2

Given potential $V(x) = Asec(x)$ for $x > 0$. I want to calculate the ground-state energy $E_0$ via the Schrödinger equation.

I'm completely stuck on this one. I've set up the time-independent Schrödinger equation, but it can't be solved without using special functions. I don't see how I can calculate the energy without solving the schrodinger equation. Any hints?

richard_stone
  • 21

1 Answers1

1

The energy spectrum of the problem(v3) with potential

$$\Phi(x)~=~\frac{A}{\cos x},\qquad\qquad x\in\mathbb{R}_{+}, $$

is unbounded from below, i.e., there is no ground state.

This can e.g. be seen using semiclassical methods a la this answer. Semiclassically, the reason is:

  1. because the potential $\Phi$ has infinitely many periods, and
  2. because the classically accessible length within one period is non-zero for any (potential) energy-level $V$.

The total accessible length $\ell(V)$ is therefore infinite for any (potential) energy-level $V$, no matter how negative $V$ is. In other words, the accessible region of phase space is always bigger than Planck constant $h$, and we can hence fit a semiclassical state, for any energy-level $E$, no matter how negative $E$ is.

Community
  • 1
Qmechanic
  • 220,844