Discussion
Consider a particle of mass m in a one-dimensional box of length L. A time-independent perturbation H1 = α δ(x - L/2) is applied to the system. The first order correction to the ground state energy is :
Answer: C
The first order correction is \(\frac{2\alpha}{L}\).
The ground state of the box is \(\psi_{1} = \sqrt{2/L}\sin(\pi x/L)\). First-order perturbation theory gives the expectation value of the perturbation, and the delta function picks out the value at \(L/2\):
\[ \begin{aligned} E^{(1)} &= \alpha|\psi_{1}(L/2)|^{2} \\ &= \alpha\cdot\frac{2}{L}\sin^{2}\frac{\pi}{2} \\ &= \frac{2\alpha}{L} \end{aligned} \]
The correction would be 0 for the first excited state, which has a node at the centre. \(\frac{\pi^{2}\hbar^{2}}{2mL^{2}}\alpha\) multiplies the unperturbed energy by \(\alpha\), and \(L/\alpha\) has the wrong dimensions.
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