Question

The following theorem concerns the energy eigenvalues E_n (E₁ < E₂ < E₃ < . . . ) of the Schrodinger equation in one dimension:

Theorem: If the potential V₁(x) gives the eigenvalues E_1n, and the potential V₂(x) gives the eigenvalues E_2n, and V₁(x) ≤ V₂(x) for all x, then E_1n ≤ E_2n.

(a) Prove this theorem

Hint: Consider a potential V(\(\lambda\), z), where V(0, x) = V₁(x) and V(l, x) = V₂(x) and \(\partial V/\partial \lambda \ge 0\) (for all x), and calculate \(\partial E_n/\partial \lambda\).

(b) Now consider the potential 

We want to determine the number of bound states that this potential can hold. Assume this number N is >> 1. It may be helpful to draw a qualitative picture of the wave function for the highest bound state.

Choose a solvable comparison potential and use the theorem above to determine either a rigorous upper bound to N or a rigorous lower bound to N. (Both can be done but you are asked for only one.)

Answer 1

(a) Define \(V(\lambda, x) = \lambda V_2 (x) +(1 - \lambda )V_1(x).\) Obviously V(0, x) = V₁(x), V(1, x) = V₂(x), \(\partial V/\partial \lambda = V_2(x) - V_1(x) \ge 0.\)

The Hamiltonian is then

we have \(E_{1n} = E_n (0) \le E_n(1)\), and the theorem is proved. Note that we have used \((n \lambda|n \lambda ) = 1\).

(b) Let V(x) = kx²/2. Then V(x) ≥ U(x). If E, is an energy level for the potential U(x), then \(E_n\le (n + 1/2)h \omega\), where \(\omega = \sqrt{k /m}\). For a bound state, E_n ≤ ka²/2. Solving (N + 1/2)h\(\omega\) ≤ ka²/2, we find

\(N \le \frac{m \omega ^2}{2h} - \frac 12 = \left[\frac{m\omega a}{2h}\right]^2\)

where [A] indicates the maximum integer that is less than A.

We now choose for V(x) a square well of finite depth,

V(x) = ka²/2, |x| > a,

V(x) = 0, |x| ≤ a

The number of bound states of U(x) is less than that of V(x), which for the latter is [2m\(\omega\)a²/7rh] + 1. We can take the upper bound to the number of bound states of U(x) as [2m\(\omega\)a²/πh] as for N >> 1 the term 1 can be neglected. Taken together, we get that the number of bound states is between [m\(\omega\)a²/2h] and [2m\(\omega\)a²/πh].