2.6: Exercises

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Exercises

Exercise \(\PageIndex{1}\)

Use the variational theorem to prove that a 1D potential well has at least one bound state. Assume that the potential \(V(x)\) satisfies (i) \(V(x) < 0\) for all \(x\), and (ii) \(V(x)\rightarrow 0\) for \(x\rightarrow\pm\infty\). The Hamiltonian is

\[\hat{H} = - \frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x).\]

Consider a (real) trial wavefunction

\[\psi(x;\gamma) = \left(\frac{2\gamma}{\pi}\right)^{1/4} \,e^{-\gamma x^2}.\]

Note that this can be shown to be normalized to unity, using Gauss’ integral

\[\int_{-\infty}^{\infty} dx\; e^{-2\gamma x^2} = \sqrt{\frac{\pi}{2\gamma}}.\]

Now prove that

\[\begin{align} \begin{aligned} \langle E \rangle &= \int_{-\infty}^\infty dx \; \psi(x) \, \hat{H} \,\psi(x) \\ &= \frac{\hbar^2}{2m} \int_{-\infty}^\infty dx\, \left(\frac{d\psi}{dx}\right)^2 \;+\; \int_{-\infty}^\infty dx\; V(x) \,\psi^2(x) \\ &= A\sqrt{\gamma} \left[\sqrt{\gamma} \;+\, B \int_{-\infty}^\infty dx\; V(x) \;e^{-\gamma x^2} \right], \end{aligned} \end{align}\]

where \(A\) and \(B\) are positive real constants to be determined. By looking at the quantity in square brackets in the limit \(\gamma \rightarrow 0\), argue that \(\langle E \rangle < 0\) in this limit. Hence, explain why this implies the existence of a bound state.

Finally, try generalizing this approach to the case of a 2D radially-symmetric potential well \(V(x,y) = V(r)\), where \(r = \sqrt{x^2+y^2}\). Identify which part of the argument fails in 2D. [For a discussion of certain 2D potential wells that do always support bounds states, similar to 1D potential wells, see Simon (1976).]

Exercise \(\PageIndex{2}\)

In this problem, you will investigate the existence of bound states in a 3D potential well that is finite, uniform, and spherically-symmetric. The potential function is

\[V(r,\theta,\phi) = -U\Theta(a-r),\]

where \(a\) is the radius of the spherical well, \(U\) is the depth, and \((r,\theta,\phi)\) are spherical coordinates defined in the usual way.

The solution involves a variant of the partial wave analysis discussed in Appendix A. For \(E < 0\), the Schrödinger equation reduces to

\[\begin{cases}\Big(\nabla^2 + q^2\Big) \psi(r,\theta,\phi) = 0 \;\;\mathrm{where}\;\; q = \sqrt{2m(E+U)/\hbar^2}, \;\;&\mathrm{for} \; r \le a \\ \Big(\nabla^2 - \gamma^2\Big) \psi(r,\theta,\phi) = 0 \;\;\mathrm{where}\;\; \gamma = \sqrt{-2mE/\hbar^2}, \;\;&\mathrm{for} \; r \ge a. \end{cases}\]

For the first equation (called the Helmholtz equation), we seek solutions of the form

\[\psi(r,\theta,\phi) = f(r) \, Y_{\ell m}(\theta,\phi),\]

where \(Y_{\ell m}(\theta,\phi)\) are spherical harmonics, and the integers \(l\) and \(m\) are angular momentum quantum numbers satisfying \(l \ge 0\) and \(-l \le m \le l\). Substituting into the Helmholtz equation yields

\[r^2\frac{d^2f}{dr^2} + 2r \frac{df}{dr}+\left[q^2r^2-l(l+1)\right] f(r) = 0,\]

which is the spherical Bessel equation. The solutions to this equation that are non-divergent at \(r = 0\) are \(f(r) = j_\ell(qr)\), where \(j_\ell\) is called a spherical Bessel function of the first kind. Most numerical packages provide functions to calculate these (e.g., scipy.special.spherical_jn in Scientific Python).

Similarly, solutions for the second equation can be written as \(\psi(r,\theta,\phi) = g(r) \, Y_{\ell m}(\theta,\phi),\) yielding an equation for \(g(r)\) called the modified spherical Bessel equation. The solutions which do not diverge as \(r\rightarrow \infty\) are \(g(r) = k_\ell(\gamma r)\), where \(k_\ell\) is called a modified spherical Bessel function of the second kind. Again, this can be computed numerically (e.g., using scipy.special.spherical_kn in Scientific Python).

Using the above facts, show that the condition for a bound state to exist is

\[\frac{qj_\ell'(qa)}{j_\ell(qa)} = \frac{\gamma k_\ell'(\gamma a)}{k_\ell(\gamma a)},\]

where \(j_\ell'\) and \(k_\ell'\) denote the derivatives of the relevant special functions, and \(q\) and \(\gamma\) depend on \(E\) and \(U\) as described above. Write a program to search for the bound state energies at any given \(a\) and \(U\), and hence determine the conditions under which the potential does not support bound states.