Past exam of the physics A course of the University of Cambridge / 2016 / Paper 1 / Section A

Use the unitary position-to-momentum Fourier transform $\varphi (p)=(2\pi \hslash {)}^{-1/2}\int \psi (x)e^{-ipx/\hslash }dx$. Integration giveswhere $\mathrm{sinc}u=\sin u/u$ with its continuous value one at zero. The amplitude is real and even, with central maximum $\sqrt{a/(2\pi \hslash )}$, zeros at $p=2n\pi \hslash /a$ for nonzero integer $n$, and alternating-sign sidelobes decaying as $1/|p|$.

The probability density is $|\varphi {|}^2$, not the signed Sinc function amplitude. Its integral is one by the unitarity of the Fourier transform.

Nonrelativistic acceleration from rest gives $p=\sqrt{2m_{e}eV}$ and $\lambda =h/p$. The observed deflection is the angle between incident and diffracted beams, $2\theta =22^{\circ }$, so Bragg's law at first order givesHere $\lambda \simeq 0.100\mathrm{nm}$. The numerical evaluation uses the externally documented CODATA 2014 values $h=6.626070040\times 10^{-34}\mathrm{J}\mathrm{s}$, $m_e=9.10938356\times 10^{-31}\mathrm{kg}$ and $e=1.6021766208\times 10^{-19}\mathrm{C}$; it does not assert which rounded constants were in the exam's handbook. Relativistic corrections at this voltage are negligible at the precision of the angle.^[ref]

If $\hat{A}|a⟩=\lambda |a⟩$, the Hermitian operator property makes $\lambda$ real. Applying the operator identity to this eigenvector gives ${\lambda }^3=1$. Its only real root is $\lambda =1$; the other cube roots of unity are nonreal. In an orthonormal eigenbasis, $\hat{A}={\sum }_a|a⟩⟨a|=\boxed{\hat{I}}$. Equivalently, the spectral theorem places the entire spectrum of the self-adjoint observable at one, giving the same conclusion. The quantum-mechanical setting assumes the usual well-defined observable and operator domains.

At the central values, $z=\sqrt{\tan 45^{\circ }}+0.10^2=1.010$. For independent input standard errors, linear propagation givesThe angular derivative here is with respect to radians, so ${\sigma }_x=2\pi /180=0.03491$. At $x=\pi /4$, the derivative is one, andThus $\boxed{z=1.010\pm 0.035}$, with the stated measurement uncertainty interpreted as one standard error. Correlated inputs would require a covariance term; their correlation is not supplied.

For independent charge arrivals described by a Poisson distribution, the one-sided Shot noise current Power spectral density is $S_I=2eI$. Treating the stated bandwidth as the Effective noise bandwidth,The expected rms noise is $5.66\mathrm{pA}$, about $0.566\%$ of the mean current. This is the full Shot noise value with Fano factor one; correlations can suppress or enhance it. The elementary charge is from the external CODATA 2014 table.^[ref]