Time-Dependent Schrödinger Equation in Coordinate Space

The time-dependent Schrödinger equation (TDSE) is the nonrelativistic equation of motion for a quantum state. In coordinate space it is an initial-value PDE:

$$i\hbar\frac{\partial\psi(\mathbf r,t)}{\partial t} = \hat H(t)\psi(\mathbf r,t).$$

This page is the canonical owner of the coordinate-space law: how the Hamiltonian acts on a wavefunction, which initial and boundary data define a problem, how probability conservation is checked, and where the nonrelativistic one-particle description ceases to apply. Abstract propagator theory, time ordering, pictures of motion, and open-system evolution belong to their broader dynamics treatments.

Required background. Coordinate Representation supplies $\psi(\mathbf r,t)=\langle\mathbf r|\psi(t)\rangle$. Wavefunctions and Probability Density supplies the Born interpretation of $|\psi|^2$.

Helpful background. Self-Adjoint Operators supplies the operator-domain condition behind probability-preserving evolution.

The coordinate-space evolution law

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

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

In one dimension this becomes

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

Every term has units of energy times wavefunction. The equation is linear, so if $\psi_1$ and $\psi_2$ solve the same equation with the same Hamiltonian, then $c_1\psi_1+c_2\psi_2$ is also a solution. This superposition principle does not say that solutions of two different potentials can be added to solve either original problem.

The familiar differential expression assumes a nonrelativistic particle with the standard quadratic kinetic energy and a sufficiently regular local scalar potential. Spin, position-dependent mass, curved coordinates, nonlocal interactions, or effective band Hamiltonians change the coordinate operator and sometimes the inner-product measure.

Initial data, spatial domain, and operator domain

Because the TDSE is first order in time, one specifies a single initial state,

$$\psi(\mathbf r,t_0)=\psi_0(\mathbf r).$$

The value of $\partial_t\psi$ at $t_0$ is then fixed by the equation; it is not independent initial data. The common kinetic term is second order in space, so the spatial problem also needs endpoint, interface, periodic, asymptotic, or regularity conditions.

A complete specification contains

$$(\mathcal H,\,D,\,\hat H(t),\,\psi_0),$$

not merely the displayed differential expression. Here $\mathcal H$ fixes the measure and state components, while a common invariant domain $D\subseteq D(\hat H(t))$ fixes the admissible spatial behavior over the time interval under consideration. A hard-wall wavefunction, a periodic wavefunction, and a wavefunction on the full line can share the expression $-\hbar^2\partial_x^2/(2m)$ while belonging to different Hamiltonians.

For a time-independent self-adjoint Hamiltonian, the unitary evolution

$$|\psi(t)\rangle = e^{-i\hat H(t-t_0)/\hbar}|\psi(t_0)\rangle$$

exists for every Hilbert-space state. If $|\psi(t_0)\rangle\in D(\hat H)$, the state is differentiable in norm and satisfies the TDSE as a strong equation. For a general square-integrable state outside $D(\hat H)$, the unitary evolution still exists, but the pointwise differential equation may not be a legitimate statement at every instant.

For a time-dependent unbounded Hamiltonian, self-adjointness at each frozen time is not by itself a complete existence theorem. A shared domain and sufficient regularity are standard working assumptions; changing domains or singular time dependence requires more careful propagator theory.

Probability conservation

For the standard kinetic term and a real scalar potential, define

$$\rho=|\psi|^2, \qquad \mathbf j = \frac{\hbar}{2mi} \left( \psi^*\nabla\psi -\psi\nabla\psi^* \right).$$

The TDSE and its complex conjugate imply

$$\partial_t\rho+\nabla\cdot\mathbf j=0.$$

Integrating over a fixed region $\Omega$ gives

$$\frac{d}{dt}\int_\Omega\rho\,d^3r = -\int_{\partial\Omega}\mathbf j\cdot d\mathbf S.$$

Thus probability inside a region changes through boundary flux. Total norm is conserved when the Hamiltonian generates unitary evolution and the boundary or asymptotic conditions eliminate net probability leakage from the modeled space. Probability Current and the Continuity Equation own the detailed derivation and interpretation.

A complex absorbing potential or another non-Hermitian effective term can produce a source or sink. That can be intentional—for example, to emulate an open numerical boundary—but it is not closed-system Schrödinger evolution.

Time-independent and time-dependent Hamiltonians

If $\hat H$ is time independent and has an energy eigenfunction $\hat H\varphi_n=E_n\varphi_n$, then

$$\psi_n(\mathbf r,t) = \varphi_n(\mathbf r)e^{-iE_nt/\hbar}$$

solves the TDSE. A general state is assembled from the discrete and continuum spectral sectors with their appropriate measures. The stationary eigenvalue problem and its boundary-value character are developed on Time-Independent Schrödinger Equation.

If $\hat H(t)$ depends explicitly on time, one generally cannot replace the evolution by $e^{-i\hat Ht/\hbar}$ or attach only a dynamical phase to an instantaneous eigenvector. Hamiltonians at different times need not commute, and the eigenbasis itself may move.

For a normalized solution and sufficiently regular self-adjoint $\hat H(t)$, the energy expectation satisfies

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

Energy is conserved when the Hamiltonian has no explicit time dependence. Norm conservation and energy conservation are different statements: a time-dependent Hermitian potential can change the mean energy while preserving the norm exactly.

Electromagnetic coupling and gauge covariance

For charge $q$ in electromagnetic potentials $(\Phi,\mathbf A)$, minimal coupling gives

$$i\hbar\,\partial_t\psi = \left[ \frac{1}{2m} \left(-i\hbar\nabla-q\mathbf A\right)^2 +q\Phi+V \right]\psi.$$

The conserved current is then

$$\mathbf j = \frac{1}{m} \operatorname{Re} \left[ \psi^*\left(-i\hbar\nabla-q\mathbf A\right)\psi \right].$$

Under

$$\mathbf A' = \mathbf A+\nabla\chi, \qquad \Phi' = \Phi-\partial_t\chi, \qquad \psi'=e^{iq\chi/\hbar}\psi,$$

the equation, density, and physical current are unchanged. The detailed operator ordering, gauge transformation, and examples belong to Minimal Coupling in Wave Mechanics.

Example: free Gaussian spreading

Take the normalized initial packet

$$\psi(x,0) = \frac{1}{(2\pi\sigma_0^2)^{1/4}} \exp\left[ -\frac{(x-x_0)^2}{4\sigma_0^2}+ik_0x \right].$$

For the free Hamiltonian $-\hbar^2\partial_x^2/(2m)$, its probability density at time $t$ remains Gaussian:

$$|\psi(x,t)|^2 = \frac{1}{\sqrt{2\pi}\,\sigma_t} \exp\left[ -\frac{(x-x_0-v_0t)^2}{2\sigma_t^2} \right],$$

where

$$v_0=\frac{\hbar k_0}{m}, \qquad \sigma_t = \sigma_0 \sqrt{1+\left(\frac{\hbar t}{2m\sigma_0^2}\right)^2}.$$

The center moves at the group velocity while the width grows. The integral of the density remains one. The spreading time $2m\sigma_0^2/\hbar$ is a useful scale and a direct dimensional check.

Numerical and analytical verification

A credible TDSE solution should report checks appropriate to its method:

• initial data are reproduced at $t=t_0$;
• boundary and interface conditions remain satisfied;
• the physical norm and, when applicable, the energy have the expected time behavior;
• the integral probability balance agrees with the measured boundary flux;
• results converge as the time step, grid spacing, domain size, or basis cutoff is refined;
• limiting cases reproduce a known free, constant-potential, or stationary solution.

Crank–Nicolson and split-operator methods can preserve norm under their stated assumptions, while generic explicit time steppers need not. Absorbing layers deliberately remove norm; their performance should be assessed by reflection error and flux accounting rather than by demanding exact unitarity.

Where the equation is not enough

The one-particle TDSE is not a universal dynamical law for every quantum problem. Relativistic particles, particle creation, radiative quantum fields, open-system reduced states, measurements conditioned on outcomes, and many-particle configuration-space dynamics require additional structure. Even within nonrelativistic wave mechanics, the PDE is incomplete until its Hilbert space, domain, boundary conditions, initial state, and normalization convention are stated.

Common pitfalls

Specifying two independent time data. The Schrödinger equation is first order in time; $\psi(t_0)$ fixes $\partial_t\psi(t_0)$ through the Hamiltonian.

Equating a real potential with a complete proof of unitarity. Boundary conditions and the operator domain still matter.

Using $e^{-i\hat Ht/\hbar}$ for an arbitrary $\hat H(t)$. Noncommuting Hamiltonians at different times require time-ordered evolution.

Confusing norm and energy conservation. A time-dependent self-adjoint Hamiltonian preserves norm but may exchange energy with the drive.

Checking only array norm. A coordinate grid may require quadrature weights, and a nonorthogonal basis requires its overlap matrix.

Exercises

1. Explain why the TDSE does not permit independent choices of both $\psi(x,t_0)$ and $\partial_t\psi(x,t_0)$.

Solution

At $t=t_0$ the equation gives

$$\partial_t\psi(x,t_0) = -\frac{i}{\hbar}\hat H(t_0)\psi(x,t_0).$$

Once the initial state and Hamiltonian are fixed, the time derivative is fixed for a state in the operator domain.

2. Show directly that a linear combination of two solutions of the same TDSE is another solution.

Solution

If $i\hbar\partial_t\psi_a=\hat H\psi_a$ for $a=1,2$, then linearity gives

$$i\hbar\partial_t(c_1\psi_1+c_2\psi_2) = c_1\hat H\psi_1+c_2\hat H\psi_2 = \hat H(c_1\psi_1+c_2\psi_2).$$

Both component solutions must use the same Hamiltonian and domain.

3. A spatially constant potential $V_0$ is added to a time-independent Hamiltonian. How does the wavefunction change, and what happens to the density?

Solution

Because $V_0I$ commutes with the original Hamiltonian,

$$\psi_{V_0}(t) = e^{-iV_0(t-t_0)/\hbar}\psi_0(t).$$

The extra factor is a global phase, so $|\psi_{V_0}|^2=|\psi_0|^2$. Energy eigenvalues shift by $V_0$, while energy differences do not.

4. Derive the energy-balance identity for a normalized solution of a regular self-adjoint $\hat H(t)$.

Solution

Differentiate the expectation value:

$$\frac{d}{dt}\langle H\rangle = \langle\dot\psi|H|\psi\rangle +\langle\psi|\dot H|\psi\rangle +\langle\psi|H|\dot\psi\rangle.$$

Using $|\dot\psi\rangle=-iH|\psi\rangle/\hbar$ and its adjoint, the first and third terms cancel. Hence

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

The manipulations assume the relevant vectors lie in the required domains.

5. Find the time at which the free Gaussian width is $\sigma_t=\sqrt2\,\sigma_0$.

Solution

From

$$\frac{\sigma_t^2}{\sigma_0^2} = 1+\left(\frac{\hbar t}{2m\sigma_0^2}\right)^2,$$

setting the left side to $2$ gives

$$|t|=\frac{2m\sigma_0^2}{\hbar}.$$

This has units of time and is the packet’s characteristic spreading time.

6. For an energy eigenstate with energy $E$, compare one forward-Euler step with one Crank–Nicolson step of size $\Delta t$. Which preserves norm?

Solution

Forward Euler multiplies the amplitude by

$$g_{\mathrm{FE}}=1-i\frac{E\Delta t}{\hbar},$$

so

$$|g_{\mathrm{FE}}|^2 = 1+\left(\frac{E\Delta t}{\hbar}\right)^2>1$$

for nonzero $E\Delta t$. Its norm error accumulates even though the method may converge as $\Delta t\to0$. Crank–Nicolson gives

$$g_{\mathrm{CN}} = \frac{1-iE\Delta t/(2\hbar)} {1+iE\Delta t/(2\hbar)},$$

whose numerator and denominator have equal modulus. Thus $|g_{\mathrm{CN}}|=1$ for a self-adjoint time-independent Hamiltonian.

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.