Past exam of the physics A course of the University of Cambridge / 2013 / Paper 1 / Section C

A pure quantum state is a unit ray in a complex Hilbert space: normalization fixes total probability, a common phase is unobservable, and superpositions retain relative phases. Observables are self-adjoint operators. For an ideal projective measurement with spectral projector $P_a$, the Born rule gives $p(a)=⟨\psi |P_a|\psi ⟩$ and the Lüders rule gives conditional state $P_a\psi /\sqrt{p(a)}$. This preserves an eigenspace, not necessarily a unique vector when the outcome is degenerate. Between measurements, a closed system evolves unitarily according to $i\hslash {\partial }_t\psi =H\psi$. Composite systems use tensor products; quantum statistics additionally restrict identical-particle states. These are a framework supported by many experiments, not separate logical deductions from any one experiment.

In the photoelectric effect, monochromatic light illuminates a clean metal and a retarding potential measures the largest emitted electron kinetic energy. frequency controls the stopping energy, $e{V}_s=h\nu -\Phi$, while in the one-photon regime intensity mainly controls the emission rate. A threshold frequency and a stopping-energy slope $h/e$ support quantized energy exchange rather than gradual absorption of arbitrary classical-wave energy by each electron. Contact potentials, surface contamination, dark current and spectral impurities require controls. Robert Andrews Millikan tested the frequency–stopping-potential relation with careful surface and electrical control.^[ref]

The Stern-Gerlach experiment sends a collimated neutral silver beam through an inhomogeneous magnetic field onto a detector. The force is approximately $F_z={\mu }_z{\partial }_{z}B$ in an appropriate strong-field electronic description. Two resolved components rather than a continuous classical orientation spread support discrete angular-momentum projections. Field-off beam profiles and magnetic-gradient/beam-width controls distinguish splitting from ordinary broadening. In the modern description the silver ground electronic state has $J=1/2$; the 1922 experiment preceded the electron-spin interpretation. Ideal sequential analyzers along different axes illustrate incompatible observables and state preparation, but were not the original apparatus.^[ref]

In the Davisson–Germer experiment, accelerated electrons scatter from a nickel single crystal and an angularly scanned collector measures the scattered intensity. Changing voltage and crystal orientation changes narrow diffraction maxima. The inferred wavelength agrees with $\lambda =h/p$, with nonrelativistic $p=\sqrt{2meV}$. vacuum, crystal cleaning, energy selection and background measurements matter: adsorbed surface gas can produce additional diffraction features. This supports matter-wave amplitudes and interference, with measured intensities related to their squared magnitude; it does not mean an electron must follow a classical orbit through the apparatus.^[ref] Together these experiments motivate quantized outcomes, wave amplitudes and probabilistic measurement while leaving a distinction between observed evidence and the general postulates used to organize it.

A continuous symmetry is represented by a unitary family $U(ϵ)=e^{-iϵG/\hslash }$ with a self-adjoint conserved generator of a continuous quantum symmetry $G$. If $UH{U}^{†}=H$ for all $ϵ$, differentiating gives $[H,G]=0$. Thus, for no explicit time dependence,translation invariance conserves the corresponding momentum component; rotational invariance about one axis conserves that angular-momentum component. A central potential conserves all components and $L^2$, but the components need not be simultaneously measurable because $[L_x,L_y]=i\hslash L_z$. Time-translation invariance of a closed system gives a time-independent Hamiltonian and energy conservation.

A discrete symmetry need not have a continuously varying generator. For parity, $[H,P]=0$ lets stationary states be chosen even or odd, and preserves parity sectors. symmetry does not force every state to be an eigenstate of its conserved observable. Explicit time dependence adds $⟨\partial G/\partial t⟩$ to the conservation equation, and an external symmetry-breaking potential generally permits exchange with the environment. Degenerate eigenspaces require choosing a compatible common basis rather than assuming every eigenvector already diagonalizes every commuting operator.

For a dilute linear diatomic gas with fixed molecular number, translation contributes molar $C_V=3R/2$. Rotational levels are $E_J={\hslash }^{2}J(J+1)/(2I)$ with degeneracy $2J+1$:At $T\ll {\theta }_r$ rotational excitation is frozen out; at $T\gg {\theta }_r$ the partition function is proportional to $T$, giving $U_{\mathrm{rot}}=N{k}_{B}T$ and $C_{V,\mathrm{rot}}=N{k}_B$. Nuclear-spin populations and molecular symmetry modify detailed low-temperature behaviour. The high-temperature result corresponds to two quadratic rotational degrees of freedom; a nonlinear molecule has three and contributes $3N{k}_B/2$.

For each independent harmonic vibration, write $x=\hslash \omega /(k_{B}T)$. Its energy and heat capacity areThe contribution is exponentially small at low temperature and approaches $N{k}_B$ per mode at high temperature. zero-point energy does not contribute to $dU/dT$. Typically rotations activate before bond stretching, producing diatomic plateaus $C_V/R=3/2,5/2,7/2$ as temperature increases within the molecular regime; ideal-gas $C_P=C_V+R$. Dissociation, electronic excitation and nonideal interactions eventually invalidate this simple sequence. Polyatomic gases sum over their distinct vibrational frequencies; no single activation temperature describes all modes.

A black body absorbs all incident radiation; a small opening into an equilibrated opaque cavity approximates it because internal reflections strongly suppress escape without absorption. Equilibrium radiation has zero photon chemical potential. Counting electromagnetic modes and assigning mean occupation $(e^{h\nu /(k_{B}T)}-1{)}^{-1}$ gives the spectral energy densityAt low frequency this becomes the Rayleigh-Jeans law, $u_{\nu }\simeq 8\pi k_{B}T{\nu }^2/c^3$; using that limit at all frequencies gives the ultraviolet catastrophe. At high frequency the exponential suppresses the spectrum. Quantized mode energy resolves the classical divergence.

Integration gives $u=a_{\mathrm{rad}}{T}^4$. An isotropic radiation field sends outward flux $cu/4$ through a small aperture, soDifferentiating the wavelength spectrum gives ${\lambda }_{\max }T\simeq 2.90\times 10^{-3}\mathrm{m}\mathrm{K}$; the frequency-spectrum maximum is a different transformed peak, not simply $c/{\lambda }_{\max }$. A real surface has emissivity less than one and possibly dependent on wavelength. Cavity temperature uniformity, aperture geometry, calibrated detector response and background subtraction are necessary when testing the spectrum or inferring temperature from it.