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2. In this problem you are asked to show that the eigenfunctions, \( \psi_{m} \) and \( \psi_{n} \), belonging to different discrete eigenvalues \( E_{m} \) and \( E_{n} \), are orthogonal. For a real potential, the energy eigenfunction \( \psi_{n}(x) \) and the complex conjugate of \( \psi_{m}(x) \) satisfy the equations \[ \left[-\frac{\hbar^{2}}{2 m} \frac{\mathrm{d}^{2}}{\mathrm{~d} x^{2}}+V(x)\right] \psi_{n}=E_{n} \psi_{n} \] and \[ \left[-\frac{\hbar^{2}}{2 m} \frac{\mathrm{d}^{2}}{\mathrm{~d} x^{2}}+V(x)\right] \psi_{m}^{*}=E_{m} \psi_{m}^{*} \] Multiply the first equation by \( \psi_{m}^{*} \) and the second by \( \psi_{n} \), subtract and show that
Submitted by Sokhela V. Apr. 02, 2023 07:10 a.m.
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