Existence of Bound States. A potential "well" (in one dimension) is a function V(x) that is never positive (V(x)≤0 for all x), and goes to zero at infinity (V(x)→0 as x→±∞).
(a) Prove the following Theorem: If a potential well V1(x) supports at least one bound state, then any deeper/wider well (V2(x)≤V1(x) for all x) will also support at least one bound state. *Hint:* Use the ground state of V1, ψ1(x), as a variational test function.
(b) Prove the following Corollary: Every potential well in one dimension has a bound state. *Hint:* Use a finite square well (Section 2.6) for V1.
(c) Does the Theorem generalize to two and three dimensions? How about the Corollary? *Hint:* You might want to review Problems 4.11 and 4.51.