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.
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!}} ______ __________ ________ _____ _______.
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Determine the WKB approximation for the bound state energy of a particle of mass m in the potential:
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and
. Show that
and determine the eigenvalues of
.
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