Wave Mechanics: Heisenberg Uncertainty Principle, Schrödinger's wave equation (concept & particle-in-a-box)

Heisenberg Principle, Schrödinger Equation & Particle in a Box

1. Heisenberg Uncertainty Principle (1927)

Impossible to determine position ($x$) and momentum ($p$) of an electron simultaneously with perfect accuracy:

where $\hbar = h/2\pi$.

• Effect is negligible for large objects (cars, balls) but decisive for electrons.
• Consequence: Bohr's fixed orbits are wrong — we can only talk about probability clouds (orbitals).
• Energy–time form: $\Delta E \cdot \Delta t \ge h/4\pi$ — explains natural line broadening in spectra.

Numerical: electron with $\Delta x = 10^{-10}$ m (atomic size):

so $\Delta v = \Delta p/m \approx 5.8 \times 10^5$ m/s — huge uncertainty, orbit concept collapses.

2. Schrödinger's wave equation (concept)

Treats electron as a 3D standing wave. Time-independent form:

In 1D:

Symbol	Meaning
$\psi$ (psi)	Wave function — amplitude of electron wave
$\|\psi\|^2$	Probability density of finding electron at that point
$E, V$	Total and potential energy
Quantum numbers $n, l, m, s$	Emerge as boundary conditions on $\psi$

• $\psi$ itself has no physical meaning; $\|\psi\|^2$ does (Born interpretation).
• Nodes are points where $\psi = 0$ (zero probability).

3. Particle in a 1D box (key model)

Electron confined to length $L$ with $V = 0$ inside, $V = \infty$ outside. Solution:

E3 ───  ψ: two nodes
E2 ───  ψ: one node at centre
E1 ───  ψ: no node (ground state, zero-point energy ≠ 0)
     |_______________|
     0               L

Key lessons:

• Energy is quantized by confinement alone — no quantization postulate needed.
• Zero-point energy: $E_1 = h^2/8mL^2
  e 0$ — particle can never be at rest (matches uncertainty principle).
• Spacing grows as $n^2$; gap narrows as box gets bigger (why nano-particles show size-dependent colour — quantum confinement).
• Model for conjugated dyes, quantum dots, electrons in metals.

Numerical: electron in $L = 1$ nm box: $E_1 = 6.0 \times 10^{-20}$ J $\approx 0.38$ eV; $E_1 \to E_2$ transition $\Delta E = 3E_1 \approx 1.13$ eV → $\lambda \approx 1100$ nm.

Exam Points

• State both forms of uncertainty principle + one numerical (3 marks standard).
• Write Schrödinger equation and explain $\psi$ vs $\|\psi\|^2$ (3 marks); derive/quote $E_n = n^2h^2/8mL^2$ and sketch $\psi_1, \psi_2$ (5 marks favourite).