Quantum Wave Phenomena
Quantum Wave Phenomena
Quantum mechanics begins with a radical idea: matter behaves like waves. An electron, a photon, even a molecule can spread through space, interfere with itself, and be detected only as a probability. The mathematics behind these effects — complex amplitudes, wavefunctions, and Fourier transforms — reveals a universe far stranger and richer than classical physics imagined.
Every quantum system is described by a complex-valued wavefunction $\psi(x, t)$. The probability of finding the particle at position $x$ is:
$$P(x) = |\psi(x,t)|^2$$
The wavefunction must be normalized: $\int_{-\infty}^{+\infty} |\psi|^2\, dx = 1$.
Double-Slit Interference
Young's double-slit experiment, performed with single electrons, is the clearest demonstration of wave-particle duality. When two slits of separation $d$ are illuminated by particles of de Broglie wavelength $\lambda$, a screen at distance $L$ shows bright fringes spaced by:
$$\Delta y = \frac{\lambda L}{d}$$
Each particle passes through both slits simultaneously as a wave — and arrives at the screen as a point particle. No classical picture can explain the pattern.
Particle in a Box
The simplest exactly-solvable quantum system: a particle confined to a 1D box of length $L$ with infinite potential walls. The allowed wavefunctions and energies are:
$$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right), \qquad E_n = \frac{n^2 \pi^2 \hbar^2}{2 m_e L^2}$$
The integer $n = 1, 2, 3, \ldots$ is the quantum number. Note that the zero-point energy $E_1 \neq 0$ — the particle can never be perfectly at rest, in agreement with the uncertainty principle.
The Heisenberg Uncertainty Principle
Quantum mechanics forbids the simultaneous precise knowledge of position and momentum:
$$\sigma_x \cdot \sigma_p \geq \frac{\hbar}{2}$$
A particle localized sharply in space (small $\sigma_x$) must have a widely spread momentum distribution (large $\sigma_p$), and vice versa. This is not a statement about measurement disturbance — it is a fundamental property of waves.