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Schrödinger equation numerically

The time-dependent Schrödinger equation determines how a nonrelativistic quantum state evolves. Except for specially chosen potentials, its solutions are rarely available as simple formulas. Numerical solution replaces continuous space with a finite representation and replaces exact time evolution with a sequence of controlled approximations.

A smooth animation is not evidence that the calculation is correct. The solver must preserve the structure of quantum mechanics, state its boundary conditions, and show convergence as space and time resolution improve.

The continuous equation

For one spinless particle of mass \(m\) moving in a one-dimensional, possibly time-dependent potential \(V(x,t)\),

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

with Hamiltonian

\[\hat H=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x,t)\]

The kinetic term depends on spatial curvature of the wavefunction; the potential term multiplies the wavefunction at each point. The state is complex even when the potential is real. Discarding the imaginary component prevents correct phase evolution.

For a time-independent Hamiltonian, a stationary spatial state satisfies

\[\hat H\phi_n(x)=E_n\phi_n(x)\]

and evolves by a phase:

\[\psi_n(x,t)=\phi_n(x)e^{-iE_nt/\hbar}\]

Its probability density does not change because the phase has magnitude one. This gives an important exact test for a numerical solver.

Create the spatial grid

Choose grid points

\[x_j=x_{\min}+j\Delta x,\qquad j=0,1,\ldots,N-1\]

and store complex samples \(\psi_j\approx\psi(x_j)\). On an interior point, the centered second derivative is

\[\frac{\partial^2\psi}{\partial x^2}(x_j)\approx\frac{\psi_{j+1}-2\psi_j+\psi_{j-1}}{\Delta x^2}\]

The approximation has leading error proportional to \(\Delta x^2\) for a sufficiently smooth wavefunction. Substitution makes the discrete Hamiltonian tridiagonal:

\[H_{jj}=\frac{\hbar^2}{m\Delta x^2}+V_j,\qquad H_{j,j\pm1}=-\frac{\hbar^2}{2m\Delta x^2}\]

These entries describe interior points. The first and last rows depend on the boundary condition.

import numpy as np


def box_hamiltonian(interior_points):
    dx = 1 / (interior_points + 1)
    # Infinite well from 0 to 1; psi=0 at both endpoints, hbar=mass=1.
    H = np.diag(np.full(interior_points, 1 / dx**2))
    off_diagonal = np.full(interior_points - 1, -0.5 / dx**2)
    H += np.diag(off_diagonal, 1) + np.diag(off_diagonal, -1)
    return H, dx


H, dx = box_hamiltonian(40)
energies, states = np.linalg.eigh(H)
print("Lowest energies:", energies[:3])

Boundary conditions are part of the physics

Several boundary choices answer different questions:

An absorbing boundary is deliberately nonunitary within the visible domain because probability leaves that domain. It should not be confused with numerical norm loss. Record absorbed probability or flux so the full probability budget remains understandable.

The grid also needs enough points per shortest relevant wavelength. A packet that is barely above the two-samples-per-wavelength aliasing limit has severe phase and dispersion error even if it does not crash.

Why forward Euler fails

The differential equation can be rearranged as

\[\frac{\partial\psi}{\partial t}=-\frac{i}{\hbar}\hat H\psi\]

A tempting forward-Euler step is

\[\psi^{n+1}=\psi^n-\frac{i\Delta t}{\hbar}H\psi^n\]

For a Hamiltonian eigenstate with energy \(E\), this multiplies the amplitude by \(1-iE\Delta t/\hbar\). Its magnitude is

\[\left|1-iE\Delta t/\hbar\right|=\sqrt{1+(E\Delta t/\hbar)^2}>1\]

so the norm grows at every nonzero step. A smaller time step slows the failure but does not make forward Euler unitary. Renormalizing after each step conceals this defect and distorts the state.

Crank–Nicolson time stepping

For a time-independent Hermitian Hamiltonian, Crank–Nicolson averages the right side between the old and new state:

\[\left(I+\frac{i\Delta t}{2\hbar}H\right)\psi^{n+1}=\left(I-\frac{i\Delta t}{2\hbar}H\right)\psi^n\]

The update is unitary up to linear-solver and floating-point error. It is second-order accurate in time and, with the centered stencil, second-order in space. “Unconditionally stable” does not mean “accurate for any time step.” A large \(\Delta t\) can still give large phase error.

dt = 0.001
identity = np.eye(len(H))
left = identity + 0.5j * dt * H
right = identity - 0.5j * dt * H
psi = states[:, 0].astype(complex) / np.sqrt(dx)
next_psi = np.linalg.solve(left, right @ psi)
assert np.isclose(np.sum(abs(next_psi) ** 2) * dx, 1.0)
print("Crank-Nicolson probability:", np.sum(abs(next_psi) ** 2) * dx)

For a tridiagonal finite-difference Hamiltonian, use a complex tridiagonal solver rather than a dense inverse. Never explicitly calculate a matrix inverse for each step. If the potential changes with time, evaluate it according to the chosen time-centering rule and rebuild or update the operator as required.

Split-operator evolution

For suitable periodic grids, the formal short-time operator can be split into potential and kinetic pieces:

\[e^{-iH\Delta t/\hbar}\approx e^{-iV\Delta t/(2\hbar)}e^{-iT\Delta t/\hbar}e^{-iV\Delta t/(2\hbar)}\]

The potential factor is diagonal in position space. The kinetic factor is diagonal in momentum space, so a fast Fourier transform can move between the representations. The symmetric split is second-order accurate in time for smooth conditions. Periodicity, FFT normalization, wave-number ordering, and aliasing must all be explicit.

# A separate periodic grid for split-operator evolution.
x = np.linspace(-10, 10, 256, endpoint=False)
dx = x[1] - x[0]
k = 2 * np.pi * np.fft.fftfreq(len(x), d=dx)
psi = np.exp(-x * x / 2).astype(complex)
psi /= np.sqrt(np.sum(abs(psi) ** 2) * dx)
potential = 0.5 * x * x
half_potential_phase = np.exp(-0.5j * potential * dt)
kinetic_phase = np.exp(-0.5j * k * k * dt)
psi = half_potential_phase * np.fft.ifft(
    kinetic_phase * np.fft.fft(half_potential_phase * psi)
)
assert np.isclose(np.sum(abs(psi) ** 2) * dx, 1.0)

Probability and expectation diagnostics

For a normalized state,

\[\int|\psi(x,t)|^2\,dx=1\]

and on a uniform grid,

\[\mathcal N=\sum_j|\psi_j|^2\Delta x\approx1\]

Monitor \(\mathcal N-1\) without rounding it away. A Hermitian time-independent Hamiltonian also conserves the energy expectation

\[\langle H\rangle=\frac{\sum_j\psi_j^*(H\psi)_j\Delta x}{\sum_j|\psi_j|^2\Delta x}\]

Small imaginary parts in the computed expectation can arise from roundoff. A substantial imaginary component indicates a non-Hermitian operator, inconsistent boundaries, or an implementation error unless non-Hermitian physics was deliberately introduced.

Other useful diagnostics include

\[\langle x\rangle=\sum_j x_j|\psi_j|^2\Delta x,\qquad P_{[a,b]}=\sum_{x_j\in[a,b]}|\psi_j|^2\Delta x\]

Use quadrature weights appropriate to the grid. On a nonuniform grid, \(\Delta x\) cannot be treated as one global constant.

Analytic benchmark: the infinite square well

For a box from \(0\) to \(L\) with zero wavefunction at both walls, the exact stationary states and energies are

\[\phi_n(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right),\qquad E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}\]

A numerical Hamiltonian should reproduce the low-lying energies increasingly well as \(\Delta x\) decreases. If an eigenstate is time-evolved, its density should remain fixed and its phase should advance at \(E_n/\hbar\).

exact = np.pi**2 * np.arange(1, 4) ** 2 / 2
previous_error = float("inf")
for points in [20, 40, 80]:
    H, dx = box_hamiltonian(points)
    numerical = np.linalg.eigvalsh(H)[:3]
    error = np.max(abs(numerical - exact) / exact)
    assert error < previous_error
    previous_error = error
    print(points, "largest relative energy error:", error)

High-energy grid modes are poorly resolved, so agreement for the first few eigenvalues does not validate every eigenvalue near the grid cutoff.

Convergence and error budget

Run the same physical problem on at least three space and time resolutions. Compare complex state only after removing an irrelevant global phase, or compare phase-insensitive observables such as density when appropriate. A useful state error is based on overlap:

\[\mathcal F=|\langle\psi_{\mathrm{reference}}|\psi_{\mathrm{test}}\rangle|^2\]

Values near one indicate close normalized states up to global phase, but the comparison requires interpolation onto a common grid and a consistent inner product.

Vary one numerical choice at a time:

Keep floating-point error separate from discretization and finite-domain error. A curve that looks smoother on a finer grid may still represent a different physical domain or improperly rescaled initial state.

Scope

This equation describes a nonrelativistic single particle without explicit spin. It does not include particle creation, quantized electromagnetic fields, measurement apparatus, or many-body exchange unless those are deliberately added. A potential with electromagnetic fields requires gauge-consistent coupling, not merely inserting any desired force into \(V(x,t)\).

The numerical methods here are principles, not an embedded simulation. A production solver also needs tested complex linear algebra, unit handling, performance limits, checkpointing, and reproducible metadata.

A worked Python notebook

Use the infinite-square-well ground state as a numerical eigenvalue benchmark with mass and reduced Planck constant set to one. The question for this calculation is precise: how does ground-state energy change when we change well width, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The wavefunction vanishes at both walls and the potential is zero inside. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0.5 to 2 scaled length units. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for well width, expressed in scaled length units. The output is ground-state energy, expressed in scaled energy units. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = \frac{\pi^{2}}{2 q^{2}}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 1 scaled length units, the expression gives 4.9348 scaled energy units. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("pi**2/(2*q**2)", locals={"q": q})
reference_input = 1.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "scaled length units")
print("Predicted ground-state energy:", reference_output, "scaled energy units")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 19.7392 scaled energy units. The midpoint, at 1.25 scaled length units, gives 3.15827 scaled energy units. The upper endpoint gives 1.2337 scaled energy units. Read these together with the assumptions: they describe different choices of well width within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.5, 1.25, 2.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(
        f"{input_value:.6g} scaled length units -> "
        f"{output_value:.6g} scaled energy units"
    )
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows well width; the vertical axis shows ground-state energy. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.5, 2.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Well width (scaled length units)")
ax.set_ylabel("Ground-state energy (scaled energy units)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does ground-state energy change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1 scaled length units is -9.8696 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = - \frac{\pi^{2}}{q^{3}}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 1.0075
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "scaled energy units")
print("Full expression:", full_prediction, "scaled energy units")
print(
    "Approximation difference:",
    full_prediction - linear_prediction,
    "scaled energy units",
)

Here the changed input is 1.0075 scaled length units. The tangent estimate is 4.86078 scaled energy units, while the full expression gives 4.8616 scaled energy units. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 1.5
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of well width inside the stated interval, predict whether ground-state energy will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The wavefunction vanishes at both walls and the potential is zero inside. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of scaled energy units: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(2.0) - evaluate(0.5))
for points in [101, 201, 401]:
    grid = np.linspace(0.5, 2.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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