Past exam of the physics A course of the University of Cambridge / 2017 / Paper 1 / Section C / C11

The Bohr model combines classical circular motion in the Coulomb potential with explicitly quantum postulates. A one-electron atom of nuclear charge $+Ze$ has electron–nucleus reduced mass $\mu$. With $C=Z{e}^2/(4\pi {\epsilon }_0)$, the relative circular orbit obeysBohr postulates permitted nonradiating stationary orbits with $L=n\hslash$, for $n=1,2,\dots$. These assumptions giveFor hydrogen with the nucleus taken infinitely heavy, the $n=1$ radius is the Bohr radius, $a_0\simeq 5.29\times 10^{-11}\mathrm{m}$. The finite reduced mass gives the small nuclear-motion correction, and increasing $Z$ contracts the one-electron orbit.

The circular-motion equation gives $T=C/(2r)$ while the potential energy is $-C/r$, so the total energy is $-C/(2r)$. Substituting the quantized radius yieldsFor the heavy-nucleus hydrogen approximation, $E_n\simeq -13.6\mathrm{eV}/n^2$. The ionization threshold is zero energy, so removing the ground-state electron requires about $13.6\mathrm{eV}$.

The model further postulates that radiation is emitted or absorbed only in transitions between stationary orbits, with photon energy $h\nu =|E_i-E_f|$. Emission from $n_i>n_f$ givesThus it reproduces the principal hydrogen spectral series, their limiting wavelengths, and the gross hydrogenic energy levels. For example, $n=3\to 2$ in hydrogen has $\Delta E\simeq 1.89\mathrm{eV}$ and $\lambda \simeq 656\mathrm{nm}$, a visible line. Absorption reverses the transition when the photon matches the energy gap.

The achievement is that quantized energies explain discrete spectra, but the model's stationary orbits are postulates rather than consequences of classical electrodynamics: an accelerating classical charge would radiate. It does not provide a general calculation of line intensities or transition selection rules and fails as a quantitative theory of many-electron atoms. Spin, fine structure and the full Zeeman effect require further physics. Modern quantum mechanics replaces definite electron orbits by wavefunctions and quantum probabilities. In particular, the true hydrogen ground state has $\ell =0$, so its orbital angular momentum vanishes; the model's rule $L=n\hslash$ should not be identified with the modern orbital eigenvalue rule $L^2=\ell (\ell +1){\hslash }^2$. Bohr's energy quantization remains a successful hydrogenic approximation, while its literal orbital picture does not.

Ehrenfest theorem relates the dynamics of quantum expectation values to operator commutators. For a normalized state obeying $i\hslash |\dot{\psi }⟩=\hat{H}|\psi ⟩$ and a possibly time-dependent observable $\hat{A}$, differentiate $⟨A⟩=⟨\psi |\hat{A}|\psi ⟩$. Using the adjoint Schrödinger equation for the bra givesThusThe identity assumes a common domain on which the required products and derivatives are defined.

For $H=p^2/(2m)+V(x)$, the canonical commutation relation gives $[x,H]=i\hslash p/m$ and $[p,H]=-i\hslash V^{\prime }(x)$. ThereforeThese resemble Hamilton's equations and Newton's second law, but generally $⟨V^{\prime }(x)⟩\ne V^{\prime }(⟨x⟩)$. If the state is localized about $\overline{x}=⟨x⟩$, a local expansion givesThe classical trajectory equation for the mean is accurate when the force changes little across the wave packet. For a quadratic harmonic potential, $V^{\prime }$ is linear and this closure is exact for every state with the relevant finite moments: $m{d}^2⟨x⟩/d{t}^2=-m{\omega }^2⟨x⟩$. This exact classical evolution of the centroid does not eliminate its quantum position uncertainty or make every state a point particle.

The correspondence principle is broader: quantum predictions recover classical ones in an appropriate limit, such as large action compared with $\hslash$ and observations that cannot resolve individual quantum levels. It is not the assertion that every large quantum number automatically makes every possible state classical. Delocalized superpositions and interference can remain significant unless the physical preparation and measurement justify a classical description.

A spectral example is useful. For a quantum harmonic oscillator, neighboring levels differ by $\hslash \omega$, giving transition frequency $\omega /(2\pi )$, equal to its classical oscillation frequency. A coherent state makes the connection more concrete: its mean position and momentum oscillate classically while the finite-width Gaussian profile retains its quantum uncertainty. For hydrogenic Bohr levels $E_n=-A/n^2$, transitions between nearby large $n$ have $\Delta E/\hslash \simeq 2A/(\hslash n^3)$, which equals the classical circular orbital angular frequency in that model. Ehrenfest's theorem supplies an exact expectation-value bridge; a classical trajectory follows only when the force can also be evaluated at the mean position.

A Hermitian operator satisfies the adjoint symmetry conditionfor vectors in its specified domain. For a finite matrix this means $A=A^{†}$. For an unbounded differential operator, boundary conditions and domains matter: symmetry on a chosen domain is not alone sufficient for self-adjointness. Physical observables require the corresponding self-adjoint operator and its spectral measure. This distinction prevents purely formal integration by parts from being mistaken for a complete operator definition.

Real eigenvalues. If $A|a⟩=a|a⟩$ with nonzero norm, Hermiticity impliesso $a=a^{\ast }$. Thus eigenvalues representing measurement outcomes are real.

Orthogonality. For eigenvectors $|a⟩,|b⟩$,Hence $(a-b)⟨b|a⟩=0$, and distinct eigenvalues have orthogonal eigenvectors. Within a degenerate eigenspace, Gram-Schmidt orthogonalization can choose an orthonormal basis. In finite dimensions the spectral theorem gives a complete orthonormal eigenbasis andwhere $P_a$ projects onto each eigenspace. In infinite dimensions a continuous spectrum may require a spectral integral rather than a normalizable eigenvector for every possible outcome; a position operator on the full line is a familiar example.

Measurements and probabilities. In the discrete case the Born rule assigns probability $⟨\psi |P_a|\psi ⟩$ to the outcome $a$. For a nondegenerate eigenbasis, $|\psi ⟩={\sum }_a{c}_a|a⟩$ gives probabilities $|c_a{|}^2$, withThe variance vanishes precisely when the state lies in an eigenspace with a single sharp outcome. A repeatable ideal projective measurement prepares the corresponding eigenspace; superpositions of different eigenvalues do not have a preassigned sharp value of that observable in this description.

Compatibility and dynamics. Commuting Hermitian matrices admit simultaneous diagonalization, permitting a common set of sharp labels. For example $L^2,L_z$ commute, whereas $L_x,L_y$ do not. In the differential-operator setting the corresponding spectral compatibility requires appropriate domain conditions. Noncommuting observables obey the Robertson uncertainty principleThe canonical commutation relation then gives $\Delta x\Delta p\ge \hslash /2$.

The Hamiltonian operator represents energy. A self-adjoint $H$ generates unitary time evolution, $U(t)=e^{-iHt/\hslash }$, preserving norm and total probability; its eigenvalues determine stationary-state phases and transition energies. Hermiticity supplies real outcomes and orthogonal eigenspaces, while self-adjointness and the spectral theorem make these measurement and evolution rules mathematically complete.