Time-Dependent Schrödinger Equation

The time-dependent Schrödinger equation is the closed-system equation of motion for a quantum state:

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

It says that the Hamiltonian generates time translations.

First-Order Time Evolution

The equation is first order in time. Given an initial state $\lvert\psi(t_0)\rangle$ and a Hamiltonian $H(t)$, it determines the later state within the domain where the Hamiltonian describes the system.

This is different from the time-independent Schrödinger equation,

$$H\lvert E\rangle=E\lvert E\rangle,$$

which is an eigenvalue problem. The eigenvalue equation is a tool for solving time evolution when $H$ is time independent; it is not the whole dynamics.

For an unbounded Hamiltonian, the differential equation also has a domain statement. A self-adjoint, time-independent $H$ defines unitary evolution for every Hilbert-space vector, but a state must lie in $D(H)$ for $H|\psi\rangle$ and the strong time derivative to exist at the instant in question. In coordinate representation, boundary and matching conditions are part of this domain. Less regular vectors may still evolve unitarily even when the TDSE is not an ordinary pointwise PDE.

Hamiltonian as Generator

For a short time step $dt$, the equation implies

$$\lvert\psi(t+dt)\rangle = \left(I-\frac{i}{\hbar}H(t)\,dt\right) \lvert\psi(t)\rangle +O(dt^2).$$

The operator multiplying the state is an infinitesimal time translation. This is the sense in which $H$ generates time evolution.

Norm Preservation

If $H(t)$ is self-adjoint, Schrödinger evolution preserves the norm. Using the equation and its adjoint,

$$\frac{d}{dt} \langle\psi(t)\rvert\psi(t)\rangle = \frac{i}{\hbar} \langle\psi(t)\rvert H(t)\lvert\psi(t)\rangle - \frac{i}{\hbar} \langle\psi(t)\rvert H(t)\lvert\psi(t)\rangle =0.$$

Thus closed-system time evolution preserves total probability. Coordinate-space probability current is a representation-specific form of the same conservation law.

Time-Independent Hamiltonians

When $H$ does not depend on time,

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

and

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

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

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

The phases of different energy components are what produce beats, oscillations, and time-dependent expectation values.

Time-Dependent Hamiltonians

When $H(t)$ depends on time, the equation still holds:

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

But the solution is not usually a simple exponential of the integral of $H(t)$. If

$$[H(t_1),H(t_2)]\ne0,$$

then operator order matters, and time ordering is required.

Under the usual existence assumptions, the formal solution is

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

If $[H(t_1),H(t_2)]=0$ throughout the interval, the time-ordering symbol may be dropped. In either case, the propagator obeys

$$U(t_2,t_0) = U(t_2,t_1)U(t_1,t_0),$$

and is unitary when the time-dependent generator is self-adjoint with a suitable common-domain or propagator construction. Time Ordering and the Dyson Expansion develop these qualifications.

Coordinate Representation

Taking the overlap with a position eigenket gives the wave-mechanics form. For a particle in a scalar potential,

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

The coordinate-space page explains boundary conditions, probability current, and wavefunction interpretation in more detail.

Two-Level Example

For

$$H=\frac{\hbar\omega}{2}\sigma_z,$$

an initial state

$$\lvert\psi(0)\rangle =c_+\lvert+\rangle+c_-\lvert-\rangle$$

evolves as

$$\lvert\psi(t)\rangle =c_+e^{-i\omega t/2}\lvert+\rangle +c_-e^{i\omega t/2}\lvert-\rangle.$$

The overall phase is not observable, but the relative phase between the two components is observable.

Energy and Observable Balance

Norm conservation does not imply energy conservation. For a state satisfying the TDSE and the required domain assumptions,

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

Thus a closed system can remain perfectly unitary while a time-dependent drive exchanges energy with it. More generally, an observable $A(t)$ satisfies

$$\frac{d}{dt}\langle A\rangle = \frac{i}{\hbar} \langle[H,A]\rangle + \left\langle \frac{\partial A}{\partial t} \right\rangle.$$

This identity links Schrödinger evolution to the Heisenberg equation and Ehrenfest dynamics. Adding $C(t)I$ to the Hamiltonian only multiplies every state by a common phase and does not change closed-system expectation values.

Numerical Propagation

Numerical methods should approximate the unitary structure rather than only the differential equation. Common choices include:

Method	Natural setting	Main audit
Crank–Nicolson	fixed Hermitian matrix from a spatial discretization	linear-solve and spatial convergence
split operator	$H=T+V$ with efficient Fourier transforms	time-step error from $[T,V]\ne0$
Krylov exponential action	large sparse Hamiltonians	Krylov dimension and time-step convergence
spectral propagation	sufficiently complete eigensystem	basis truncation

Forward Euler,

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

is not unitary even for a Hermitian $H$ and generally produces norm drift. A defensible calculation reports norm error, time-step convergence, spatial or basis convergence, energy conservation when $\partial_tH=0$, and comparison with an exact limit.

Scope and Limits

The TDSE is a general state equation, but the scalar one-particle PDE is not a universal model. Spin and internal levels require multicomponent states; vector potentials change the coordinate Hamiltonian and current; open systems usually require density operators and master equations; relativistic particle creation requires quantum fields. These are changes of state space or dynamics, not corrections obtained by silently reusing the scalar free-particle equation.

Common Mistakes

• Treating every solution as an energy eigenstate.
• Confusing the time-dependent and time-independent Schrödinger equations.
• Dropping the factor of $i$.
• Using $e^{-iH(t-t_0)/\hbar}$ when $H$ is time dependent.
• Confusing global phase with relative phase.
• Forgetting that a time-dependent Hamiltonian need not conserve energy.
• Assuming that norm conservation implies energy conservation.
• Choosing coordinate initial data that violate the Hamiltonian’s boundary conditions.
• Treating forward Euler as a unitary propagator.

Cross-Links

• Hamiltonians as Generators
• Time as a Parameter
• Time-Evolution Operator
• Time-Dependent Schrödinger Equation in Coordinate Space
• Schrödinger Equation Formula Card
• Commutators

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.

Exercises

1. Show that a single energy eigenstate changes only by a phase when $H$ is time independent.

Solution

If $H\lvert E\rangle=E\lvert E\rangle$, then

$$\lvert\psi(t)\rangle =e^{-iE(t-t_0)/\hbar}\lvert E\rangle$$

satisfies the time-dependent Schrödinger equation. Its ray is unchanged because the time dependence is a global phase for that single component.

2. A state is a superposition of two nondegenerate energy eigenstates. Show explicitly why its position-space density can depend on time even though each eigenstate is stationary.

Solution

For

$$\psi(x,t) = c_1\psi_1(x)e^{-iE_1t/\hbar} +c_2\psi_2(x)e^{-iE_2t/\hbar},$$

$|\psi|^2$ contains a cross term proportional to

$$c_1^*c_2 \psi_1^*(x)\psi_2(x) e^{-i(E_2-E_1)t/\hbar}$$

and its complex conjugate. Unless the overlap vanishes pointwise or $E_1=E_2$, the changing relative phase produces a time-dependent density.

3. Derive $d\langle H\rangle/dt=\langle\partial_tH\rangle$ for a time-dependent self-adjoint Hamiltonian.

Solution

Differentiate $\langle\psi|H|\psi\rangle$. The derivatives of the bra and ket contribute

$$\frac{i}{\hbar}\langle H^2\rangle - \frac{i}{\hbar}\langle H^2\rangle =0,$$

while the explicit operator derivative contributes $\langle\partial_tH\rangle$.

4. State a sufficient condition for replacing the time-ordered exponential by an ordinary exponential of $\int H(t)dt$.

Solution

It is sufficient that

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

for every pair of times in the interval. Then all infinitesimal evolution factors commute, so their chronological order does not affect the product.

5. Explain why checking only norm conservation is insufficient to validate a numerical propagation with a time-independent Hamiltonian.

Solution

A method can preserve or renormalize the norm while accumulating phase, dispersion, spatial-discretization, or basis-truncation errors. One should also test time-step and spatial convergence, energy conservation, reversibility, and an exact benchmark such as an energy eigenstate or a free Gaussian packet.