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Photons and quantization

Classical electromagnetic waves correctly describe interference, diffraction, polarization, and the average transport of energy. Some exchanges between light and matter, however, occur in discrete amounts. A photon is one quantum of the electromagnetic field. Its energy depends on frequency, so brightness and color answer different physical questions.

A brighter monochromatic beam usually delivers more photons per second. A higher-frequency beam gives more energy to each photon. Increasing intensity cannot replace insufficient photon energy in a one-photon photoelectric process: many packets that are individually below threshold still remain below threshold.

Planck's quantum

Planck introduced discrete oscillator energies while explaining thermal radiation. For an oscillator of frequency \(f\), allowed energies occur in steps:

\[E_n=nhf,\qquad n=0,1,2,\ldots\]

Here \(h\) is Planck's constant. In the modern photon description, one photon of frequency \(f\) carries energy

\[E_\gamma=hf=\hbar\omega\]

where

\(\omega=2\pi f\)

and \(\hbar=h/(2\pi)\). The exact SI value is

\[h=6.62607015\times10^{-34}\ \mathrm{J\,s}\]

Frequency must be non-negative in this energy relation. The zero-frequency limit has zero energy per quantum, but a freely propagating photon is not a static zero-frequency particle.

Energy from wavelength

For light in vacuum, \(c=f\lambda\), so

\[E_\gamma=\frac{hc}{\lambda}\]

The wavelength in this formula is the vacuum wavelength. In a transparent material, frequency stays fixed at a stationary boundary while wavelength changes with phase speed. The photon energy \(hf\) therefore does not change merely because the wavelength inside the material is shorter. Energy can of course be exchanged during absorption, emission, or inelastic scattering.

At \(f=8.0\times10^{14}\,\mathrm{Hz}\),

\[E_\gamma=(6.62607015\times10^{-34})(8.0\times10^{14})\ \mathrm J\approx5.30\times10^{-19}\ \mathrm J\]

Using \(1\,\mathrm{eV}=1.602176634\times10^{-19}\,\mathrm J\), this is about \(3.31\,\mathrm{eV}\).

h, elementary_charge, frequency = 6.62607015e-34, 1.602176634e-19, 8.0e14
energy_j = h * frequency
energy_ev = energy_j / elementary_charge
print(f"Photon energy: {energy_j:.6g} J = {energy_ev:.3f} eV")

Photon momentum

A photon has zero invariant mass but carries momentum. The relativistic energy–momentum relation gives

\[p_\gamma=\frac{E_\gamma}{c}=\frac{h}{\lambda}\]

The momentum points along the direction of propagation. Absorbing, reflecting, or redirecting photons transfers momentum to matter and produces radiation pressure. A reflecting surface can receive more momentum per incident photon than a perfectly absorbing one because the photon momentum reverses direction.

The statement “photons are massless” therefore does not mean they have no physical effect. They carry energy, momentum, and angular momentum, and they contribute to the total invariant mass of a multi-particle system.

Photon rate, energy, and optical power

If a monochromatic beam delivers \(\dot N\) photons per second, its average power is

\[P=\dot N E_\gamma=\dot N hf\]

Conversely,

\[\dot N=\frac{P}{hf}\]

This relation concerns an average rate. Over a finite interval \(\Delta t\), the expected number is \(\bar N=P\Delta t/(hf)\). An expected value may be fractional even though an actual ideal photon count is an integer.

power, exposure = 1e-3, 1.0
expected_photons = power * exposure / energy_j
print(f"Expected photons in one second: {expected_photons:.6g}")

For a broadband source, there is no single photon energy. Divide the spectrum into frequency intervals and combine their energy or photon-rate contributions. A quoted central wavelength alone is insufficient when the bandwidth is large.

The photoelectric effect

When light illuminates a suitable surface, electrons can be emitted. Einstein's one-photon energy balance is

\[K_{\max}=hf-\phi\]

where \(\phi\) is the material's work function and \(K_{\max}\) is the maximum emitted-electron kinetic energy. The threshold frequency is

\[f_0=\frac{\phi}{h}\]

Below \(f_0\), this single-photon model predicts no photoelectrons regardless of intensity. Above threshold, raising frequency increases the maximum kinetic energy. At fixed frequency above threshold, raising intensity generally increases the emission rate because more photons arrive; it does not change \(K_{\max}\).

If a reverse stopping potential \(V_s\) just prevents the fastest electrons from reaching the collector, then

\[eV_s=K_{\max}=hf-\phi\]

This provides an experimental route to the photon-energy relation. Real surfaces have distributions of electron states, contamination, contact potentials, and multiphoton effects at extreme intensity, so the simple equation describes an idealized maximum-energy boundary.

work_function_j = 2.0 * elementary_charge
if energy_j < work_function_j:
    print("Below the one-photon emission threshold.")
else:
    maximum_kinetic_energy = energy_j - work_function_j
    stopping_potential = maximum_kinetic_energy / elementary_charge
    print(f"Stopping potential: {stopping_potential:.3f} V")

Compton scattering

Photons also behave as particles in collisions. When an X-ray photon scatters from an approximately free electron initially at rest, conservation of relativistic energy and momentum gives the wavelength shift

\[\lambda'-\lambda=\frac{h}{m_ec}(1-\cos\theta)\]

where \(m_e\) is the electron mass and \(\theta\) is the photon's scattering angle. The quantity

\(h/(m_ec)\)

is the electron Compton wavelength.

At \(\theta=0\), there is no wavelength change. At \(\theta=\pi\), the shift is largest: \(2h/(m_ec)\). Because a longer wavelength means lower photon energy, the scattered electron receives the lost energy and balancing momentum.

This effect cannot be explained by treating the incoming light only as a classical continuous wave interacting with a free electron. Its angle-dependent wavelength shift directly uses photon momentum.

Quantization does not eliminate wave behavior

A photon is not a tiny classical bead following a known trajectory while a separate classical wave guides it. Quantum electrodynamics predicts probabilities for preparation, propagation, and detection. Single-photon detection events are localized, yet many repeated events can build an interference pattern. Which-path information can remove interference even when photons arrive one at a time.

Likewise, a classical electromagnetic field can be a very accurate approximation when many photons occupy the relevant modes and quantum fluctuations are negligible. “Wave” and “particle” describe complementary experimental features; neither classical picture alone is the full quantum state.

Measurement and detector efficiency

An ideal detector records every incident photon in the relevant band. A real detector has quantum efficiency \(0\leq\eta\leq1\). If \(N_{\mathrm{incident}}\) photons arrive independently under a simple model, the mean detected count is

\[\mathbb E[N_{\mathrm{detected}}]=\eta N_{\mathrm{incident}}\]

This expectation is not an actual fractional count. Dark counts, dead time, afterpulsing, saturation, spectral response, and correlated light can all invalidate a simple independent-detection model. Keep those detector assumptions separate from \(E=hf\).

Checks and boundaries

A reliable calculation should pass several independent checks:

Do not use a vacuum wavelength with a material phase speed without explaining the conversion. Do not infer exact photon number from classical average power without specifying an interval and quantum state. Do not apply the free-electron Compton formula unchanged when binding energies are important. These are boundaries of the simple models, not failures of quantization.

A worked Python notebook

Convert one photon frequency into its energy using Planck's constant. The question for this calculation is precise: how does photon energy change when we change photon frequency, 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.

Power also depends on photon rate; frequency alone does not specify beam intensity. 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 4e+14 to 1e+15 Hz. 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 photon frequency, expressed in Hz. The output is photon energy, expressed in J. 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.

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

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 8e+14 Hz, the expression gives 5.30086e-19 J. 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 = 800000000000000.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "Hz")
print("Predicted photon energy:", reference_output, "J")

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 2.65043e-19 J. The midpoint, at 7e+14 Hz, gives 4.63825e-19 J. The upper endpoint gives 6.62607e-19 J. Read these together with the assumptions: they describe different choices of photon frequency 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([400000000000000.0, 700000000000000.0, 1000000000000000.0])
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} Hz -> {output_value:.6g} J")
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 photon frequency; the vertical axis shows photon 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(400000000000000.0, 1000000000000000.0, 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("Photon frequency (Hz)")
ax.set_ylabel("Photon energy (J)")
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 photon energy change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 8e+14 Hz is 6.62607e-34 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

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

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 = 803000000000000.0
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, "J")
print("Full expression:", full_prediction, "J")
print("Approximation difference:", full_prediction - linear_prediction, "J")

Here the changed input is 8.03e+14 Hz. The tangent estimate is 5.32073e-19 J, while the full expression gives 5.32073e-19 J. 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 = 600000000000000.0
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 photon frequency inside the stated interval, predict whether photon 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: Power also depends on photon rate; frequency alone does not specify beam intensity. 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 J: 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(1000000000000000.0) - evaluate(400000000000000.0))
for points in [101, 201, 401]:
    grid = np.linspace(400000000000000.0, 1000000000000000.0, 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.

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