# Variational Method and Bound States

## Homework Statement

Consider a potential function $V(x)$ such that:
$$\begin{cases} V(x)\leq 0\text{ for }x\in[-x_0,x_0] \\ V(x)=0 \text{ for }x\not\in[-x_0,x_0] \end{cases}$$
Show, using the variational method that:

(a) In the 1-dimensional case $\lambda^2V(x)$ always possesses at least one bound state.
(b) In the 3-dimensional spherically symmetric case, $V(|\vec r|)$, it possesses no bound states if $\lambda^2$ is made sufficiently small.

2. The attempt at a solution
The idea is to use the variational method, i.e.:
$$\frac{\langle\psi|H|\psi\rangle}{\langle \psi|\psi\rangle}\geq E_{ground},$$
to show that the average value of the energy is negative and hence the ground state energy is negative.

In my first attempt I used a gaussian trial function:
$$\psi(x)=\left(\frac{2a}{\pi}\right)^{1/4}e^{-ax^2}.$$
However the problem is that it turns out that the average value of the kinetic energy is $\hbar^2a/2m$ and so, in order to determine whether the average energy is negative or not we need to know $a$ explicitly. However, this is not possible since we have no insight on the actual shape of the potential.

It seems to me that this thing will turn out to be a problem for every trial function. How can I do that?

Thank you

Orodruin
Staff Emeritus