The energy eigenstates of the infinite square well are of course not unique as a basis for the Hilbert space, which has no support at the boundaries of the well, where \(x=0\) and \(x=a\). For example, the wavefunction

\[\begin{equation*}\zeta_{1}(x)=N x(a-x) \tag{5.82}\end{equation*}\]

vanishes at the boundaries of the well, but is clearly not the same as the sinusoidal ground-state wavefunction of the infinite square well. A comparison of this wavefunction with the wavefunction of the ground state of the infinite square well is shown in Fig. 5.7. In this problem, we will invert the usual problem, and identify the potential for which \(\zeta_{1}(x)\) is the ground-state wavefunction.

(a) First, normalize this wavefunction and determine the normalization constant \(N\).

(b) What is the overlap of \(\zeta_{1}(x)\) with the ground-state wavefunction of the infinite square well? That is, what fraction of \(\zeta_{1}(x)\) is described by the ground state of the infinite square well?

(c) Now, let's assume that this wavefunction \(\zeta_{1}(x)\) is an eigenstate of another Hamiltonian with potential \(V(x)\) :

\[\begin{equation*}\left(-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}}+V(x)\right) \zeta_{1}(x)=E_{1} \zeta_{1}(x) \tag{5.83}\end{equation*}\]

for some ground-state energy \(E_{1}\). Determine the potential \(V(x)\) in terms of the energy \(E_{1}\). How is this potential similar to and different from the infinite square well?
(d) Now, with this potential \(V(x)\), can you guess the wavefunction of the first excited state? How much larger is its energy than the ground state? Can you guess higher excited states?

Transcribed Image Text:

Fig. 5.7 V(2) 7 = 0 2=4 Comparison between the ground-state wavefunction (x) of the infinite square well (solid) and the wavefunction S1(x) (dashed) of Eq. (5.82).