2.6: Exercises

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.

Further Reading

[1] Bransden & Joachain, §4.4, 9.2–9.3, 13.4

[2] Sakurai, §5.6, 7.7–7.8

[3] R. Courant and D. Hilbert, Methods of Mathematical Physics vol. 1, Interscience (1953).

[4] B. Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions, Annals of Physics 97, 279 (1976).