Time-Independent Schrödinger Equation

The time-independent Schrödinger equation is the energy eigenvalue problem for a time-independent Hamiltonian:

$$\hat H\psi_n=E_n\psi_n.$$

In coordinate-space wave mechanics it is usually a differential equation plus boundary conditions. Solving it gives allowed energies and stationary modes; general state reconstruction uses the full spectral resolution, including any generalized continuum sector rather than an ordinary discrete basis alone.

Required background. Time-Dependent Schrödinger Equation in Coordinate Space supplies the evolution law from which separation is obtained. Coordinate Representation supplies the differential-operator language.

Helpful background. Self-Adjoint Operators supplies the domain-sensitive spectral facts used below.

From Separation Of Variables

Start with the time-dependent Schrödinger equation

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

and suppose $\hat H$ does not depend on time. Look for separated solutions

$$\psi(x,t)=\varphi(x)T(t).$$

Substitution gives

$$i\hbar\varphi(x)\frac{dT}{dt} =T(t)\hat H\varphi(x).$$

On regions where both factors are nonzero, division by $\varphi(x)T(t)$ separates the time and space dependence:

$$i\hbar\frac{1}{T}\frac{dT}{dt} =\frac{1}{\varphi}\hat H\varphi =E.$$

The time part is

$$T(t)=e^{-iEt/\hbar},$$

and the spatial part is

$$\hat H\varphi=E\varphi.$$

Thus the stationary-state problem is not a separate dynamical law. It is the separated form of time evolution when the Hamiltonian is time independent. The separated equations extend across isolated nodes by continuity; the derivation does not require dividing by zero at a node.

Coordinate-Space Form

For one particle in one dimension with potential $V(x)$,

$$\hat H =-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x),$$

so the time-independent equation is

$$-\frac{\hbar^2}{2m}\frac{d^2\varphi(x)}{dx^2} +V(x)\varphi(x) =E\varphi(x).$$

This is an eigenvalue equation for a differential operator. The allowed values of $E$ are determined by the potential, the coordinate domain, and the boundary conditions.

Self-Adjoint Spectral Problem

The stationary problem is physically well posed when $\hat H$ is self-adjoint on a stated domain. Self-adjointness supplies three central facts:

• energy eigenvalues are real;
• eigenfunctions belonging to distinct eigenvalues are orthogonal;
• the spectral resolution supports unitary time evolution.

If

$$\hat H\varphi_n=E_n\varphi_n,$$

then

$$\langle\varphi_n|\hat H\varphi_n\rangle = E_n\langle\varphi_n|\varphi_n\rangle.$$

The left side is real for a self-adjoint Hamiltonian, so $E_n$ is real for a nonzero eigenfunction. Likewise, for two eigenfunctions,

$$\begin{aligned} \langle\varphi_m|\hat H\varphi_n\rangle &=E_n\langle\varphi_m|\varphi_n\rangle,\\ \langle\hat H\varphi_m|\varphi_n\rangle &=E_m\langle\varphi_m|\varphi_n\rangle. \end{aligned}$$

Subtracting gives

$$(E_n-E_m)\langle\varphi_m|\varphi_n\rangle=0.$$

Thus $E_m\ne E_n$ implies orthogonality. Degenerate eigenfunctions need not be orthogonal automatically, but an orthonormal basis can be chosen within each degenerate eigenspace.

The word “self-adjoint” includes the domain. A formally Hermitian differential expression with unsuitable boundary conditions need not define the required observable.

Stationary Does Not Mean No Time Dependence

The full stationary-state solution is

$$\psi_n(x,t)=\varphi_n(x)e^{-iE_n t/\hbar}.$$

It has a time-dependent phase. However, its probability density is time independent:

$$\lvert\psi_n(x,t)\rvert^2 =\lvert\varphi_n(x)\rvert^2.$$

This is why energy eigenstates are called stationary states. The state vector changes by a phase, but the probability distribution of every time-independent observable defined on the state remains constant. The statement is about all such observables, not only those that commute with the Hamiltonian.

A superposition of different energies is generally not stationary:

$$\psi(x,t) =c_1\varphi_1(x)e^{-iE_1t/\hbar} +c_2\varphi_2(x)e^{-iE_2t/\hbar}.$$

The relative phase depends on $E_2-E_1$, and interference terms can make the probability density time dependent.

A superposition of states with the same degenerate energy remains stationary:

$$\psi(x,t) = \left( \sum_{\alpha}c_\alpha\varphi_{E,\alpha}(x) \right)e^{-iEt/\hbar}.$$

The degeneracy label $\alpha$ distinguishes independent states in the same energy eigenspace. Symmetries often provide additional commuting observables that organize this basis.

Shifting every energy by a constant $C$ changes the full state only by a common phase:

$$\hat H\longmapsto\hat H+CI, \qquad \psi(t)\longmapsto e^{-iCt/\hbar}\psi(t).$$

Energy differences govern observable beat frequencies and transition energies.

Discrete And Continuous Spectra

Bound systems often have discrete energies:

$$E_1,E_2,E_3,\ldots$$

with square-normalizable eigenfunctions. The infinite square well and harmonic oscillator are standard examples.

Free particles and scattering states often have continuous spectra. On the full line, a free-particle energy

$$E=\frac{\hbar^2k^2}{2m}$$

is labeled by a continuous wavenumber $k$, and the corresponding plane waves are not square-normalizable on $\mathbb R$. They are handled with delta normalization, box normalization, or wave packets.

Many physical Hamiltonians have both discrete and continuous parts: bound states below a threshold and scattering states above it.

For a one-dimensional potential approaching $V_\infty$ at spatial infinity, the large-distance equation suggests

$$\varphi''(x) +\frac{2m}{\hbar^2} \left(E-V_\infty\right)\varphi(x) \approx0.$$

When $E<V_\infty$, asymptotic solutions are exponential and square-integrability selects the decaying branch for a bound state. When $E>V_\infty$, solutions are oscillatory and are usually interpreted as continuum states. Threshold behavior at $E=V_\infty$ requires separate care.

This is a diagnostic, not a universal theorem for every long-range or singular potential. Coulomb tails, periodic potentials, and finite domains have their own spectral structures.

Continuous energies can also be degenerate. For the free particle in one dimension, $k$ and $-k$ have the same energy. In three dimensions, every momentum direction with fixed $|\mathbf k|$ shares the same energy.

Why the spectral problem is useful

For a purely discrete spectrum with a complete orthonormal eigenbasis, solving the TISE supplies the modes used to reconstruct an initial state:

$$\psi(x,t) = \sum_n c_n\varphi_n(x)e^{-iE_nt/\hbar}, \qquad c_n=\int\varphi_n^*(x)\psi(x,0)\,dx.$$

If the spectrum also has a continuum, the spectral resolution includes generalized eigenvectors, channel labels, the correct continuum measure, and integrals in addition to the discrete sum. This page uses that fact only to explain why the stationary eigenproblem matters. The separate stationary-state-expansion treatment owns convergence, mixed spectra, degeneracy bookkeeping, revivals, and detailed reconstruction workflows.

Boundary-Value Character

The time-independent Schrödinger equation is usually a boundary-value problem, not just a differential equation. For an infinite square well on $0<x<L$, acceptable solutions must satisfy

$$\varphi(0)=0, \qquad \varphi(L)=0.$$

Those two endpoint conditions are what make the allowed energies discrete:

$$E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}.$$

For finite wells, wavefunctions extend into classically forbidden regions and decay rather than vanish at the walls. For rings, periodic boundary conditions replace endpoint vanishing. The equation and the boundary conditions must be solved together.

For a finite jump in an otherwise regular potential, integrating the equation across a shrinking interval gives continuity of $\varphi$ and $\varphi'$. A delta interaction produces a derivative jump, an infinite wall imposes a domain boundary, and singular endpoints require separate analysis. Matching rules follow from the Hamiltonian; they are not universal decorations added after solving.

Practical Solution Workflow

For a piecewise one-dimensional problem:

1. State the coordinate domain, potential, Hilbert space, and boundary conditions.
2. Identify natural length and energy scales.
3. Solve the local ODE separately in each region.
4. Reject branches incompatible with regularity or asymptotic behavior.
5. Apply matching and endpoint conditions.
6. Solve the resulting quantization or scattering condition for $E$.
7. Normalize bound states or choose a stated continuum normalization.
8. Check the equation, boundaries, dimensions, limiting cases, and node structure.

The determinant of the matching equations often vanishes only at allowed bound-state energies. For scattering energies, the same matching system instead determines reflection and transmission amplitudes.

Rayleigh Quotient and Ground-State Bound

For a semibounded self-adjoint Hamiltonian, the variational principle is naturally stated on its quadratic-form domain. When the expectation is written as $\langle\phi|H|\phi\rangle$, it is sufficient to take a normalized trial state in $D(H)$:

$$\mathcal E[\phi] = \langle\phi|\hat H|\phi\rangle.$$

Expanding in energy eigenstates gives

$$\mathcal E[\phi] = \sum_n|c_n|^2E_n$$

for a purely discrete spectrum. If $E_0$ is the ground-state energy,

$$\mathcal E[\phi]\ge E_0.$$

Thus every admissible normalized trial state provides an upper bound on $E_0$. Equality holds when the trial state lies entirely in the ground eigenspace. The variational method is developed elsewhere, but the bound is an important consistency check for stationary calculations.

For an approximate numerical eigenfunction, the Rayleigh quotient

$$\widetilde E = \frac{\langle\phi|\hat H|\phi\rangle} {\langle\phi|\phi\rangle}$$

and the residual

$$r=\hat H\phi-\widetilde E\phi$$

test different features: the quotient estimates energy, while $\lVert r\rVert$ measures the failure of the eigenvalue equation. A small residual guarantees that $\widetilde E$ lies near the spectrum, but accuracy of an individual eigenvector additionally requires an isolated spectral gap. Near degeneracy, the stable object may be the whole spectral subspace rather than one chosen eigenvector.

Dimensionless Form

Choose a characteristic length $\ell$ and energy

$$E_\ell=\frac{\hbar^2}{2m\ell^2}.$$

With $x=\ell\xi$, $E=E_\ell\varepsilon$, and $V=E_\ell v$, the one-dimensional equation becomes

$$-\frac{d^2\varphi}{d\xi^2} +v(\xi)\varphi = \varepsilon\varphi.$$

Nondimensionalization exposes the independent control parameters, improves numerical conditioning, and makes limiting cases easier to compare. The choice of $\ell$ should reflect the system rather than convenience alone.

Numerical Eigenvalue Problems

Finite differences or basis truncation turn the TISE into

$$H\mathbf c=E\mathbf c$$

or, in a nonorthogonal basis,

$$H\mathbf c=ES\mathbf c.$$

A credible computation checks both the algebraic residual and physical convergence under grid spacing, box size, or basis size. High-energy states usually probe the discretization more severely than low-energy states.

Dense diagonalization and sparse eigensolvers are later numerical treatments. Neither algorithm can repair an incorrect domain or boundary condition.

What the TISE Does Not Do

The eigenvalue problem finds stationary modes and their energies. It does not choose the coefficients of a physical state, describe a measurement update, or by itself evolve arbitrary initial data. Those tasks require an initial state and the Time-Dependent Schrödinger Equation.

For explicitly time-dependent Hamiltonians, an instantaneous eigenvalue equation

$$\hat H(t)|n(t)\rangle=E_n(t)|n(t)\rangle$$

can still be useful, but its eigenvectors do not generally evolve by attaching only $e^{-i\int E_n dt/\hbar}$. Basis motion and transitions between instantaneous eigenspaces must also be considered.

Boundary of This Page

This page owns the practical stationary eigenvalue problem. Detailed endpoint and interface rules belong to Boundary Conditions. The Unbounded Spectral Theorem and Self-Adjoint Operators own the abstract spectral and domain theory. Model-specific spectra and wavefunctions remain outside this foundation.

Common Mistakes

• Thinking every state of a time-independent Hamiltonian is stationary.
• Forgetting the phase factor $e^{-iEt/\hbar}$ for energy eigenstates.
• Confusing a time-independent potential with a time-independent wavefunction.
• Applying bound-state normalization to continuum eigenfunctions.
• Solving the differential equation while postponing boundary conditions until after choosing arbitrary energies.
• Treating degeneracy as impossible; distinct eigenfunctions can share the same energy.
• Assuming every energy eigenfunction is square-normalizable.
• Omitting continuum states from a completeness relation.
• Using a local matching rule for an infinite wall, delta interaction, or singular endpoint without rederiving it.
• Reporting a numerical eigenvalue without a residual and discretization-convergence study.
• Treating instantaneous eigenstates of a time-dependent Hamiltonian as exact dynamical solutions.
• Confusing the Rayleigh quotient of a trial state with proof that the trial state is an eigenstate.

Where This Is Used

• Time-Dependent Schrödinger Equation explains the full equation of motion.
• Hamiltonians in Coordinate Space constructs the differential operators whose stationary spectra are sought.
• Boundary Conditions explains how endpoint and matching conditions select spectra.
• Normalization Conventions distinguishes discrete normalization from continuum delta normalization.
• Self-Adjoint Operators explains why the operator domain, not only the differential expression, determines the observable Hamiltonian.
• Spectral Theorem gives the operator-level framework behind discrete sums and continuum spectral integrals.

Exercises

1. Show that a single stationary state has time-independent probability density.

Solution

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

$$\lvert\psi_n(x,t)\rvert^2 =\varphi_n^*(x)e^{iE_nt/\hbar} \varphi_n(x)e^{-iE_nt/\hbar} =\lvert\varphi_n(x)\rvert^2.$$

The phase cancels.

2. Suppose a system has two energy eigenstates with energies $E_1$ and $E_2$. What angular frequency appears in interference terms of a superposition?

Solution

Cross terms contain phases such as

$$e^{-i(E_2-E_1)t/\hbar}.$$

The angular frequency is

$$\omega=\frac{E_2-E_1}{\hbar},$$

up to an absolute value if one reports a positive frequency.

3. Prove that eigenfunctions of a self-adjoint Hamiltonian with distinct eigenvalues are orthogonal.

Solution

Let

$$\hat H|\varphi_n\rangle=E_n|\varphi_n\rangle, \qquad \hat H|\varphi_m\rangle=E_m|\varphi_m\rangle.$$

Then

$$\langle\varphi_m|\hat H|\varphi_n\rangle = E_n\langle\varphi_m|\varphi_n\rangle.$$

Self-adjointness also gives

$$\langle\varphi_m|\hat H|\varphi_n\rangle = \langle\hat H\varphi_m|\varphi_n\rangle = E_m\langle\varphi_m|\varphi_n\rangle.$$

Therefore

$$(E_n-E_m)\langle\varphi_m|\varphi_n\rangle=0.$$

If $E_n\ne E_m$, the inner product must vanish.

4. Show that the Rayleigh quotient of a normalized trial state is not below the ground-state energy.

Solution

Expand the trial state in a complete discrete energy basis:

$$|\phi\rangle=\sum_n c_n|n\rangle, \qquad \sum_n|c_n|^2=1.$$

Then

$$\langle\phi|\hat H|\phi\rangle = \sum_n|c_n|^2E_n.$$

Since every $E_n\ge E_0$,

$$\langle\phi|\hat H|\phi\rangle \ge E_0\sum_n|c_n|^2 =E_0.$$

Equality requires support only in the ground eigenspace.

5. Nondimensionalize the infinite-well TISE on $0<x<L$ and recover its energy scale.

Solution

Set $\xi=x/L$ and

$$E_L=\frac{\hbar^2}{2mL^2}, \qquad E=E_L\varepsilon.$$

Inside the well, the equation becomes

$$-\frac{d^2\varphi}{d\xi^2} = \varepsilon\varphi, \qquad \varphi(0)=\varphi(1)=0.$$

The nonzero solutions are $\sin(n\pi\xi)$, so

$$\varepsilon_n=n^2\pi^2.$$

Restoring units gives

$$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}.$$

6. On $0\le x\le L$, compare the free Hamiltonian $-\hbar^2d^2/(2m\,dx^2)$ with (a) Dirichlet conditions and (b) periodic conditions. Find both spectra and explain what the comparison shows about the TISE.

Solution

For Dirichlet conditions,

$$\varphi(0)=\varphi(L)=0,$$

the normalized eigenfunctions and energies are

$$\varphi_n(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right), \qquad E_n=\frac{\hbar^2\pi^2n^2}{2mL^2}, \qquad n=1,2,\ldots .$$

These levels are nondegenerate. For periodic conditions,

$$\varphi(0)=\varphi(L), \qquad \varphi'(0)=\varphi'(L),$$

the eigenfunctions and energies are

$$\varphi_n(x)=\frac{1}{\sqrt L}e^{i2\pi nx/L}, \qquad E_n=\frac{\hbar^2}{2m}\left(\frac{2\pi n}{L}\right)^2, \qquad n\in\mathbb Z.$$

The $n=0$ level is nondegenerate, while each positive energy is twofold degenerate because $n$ and $-n$ have the same energy. The differential expression is identical in the two problems, but the operator domains differ; therefore their allowed eigenfunctions, spectra, and degeneracies differ.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. I, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018, doi:10.1017/9781316995433.
• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, doi:10.1007/978-1-4757-0576-8.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, doi:10.1090/gsm/157.