First-order perturbation theory gives $E_n^{(1)} = 0$ when $H'$ is:

If $H'$ is odd and $|\psi_n|^2$ is even, the integral $\int \psi_n^* H' \psi_n dx$ is zero by symmetry.

Correct answer: An odd function and $\psi_n$ is an even (or odd) function of $x$

The variational method provides an upper bound on the:

$\langle H \rangle \geq E_0$ — any trial state overestimates the ground state energy.

Correct answer: Ground state energy

Sketch how tunnelling current in a scanning tunnelling microscope (STM) depends on the tip-sample distance, using the WKB formula.

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Correct answer: Tunnelling probability $T \propto e^{-2\kappa d}$ where $\kappa = \sqrt{2m(V-E)}/\hbar$ and $d$ is the barrier width. Current $I \propto T$ decreases exponentially with distance — atomic-scale resolution arises from this extreme sensitivity.