IGNOU BPHET 141 2026 - The eigenfunction of a particle confined in a box of length L () is$

Question

The eigenfunction of a particle confined in a box of length L ( $0 \le x \le L$ ) is
$$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left( \frac{n \pi x}{L} \right)$ $
i) Show that $\psi_n(x)$ is not an eigenfunction of the momentum operator.
ii) Calculate the expectation value of the kinetic energy.

19 Jan 2026

Answer :

Word Count : 222

The eigenfunction of a particle in a one-dimensional box of length (L) is [ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi x}{L}\right), \quad n = 1,2,3,\dots ] The momentum operator in the (x)-direction is [ \hat{p}_x = -i \hbar \frac{d}{dx}. ] i) Check if (\psi_n(x)) is an eigenfunction of (\hat{p}_x): Apply (\hat{p}_x) to (\psi_n(x)): [ \hat{p}_x \psi_n(x) = -i \hbar \frac{d}{dx} \left[ \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi x}{L}\right) \right]. ] Differentiate: [ \frac{d}{dx} \sin\left(\frac{n \pi x}{L}\right) = \frac{n \pi}{L} \cos\left(\frac{n \pi x}{L}\right). ] Thus,

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