Classical Waves and Interference

Classical wave theory explained sound, water waves, elastic vibrations, and light interference long before quantum mechanics. It supplied the mathematical language of superposition, phase, wavelength, Fourier decomposition, diffraction, and coherence. Those ideas later became indispensable for matter waves and probability amplitudes, but their quantum use is not the same as saying that particles are ordinary classical waves.

To recognize interference, follow the relative phase: adding amplitudes before squaring produces a term that is missing if intensities are added instead.

Wave Equation

A simple scalar wave field $u(x,t)$ in one spatial dimension satisfies

$$\frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial x^2},$$

where $v$ is the wave speed. In three dimensions, a scalar wave equation has the form

$$\frac{\partial^2 u}{\partial t^2} = v^2\nabla^2u.$$

A monochromatic plane-wave solution can be written as

$$u(\mathbf r,t) = A\cos(\mathbf k\cdot\mathbf r-\omega t+\phi),$$

with

$$\lambda=\frac{2\pi}{\lVert\mathbf k\rVert}, \qquad \omega=2\pi\nu.$$

For nondispersive waves, $\omega=v\lVert\mathbf k\rVert$. In dispersive media or quantum wave mechanics, $\omega$ can depend on $\mathbf k$ in a more complicated way, and wave packets can spread.

Superposition

The wave equation is linear in many important regimes. If $u_1$ and $u_2$ are solutions, then

$$u=u_1+u_2$$

is also a solution. This is the superposition principle.

Superposition is not the same as probability. In classical wave theory, the fields add. Observed intensity or energy density is often quadratic in the field amplitude. This is why adding waves can produce cancellation in some places and enhancement in others.

Complex notation makes the phase structure compact:

$$u(\mathbf r,t) = \operatorname{Re} \left[ A e^{i(\mathbf k\cdot\mathbf r-\omega t)} \right].$$

The complex exponential is a calculation tool for a classical real field. In quantum mechanics, complex amplitudes have a deeper role because probabilities come from squared moduli of amplitudes.

Interference

Suppose two coherent waves of the same frequency contribute complex amplitudes $a_1$ and $a_2$ at one point. A quadratic detector responds to

$$I \propto \lvert a_1+a_2\rvert^2.$$

Expanding gives

$$I \propto \lvert a_1\rvert^2 + \lvert a_2\rvert^2 + 2\operatorname{Re}(a_1^*a_2).$$

The last term is the interference term. A stable relative phase can produce visible fringes. Fluctuations change the detector-averaged cross term; they erase it completely only when the averaged complex phase factor vanishes. Rapid fluctuations alone do not guarantee zero interference.

For two equal-amplitude contributions with the same polarization, no additional source or reflection phase, and path-length difference $\Delta L$, constructive interference occurs when

$$\Delta L=m\lambda,$$

and destructive interference occurs when

$$\Delta L=\left(m+\frac12\right)\lambda,$$

for integer $m$.

Diffraction

Diffraction is the spreading and reshaping of waves by apertures, obstacles, and finite source sizes. It is not a separate mystery from interference; it follows from superposing contributions from different parts of a wavefront.

For a narrow slit of width $a$, the far-field minima occur approximately at

$$a\sin\theta=m\lambda, \qquad m=\pm1,\pm2,\ldots.$$

For a grating or crystal, regularly spaced scatterers reinforce particular directions. This classical wave fact became historically important when electrons diffracted from crystals. The diffraction pattern looked wave-like, but the objects being diffracted were material particles.

To calculate how an aperture or spectrum determines a wave pattern, use Fourier Transform, Fourier Series, and Wave Packets.

Coherence

Interference requires coherence: a stable phase relation between the alternatives being combined. Coherence can be temporal, spatial, or tied to a controlled preparation.

Temporal coherence asks whether the phase remains predictable over a time delay. Spatial coherence asks whether fields at separated points have a stable phase relation. In laboratory terms, coherence determines whether fringes have high visibility or wash out.

For a two-path pattern, a common visibility measure is

$$\mathcal V = \frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}.$$

To separate coherence from unequal beam brightness, take matching polarizations and constant individual intensities $I_1,I_2>0$ at the observation point. After averaging over the fast optical oscillation, write their relative phase as a scanned offset $\phi$ plus a fluctuation $\delta$. Averaging over the detector’s exposure, or over repeated preparations with the same statistics, gives

$$\overline I(\phi)=I_1+I_2+2\sqrt{I_1I_2}\,\operatorname{Re}(e^{i\phi}\gamma), \qquad \gamma=\langle e^{i\delta}\rangle.$$

This follows by averaging the squared sum of the two fields, as in Fang’s correlation treatment, Eqs. (12)–(13). A time average equals an ensemble average only when the sampling justifies it. For fixed fluctuation statistics during the phase scan, the extrema and visibility are

$$\overline I_{\max,\min}=I_1+I_2\pm2\sqrt{I_1I_2}\,|\gamma|, \qquad \mathcal V=\frac{2\sqrt{I_1I_2}}{I_1+I_2}|\gamma|.$$

The triangle inequality gives $|\gamma|\le1$, and $2\sqrt{I_1I_2}\le I_1+I_2$ ensures nonnegative intensity. Thus low visibility need not mean weak coherence: even $|\gamma|=1$ gives $\mathcal V=0.6$ when $I_2=9I_1$. For fluctuating amplitudes, the corresponding normalized quantity is $\langle a_1^*a_2\rangle/\sqrt{\langle|a_1|^2\rangle\langle|a_2|^2\rangle}$; it need not reduce to a phase average.

For a concrete phase-noise model, let the sampled $\delta$ be Gaussian with mean zero and variance $\sigma_\phi^2$. The Gaussian integral gives

$$\gamma=\frac{1}{\sqrt{2\pi}\sigma_\phi} \int_{-\infty}^{\infty}e^{-\delta^2/(2\sigma_\phi^2)}e^{i\delta}\,d\delta =e^{-\sigma_\phi^2/2}.$$

With $I_1=I_0$, $I_2=9I_0$ and $\sigma_\phi=1$ radian, the extrema are $13.639I_0$ and $6.361I_0$, so $\mathcal V\simeq0.364$. Removing the phase noise restores visibility to $0.6$; balancing the noiseless beams restores it to one. A uniform phase over a complete $2\pi$ interval instead gives $\gamma=0$ exactly. These are different experimentally testable reasons for reduced contrast.

Broad spectra, finite source size, and environmental fluctuations can all reduce the relevant field correlation. In quantum experiments, which-way records also reduce coherence in the relevant degrees of freedom even when no person observes the record; that mechanism requires the quantum description of coherent alternatives.

Wave Packets

Localized wave disturbances are built from superpositions of many wavelengths. A one-dimensional packet can be written schematically as

$$u(x,t) = \int A(k)e^{i(kx-\omega(k)t)}\,dk.$$

For a narrow spectrum centered on $k_0$, an expansion of $\omega(k)$ to first order shows that the envelope approximately moves at the group velocity

$$v_g=\left.\frac{d\omega}{dk}\right|_{k_0},$$

while phase fronts move at

$$v_{\mathrm{ph}}=\frac{\omega}{k}.$$

This distinction became important for matter waves. A plane wave with one wavelength is not localized. A localized particle-like wave state requires a packet, and packet spreading depends on the dispersion relation $\omega(k)$. See Group Velocity and Phase Velocity.

Why Wave Behavior Became Relevant to Particles

Classical physics drew a strong practical distinction: waves spread and interfere; particles follow localized trajectories. Quantum evidence broke that division.

The key steps were:

• de Broglie’s matter-wave relation connected momentum and wavelength through $\lambda=h/p$;
• electron diffraction showed material particles producing wave-like angular patterns;
• double-slit experiments showed that coherent alternatives interfere even when detections occur one at a time;
• wave mechanics made $\psi$ a central object, but not a classical material density.

The modern lesson is not that electrons are tiny water waves. It is that quantum states carry complex amplitudes whose phases affect probabilities. Classical wave theory supplied the mathematics of phase and superposition; quantum mechanics changed the meaning of the wave-like object.

Bridge to Modern Formalism

Use classical wave theory for:

• phase, wavelength, frequency, and interference geometry;
• Fourier decomposition and packets;
• diffraction from apertures and periodic structures;
• coherence and visibility intuition.

Use quantum formalism for:

• probability amplitudes;
• Born probabilities;
• wavefunctions as state representations;
• which-way information and decoherence;
• matter-wave propagation and measurement.

Continue with Probability Amplitudes, Wavefunctions and Probability Density, Plane Waves and Delta Normalization, and the Double-Slit Experiment.

Common Mistakes

• Saying that quantum mechanics discovered interference; classical waves already interfered.
• Treating a quantum wavefunction as an ordinary classical wave in space.
• Forgetting that a single detection event can be localized even when the accumulated distribution shows interference.
• Treating loss of interference as requiring a conscious observer.
• Ignoring coherence and phase stability when comparing one-slit and two-slit patterns.
• Using wave-particle duality as a substitute for amplitude addition and the Born rule.

Exercises

1. Two equal-amplitude waves reach a point with phase difference $\Delta\phi$. Show how constructive and destructive interference arise.

Solution

Let the amplitudes be $a$ and $ae^{i\Delta\phi}$. Then

$$\lvert a+ae^{i\Delta\phi}\rvert^2 = \lvert a\rvert^2 \lvert1+e^{i\Delta\phi}\rvert^2.$$

If $\Delta\phi=2\pi m$, the two amplitudes add in phase and the intensity is $4\lvert a\rvert^2$. If $\Delta\phi=(2m+1)\pi$, they cancel and the intensity is zero.

2. Why does electron diffraction not mean that electrons are classical waves?

Solution

Electron diffraction shows that electron amplitudes carry phase and wavelength information, so probabilities can form wave-like patterns. But individual detections are localized events, and the wavefunction is not a classical material displacement field. Quantum mechanics uses amplitudes and the Born rule rather than a classical wave intensity alone.

3. Two perfectly coherent beams have intensities $I_1$ and $I_2$ and matching polarization. Show that unequal brightness can reduce the visibility even without loss of coherence. Then consider a small frequency difference $\Delta\omega$: during an exposure $0\le t\le T$, let the relative phase be $\phi+\Delta\omega t$. What visibility does the detector record when $\phi$ is scanned with the same timing in each exposure? Assume $T$ is long compared with an optical period.

Solution

The extreme intensities are $I_{\max,\min}=I_1+I_2\pm2\sqrt{I_1I_2}$, so $\mathcal V=2\sqrt{I_1I_2}/(I_1+I_2)$. Equal beams give unit visibility, but $I_2=9I_1$ gives $\mathcal V=0.6$ despite perfect coherence. Visibility therefore tests both coherence and beam balance.

For the detuned beams, integrate the phase factor over the exposure:

$$\gamma_T=\frac1T\int_0^T e^{i\Delta\omega t}\,dt =e^{i\Delta\omega T/2}\operatorname{sinc}(\Delta\omega T/2), \qquad \operatorname{sinc}x=\frac{\sin x}{x},\quad\operatorname{sinc}0=1.$$

Multiply the beam-balance factor by $|\operatorname{sinc}(\Delta\omega T/2)|$ to obtain the visibility. For $I_2=9I_1$ and $\Delta\omega T=\pi$, it is $1.2/\pi\simeq0.382$; at $2\pi$ it vanishes. The relative phase is fully deterministic, but this exposure averages over one complete beat cycle. Shorter exposures can resolve the moving fringes. Vanishing time-averaged contrast therefore does not establish random phase or a quantum which-way record.

References

• M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge University Press, 1999.
• C. Davisson and L. H. Germer, “Diffraction of Electrons by a Crystal of Nickel,” Physical Review 30, 705-740, 1927, DOI: 10.1103/PhysRev.30.705.
• L. de Broglie, Recherches sur la théorie des quanta, doctoral thesis, Paris, 1924.
• N. Fang, “Lecture Notes on Wave Optics,” Lecture 11, MIT 2.71/2.710 Optics, 12 March 2014, pp. 2–3, Eqs. (12)–(13).
• E. Hecht, Optics, 5th ed., Pearson, 2017.
• M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed., American Institute of Physics, 1989.
• T. Young, “The Bakerian Lecture: Experiments and calculations relative to physical optics,” Philosophical Transactions of the Royal Society 94, 1-16, 1804, DOI: 10.1098/rstl.1804.0001.