Past exam of the physics A course of the University of Cambridge / 2012 / Paper 1 / Section C / C11

In a finite potential barrier with $E<V$, the stationary Schrödinger equation has exponentially decaying and growing solutions, with local decay constant $\kappa =\sqrt{2m(V-E)}/\hslash$. Matching the wavefunction and its derivative at finite steps produces a transmitted probability current even though a classical particle at that energy could not cross. energy remains conserved. For a smooth thick barrier, the leading WKB transmission factor isFor a rectangular barrier of height $V_0$, width $d$ and $0<E<V_0$, exact matching gives $T=[1+V_0^2{\sinh }^2(\kappa d)/(4E(V_0-E)){]}^{-1}$. Thickness and energy therefore affect transmission exponentially in the opaque limit; prefactors and near-turning-point corrections require more than the leading exponent.

alpha decay illustrates escape from a nuclear well through a Coulomb barrier. A rough decay rate is an internal encounter frequency times barrier penetrability, multiplied by the probability of forming the outgoing cluster; penetrability alone is not a complete nuclear lifetime calculation. In a scanning tunnelling microscope, electron transmission through the vacuum gap gives an approximately exponential distance dependence $I\propto e^{-2\kappa d}$ at fixed conditions. A feedback loop adjusts tip height to maintain current, providing surface information. Bias, tip electronic structure and local electronic density of states also affect the current, so constant-current height is not purely geometric topography. Both applications exploit the same sensitivity of transmission to a classically forbidden region.

The Time-dependent Schrödinger equation $i\hslash {\partial }_t|\psi ⟩=H(t)|\psi ⟩$ governs unitary evolution of a closed system. If $H$ is independent of time, expand in its energy eigenstates:An individual energy eigenstate changes only by a phase, so its probability density and time-independent-observable expectations are stationary. Superpositions can have interference terms oscillating at $(E_n-E_m)/\hslash$; fixed energy populations do not imply that every observable is constant.

For general $H(t)$, $U(t,t_0)=\mathcal{T}\exp [-(i/\hslash )\int H(t^{\prime })d{t}^{\prime }]$ requires time ordering unless Quantum Hamiltonians at different times commute. The equivalent Heisenberg description evolves observables rather than the state, giving $d⟨A⟩/dt=(i/\hslash )⟨[H,A]⟩+⟨{\partial }_{t}A⟩$. Explicit time dependence can transfer energy from an applied perturbation.

For a weak perturbation $V(t)$ to a known $H_0$, the first-order transition amplitude from $n$ to $m\ne n$ isIts squared modulus gives the leading transition probability while depletion remains small. A nearly monochromatic perturbation emphasizes energy differences near its quantum of frequency; a continuum and suitable long-time limit yield Fermi's golden rule. These approximations have stated weak-coupling and time-resolution limits; they do not replace exact unitary dynamics in every regime.

For a photon scattering through angle $\theta$ from a free electron initially at rest, conserve energy and momentum. Eliminating the final electron momentum by its relativistic energy relation givesor $E^{\prime }=E/[1+(E/m{c}^2)(1-\cos \theta )]$. The recoil electron gains kinetic energy $E-E^{\prime }$. The shift is zero forward, one electron Compton wavelength at $90^{\circ }$, and twice that wavelength backward.

Compton's X-ray scattering measurements compared scattered spectra at different angles and identified an angle-dependent shifted component. This supports photon momentum $p=E/c$ and inelastic recoil, beyond the unchanged-frequency prediction of classical elastic Thomson scattering. Bound electrons and coherent scattering can also produce an unshifted component; initial momentum and binding broaden or modify the ideal free-electron result. Treating an electron as initially free and stationary is therefore a kinematic approximation appropriate when the relevant photon transfer dominates binding scales.^[ref]