Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section A(ii) / 14

For a nonrelativistic electron,Using the supplied constants givesThis is comparable with atomic lattice spacings. A suitable crystalline surface or sufficiently thin crystal can therefore diffract the electron beam; the reciprocal-lattice geometry and scattering conditions determine the actual angles. A grating or slit can also diffract matter waves if its scale and coherence conditions are suitable. The kinetic energy is tiny compared with the electron rest energy, so relativistic corrections are negligible at the stated precision.

Only $n=1,2$ have energies below $6ϵ$. Squaring the corresponding expansion coefficients gives $0.2^2+0.3^2=0.13$.

The printed rounded coefficients have normThus the expected rounded answer is $P(E<6ϵ)\simeq 0.13$. If the decimal coefficients are treated as literal exact inputs and the state normalized, the exact probability for that input is $0.13/S\simeq 0.129916$. This distinction avoids treating rounding error as a physical departure from wavefunction normalization.

Orthonormal energy eigenstates contribute incoherently to the expectation of energy:Therefore $⟨E⟩\simeq 13.2018ϵ$ for the normalized literal decimals. Ignoring the small rounding-induced norm error gives $13.2104ϵ$, conventionally reported at this precision as about $13.2ϵ$. The mean need not equal an individual energy eigenvalue.

Each energy component acquires its own phase:The $1/\sqrt{S}$ factor normalizes the printed rounded input; it can be omitted if the coefficients are regarded as an already normalized rounded representation.

The energy probabilities and $⟨E⟩$ remain constant under this time-independent hamiltonian because phase factors have unit modulus. position probabilities can nevertheless vary: interference between different energy components generally changes with their relative phases. A stationary energy distribution is not the same thing as a stationary spatial probability distribution.

The result $16ϵ$ projects the state onto the corresponding nondegenerate energy eigenstate. Immediately after an ideal measurement,for an unobservable overall phase $\chi$. Subsequent isolated evolution adds $e^{-16iϵt/\hslash }$ but does not mix in other energy eigenstates. A repeat ideal energy measurement gives $16ϵ$ with probability one, provided the hamiltonian is unchanged and no intervening disturbance occurs.