IGNOU MPH 4 2026 - i) Normalize the wave function:ii) Calculate the expectation val

Question

i) Normalize the wave function:
$$\psi(x,0) = N \left[ \sin\left( \frac{\pi x}{L} \right) + \sin\left( \frac{2\pi x}{L} \right) \right], 0 \le x < L.$

ii) Calculate the expectation value of x for a particle in this state.

19 Jan 2026

Answer :

Word Count : 136

The normalization condition requires (\int_0^L |\psi(x,0)|^2,dx = 1). Given (\psi(x,0)=N\left[\sin\left(\frac{\pi x}{L}\right)+\sin\left(\frac{2\pi x}{L}\right)\right]), [ \int_0^L |\psi|^2 dx = N^2 \int_0^L \left[\sin^2!\left(\frac{\pi x}{L}\right)+\sin^2!\left(\frac{2\pi x}{L}\right) +2\sin!\left(\frac{\pi x}{L}\right)\sin!\left(\frac{2\pi x}{L}\right)\right] dx. ] Using orthogonality of sine functions over ([0,L]), [ \int_0^L \sin^2!\left(\frac{n\pi x}{L}\right)dx=\frac{L}{2}, \qquad \int_0^L \sin!\left(\frac{\pi x}{L}\right)\sin!\left(\frac{2\pi x}{L}\right)dx=0. ] Hence, [ \int_0^L |\psi|^2 dx = N^2\left(\frac{L}{2}+\frac{L}{2}\right)=N^2 L.

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