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Matter waves

Light displays both wave-like and particle-like behavior. Louis de Broglie proposed the complementary idea that material particles also have wave properties. Experiments with electrons, neutrons, atoms, and even large molecules have since produced diffraction and interference patterns.

A matter wave is not a little material ball physically wriggling along a sine curve. The wavefunction is a complex probability amplitude. Its phase controls interference, while its squared magnitude determines probabilities for measurement outcomes.

The de Broglie relation

For a particle with momentum magnitude \(|\mathbf p|\), the de Broglie wavelength is

\[\lambda=\frac{h}{|\mathbf p|}\]

Equivalently, the wave vector satisfies

\[\mathbf p=\hbar\mathbf k,\qquad |\mathbf k|=\frac{2\pi}{\lambda}\]

The wavelength is positive. In one dimension, a negative momentum is represented by a negative wave number \(k\), not by a negative wavelength. A function that rejects every negative signed momentum loses physically valid left-moving states; it should use the magnitude for wavelength and preserve the sign for direction.

The units agree:

\[\frac{\mathrm{J\,s}}{\mathrm{kg\,m/s}}=\frac{\mathrm{kg\,m^2/s}}{\mathrm{kg\,m/s}}=\mathrm m\]

At zero momentum, the formula does not return a finite wavelength. The limiting wavelength grows without bound. Code should report that limit explicitly rather than divide by zero and present an accidental numeric value.

import numpy as np

h, hbar = 6.62607015e-34, 6.62607015e-34 / (2 * np.pi)
momentum = np.array([1e-24, 0.0, 0.0])
wavelength = h / np.linalg.norm(momentum)
wave_vector = momentum / hbar
print(f"Wavelength: {wavelength:.6g} m; wave vector: {wave_vector}")

A wavelength example

Let the momentum magnitude be

\[|\mathbf p|=6.62607015\times10^{-24}\ \mathrm{kg\,m/s}\]

Then

\[\lambda=\frac{6.62607015\times10^{-34}\ \mathrm{J\,s}}{6.62607015\times10^{-24}\ \mathrm{kg\,m/s}}=1.00\times10^{-10}\ \mathrm m\]

This is \(0.100\,\mathrm{nm}\), comparable to atomic spacings in crystals. That scale is why crystals can diffract electrons. If the momentum magnitude becomes four times larger, the wavelength becomes four times smaller.

For a nonrelativistic particle of mass \(m\) and speed \(v\),

\[\lambda\approx\frac{h}{mv}\]

At relativistic speed, use the relativistic momentum \(p=\gamma mv\), not \(mv\):

\[\lambda=\frac{h}{\gamma mv}\]

Plane-wave phase

An ideal one-dimensional free-particle plane wave can be written

\[\psi(x,t)=A e^{i(kx-\omega t)}\]

Its wavelength and frequency satisfy

\[p=\hbar k,\qquad E=\hbar\omega\]

The real and imaginary parts oscillate, but

\[|\psi(x,t)|^2=|A|^2\]

is uniform. An exact plane wave has definite momentum and extends through all space, so it is not a localized particle state and cannot be normalized to total probability one over an infinite domain. Plane waves are useful basis states and ideal limits, not literal finite beams.

A global phase change

\(\psi\to e^{i\phi}\psi\)

changes the displayed real and imaginary components but leaves

\(|\psi|^2\)

and all ordinary predictions unchanged. Relative phase between components, however, changes interference.

Wave packets

A localized state requires a superposition of wave numbers:

\[\psi(x,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi(k)e^{i(kx-\omega(k)t)}\,dk\]

The complex weight \(\phi(k)\) describes the momentum-space amplitude. A narrow range of wave numbers produces a broad packet in position; a broad range can produce a narrow packet. This tradeoff is quantified by the uncertainty relation

\[\Delta x\,\Delta p\geq\frac{\hbar}{2}\]

The uncertainties are standard deviations of repeated measurement outcomes for similarly prepared systems. They are not merely defects of an instrument and are not generally found by multiplying arbitrary “resolutions.”

# Scaled units hbar=mass=1; a free Gaussian wave packet.
x = np.linspace(-20, 20, 512, endpoint=False)
dx = x[1] - x[0]
k = 2 * np.pi * np.fft.fftfreq(len(x), d=dx)
initial = np.exp(-x * x / 4) * np.exp(2j * x)
psi = np.fft.ifft(np.fft.fft(initial) * np.exp(-0.5j * k * k * 1.0))
print("Peak position after one time unit:", x[np.argmax(abs(psi))])

Phase velocity and group velocity

For a component with angular frequency \(\omega\) and wave number \(k\), phase velocity is

\[v_{\mathrm{phase}}=\frac{\omega}{k}\]

The envelope of a narrow wave packet moves at group velocity

\[v_{\mathrm{group}}=\frac{d\omega}{dk}\]

For a nonrelativistic free particle, \(E=p^2/(2m)\). Using \(E=\hbar\omega\) and \(p=\hbar k\) gives

\[\omega=\frac{\hbar k^2}{2m},\qquad v_{\mathrm{group}}=\frac{\hbar k}{m}=\frac{p}{m}=v\]

while

\[v_{\mathrm{phase}}=\frac{\hbar k}{2m}=\frac{v}{2}\]

The group velocity equals the particle's classical velocity for a narrow packet. The phase velocity is different and should not be interpreted as the speed of a localized particle or signal.

With total relativistic energy, the free-particle relations give \(v_{\mathrm{phase}}=c^2/v\), which exceeds \(c\) for a massive particle. This does not transmit information faster than light; the corresponding group velocity is \(v\), below \(c\).

Probability density on a grid

For a normalized one-dimensional wavefunction,

\[\int_{-\infty}^{\infty}|\psi(x,t)|^2\,dx=1\]

and the probability of finding the particle between \(a\) and \(b\) is

\[P(a\leq x\leq b)=\int_a^b|\psi(x,t)|^2\,dx\]

On a uniform grid with cell width \(\Delta x\), approximate normalization is

\[\sum_j|\psi_j|^2\Delta x\approx1\]

and

\[|\psi_j|^2=\operatorname{Re}(\psi_j)^2+\operatorname{Im}(\psi_j)^2\]

The cell width cannot be discarded on a nonuniform grid. Even on a uniform grid, it is better to keep it visible because the samples represent a probability density, not probability per point.

norm = np.sum(abs(psi) ** 2) * dx
if norm <= 0:
    raise ValueError("The wavefunction must have nonzero norm.")
psi = psi / np.sqrt(norm)
assert np.isclose(np.sum(abs(psi) ** 2) * dx, 1.0)
print("Normalised probability:", np.sum(abs(psi) ** 2) * dx)

Evidence from diffraction

In electron diffraction, a crystalline lattice supplies regularly spaced scattering centers. Constructive interference occurs under the same geometric condition used for waves scattered from crystal planes:

\[n\lambda=2d\sin\theta\]

Substituting the de Broglie wavelength predicts how the diffraction angles change with electron momentum. Raising the accelerating voltage raises electron momentum and shortens wavelength, so the pattern shifts. At low voltage a nonrelativistic kinetic-energy relation may be used; at high voltage, relativistic momentum is required.

The Davisson–Germer experiment and electron diffraction through thin materials confirmed that measured wavelengths agree with \(\lambda=h/p\). The interference pattern emerges statistically from many detection events. It does not show that each electron split into classical fragments.

Particles accelerated through a voltage

An electron accelerated from rest through potential difference \(V\) gains kinetic energy \(K=eV\) when losses are negligible. In the nonrelativistic regime, \(K=p^2/(2m_e)\), so its wavelength is

\[\lambda=\frac{h}{\sqrt{2m_e eV}}\]

This formula predicts that multiplying the accelerating voltage by four halves the wavelength. It should not be extended indefinitely. The relativistic energy–momentum relation gives the more accurate momentum through

\[pc=\sqrt{K(K+2m_ec^2)}\]

and the wavelength remains \(\lambda=h/p\). Stating which momentum formula was used is essential when comparing a calculation with an electron microscope or diffraction measurement.

Why ordinary objects do not show obvious diffraction

The de Broglie relation applies to macroscopic objects too, but their momenta make the wavelength extraordinarily small. A moving ball has a wavelength far below atomic and nuclear length scales, and interactions with air, thermal radiation, and its own many internal degrees of freedom rapidly correlate it with the environment. Observable interference requires coherent alternatives whose relative phase is preserved.

Small wavelength alone is not a claim that quantum mechanics stops applying. It explains why an impractically fine grating would be needed, while environmental decoherence explains why phase relationships become inaccessible. Classical trajectories emerge as an excellent approximation for the restricted observables and scales of everyday mechanics.

Checks and limits

A reliable matter-wave calculation should pass these checks:

The single-particle wavefunction used here omits spin, particle creation, entanglement with an environment, detector dynamics, and many-body statistics. A nonrelativistic Schrödinger wavefunction is inappropriate when relativistic particle creation is important. The de Broglie relation remains foundational, but the full theory then requires relativistic quantum mechanics or quantum field theory.

A worked Python notebook

Compare matter-wave wavelengths for different nonzero momentum magnitudes. The question for this calculation is precise: how does de Broglie wavelength change when we change momentum magnitude, 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.

A wavelength alone describes a plane-wave component, not a localized particle trajectory. 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 1e-25 to 2e-24 kg m/s. 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 momentum magnitude, expressed in kg m/s. The output is de Broglie wavelength, expressed in m. 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.

\[f(q) = \frac{6.62607015 \cdot 10^{-34}}{q}\]

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 1e-24 kg m/s, the expression gives 6.62607e-10 m. 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("6.62607015e-34/q", locals={"q": q})
reference_input = 1e-24
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "kg m/s")
print("Predicted de Broglie wavelength:", reference_output, "m")

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 6.62607e-09 m. The midpoint, at 1.05e-24 kg m/s, gives 6.31054e-10 m. The upper endpoint gives 3.31304e-10 m. Read these together with the assumptions: they describe different choices of momentum magnitude 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([1e-25, 1.05e-24, 2e-24])
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} kg m/s -> {output_value:.6g} m")
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 momentum magnitude; the vertical axis shows de Broglie wavelength. 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(1e-25, 2e-24, 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("Momentum magnitude (kg m/s)")
ax.set_ylabel("De broglie wavelength (m)")
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 de Broglie wavelength change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1e-24 kg m/s is -6.62607e+14 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.

\[f'(q) = - \frac{6.62607015 \cdot 10^{-34}}{q^{2}}\]

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 = 1.0095e-24
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, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 1.0095e-24 kg m/s. The tangent estimate is 6.56312e-10 m, while the full expression gives 6.56371e-10 m. 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 = 1.8999999999999998e-24
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 momentum magnitude inside the stated interval, predict whether de Broglie wavelength 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: A wavelength alone describes a plane-wave component, not a localized particle trajectory. 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 m: 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(2e-24) - evaluate(1e-25))
for points in [101, 201, 401]:
    grid = np.linspace(1e-25, 2e-24, 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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