Approximate calculation of the energies for the finite potential well
For a deep but finite potential well, the wave functions corresponding to the lowest energy levels are well approximated by the simple, sinusoidal wave functions obtained in the case of an infinite well (see Figure 18.11 ). Therefore, a simple approximation for the energy spectrum of a particle in a finite well can be obtained by computing the expectation value of its Hamiltonian in the energy eigenstates for the infinite-well case. Here, we test the validity of this approximation by direct comparison with the exact energy spectrum computed in the chapter.
(a) Compute the normalized energy eigenfunctions $\psi_{n}(x)$ for a particle in an infinite-well potential of width $a$ by solving the Schrödinger equation, Equation 18.11 , with the boundary condition given by Equation $18.13,$ and imposing the normalization condition $\int_{0}^{a} \psi_{n}(x)^{*} \psi_{n}(x) \mathrm{d} x=1$
(b) Compute the approximate values of the dimensionless energies for a particle in a finite well, $\tilde{x}_{n}=-\tilde{E}_{n} / V_{0},$ where $\tilde{E}_{n}=\left\langle\psi_{n}|\hat{H}| \psi_{n}\right\rangle,$ using the wave functions $\psi_{n}$ from (a) and the Hamiltonian for the particle in a finite well described in the chapter. Express your result for $\tilde{x}_{n}$ in terms of the dimensionless parameter $\alpha$ defined in Equation 18.30
(c) Following the strategy outlined in the Computational Exploration on p. $733,$ compute the exact values of dimensionless energies $x_{n}$ for $n=1,2,3,4$ by solving Equation 18.32 for $\alpha$ that corresponds to a finite well with $V_{0}=10 \mathrm{eV}$ and $a=1 \mathrm{nm} .$ Compare these values with those obtained in (b) and comment on the validity of the approximation.