IGNOU MPH 8 2026 - Determine the wave function and energy of the ground state and first e

Question

Determine the wave function and energy of the ground state and first excited state for a system of two identical bosons in 1D simple harmonic oscillator.

19 Jan 2026

Answer :

Word Count : 217

For a 1D simple harmonic oscillator (SHO), the single-particle Hamiltonian is [ \hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2 \hat{x}^2 ] with energy eigenvalues and eigenfunctions: [ E_n = \left(n + \frac{1}{2}\right)\hbar \omega, \quad n = 0,1,2,\dots ] [ \psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{m \omega}{\pi \hbar}\right)^{1/4} e^{-m\omega x^2/(2\hbar)} H_n\left(\sqrt{\frac{m\omega}{\hbar}} x\right) ] where (H_n) are Hermite polynomials. For two identical bosons, the total wave function must be symmetric under particle exchange (x_1 \leftrightarrow x_2). --- Ground State: * Both bosons occupy the lowest single-particle state (n=0). [ \Psi_0(x_1,

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