closed quantum system or to an effective closed sector.
• The Hamiltonian and its domain are specified.
• Self-adjointness and regularity are sufficient to generate unitary evolution.
• The initial condition lies in the appropriate Hilbert space and satisfies boundary conditions.
• The scalar coordinate equation assumes a spinless nonrelativistic particle with the displayed kinetic and potential terms.
• Time-ordering is retained unless Hamiltonians commute at different times.
• A chosen representation uses consistent basis, Fourier, gauge, and unit conventions.

Validity and Limitations

The abstract equation applies broadly to closed quantum systems, including spin and many-body systems, when $H$ acts on the appropriate Hilbert space. The simple scalar PDE does not by itself include:

• spin-dependent interactions;
• electromagnetic vector potentials;
• relativistic single-particle structure;
• particle creation and annihilation;
• irreversible open-system dynamics;
• stochastic measurement records;
• nonlinear mean-field approximations unless explicitly adopted as an effective model.

A non-Hermitian effective Hamiltonian can be useful for conditional decay or resonance calculations, but it generally does not preserve norm. Its interpretation and normalization rules must be stated; it is not ordinary closed-system Schrödinger evolution.

Numerical Checks

• Norm should remain constant for closed self-adjoint evolution.
• Energy expectation should remain constant when $H$ is time independent.
• The state at $t=t_0$ must reproduce the initial condition.
• Propagators must satisfy composition and unitarity within numerical tolerance.
• A stationary eigenstate should evolve only by its phase.
• Results should converge under time-step, grid, box, or basis refinement.
• A discretized Hamiltonian should be Hermitian under the implemented inner product.
• Time-ordering errors should be tested when $H(t)$ changes or fails to commute at different times.
• Boundary flux should agree with changes in probability inside a coordinate region.

Forward Euler,

$$\psi(t+\Delta t) \approx \left( I-\frac{i\Delta t}{\hbar}H \right) \psi(t),$$

is not exactly unitary even for Hermitian $H$. Norm preservation from the continuum equation does not guarantee norm preservation by an arbitrary discretization.

Minimal Worked Use

Suppose

$$\lvert\psi(t_0)\rangle = c_1\lvert E_1\rangle +c_2\lvert E_2\rangle,$$

where $H$ is time independent. Then

$$\lvert\psi(t)\rangle = c_1 e^{-iE_1(t-t_0)/\hbar} \lvert E_1\rangle + c_2 e^{-iE_2(t-t_0)/\hbar} \lvert E_2\rangle.$$

The energy probabilities remain $\lvert c_1\rvert^2$ and $\lvert c_2\rvert^2$, but the relative phase evolves at angular frequency

$$\omega_{21} = \frac{E_2-E_1}{\hbar}.$$

Observables with matrix elements between the two energy sectors can therefore oscillate even though the Hamiltonian is time independent.

Derivation and Canonical Home

Schrödinger Equation owns the abstract initial-value law, norm preservation, domains, representation changes, and relation to unitary evolution.

Time-Dependent Schrödinger Equation owns the coordinate-space PDE and practical propagation. Time-Independent Schrödinger Equation owns the stationary boundary-value problem.

Worked Examples

• Free Particle
• Gaussian Wave Packets
• Infinite Square Well
• Quantum Harmonic Oscillator
• Two-State Hamiltonians

Common Mistakes

• Treating $H\phi=E\phi$ as the universal time-evolution equation.
• Forgetting the phase $e^{-iEt/\hbar}$ of an energy eigenstate.
• Assuming every superposition of stationary states has stationary probabilities.
• Replacing a time-ordered exponential with an ordinary exponential.
• Omitting the moving-basis connection term.
• Ignoring the Hamiltonian domain, boundary conditions, or interface conditions.
• Assuming a time-dependent self-adjoint Hamiltonian fails to preserve norm.
• Assuming norm conservation implies energy conservation.
• Using the scalar-potential PDE in a problem with spin or a vector potential.
• Confusing canonical and kinetic momentum under electromagnetic coupling.
• Supplying an independent initial time derivative to a first-order equation.
• Expecting an arbitrary time-stepping method to preserve unitarity.

Related Formulas

• Time-Evolution Operator
• Propagator Composition
• Heisenberg Equation
• Interaction-Picture Evolution
• Hamiltonian Operator Symbol
• Probability Current
• Lindblad Equation

References

• E. Schrödinger, “Quantisierung als Eigenwertproblem,” Annalen der Physik 384, 361–376 (1926), and subsequent papers in the series.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 4 and 7.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, ch. 2.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, rev. ed., Academic Press, 1980, sec. VIII.7.