Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 35B / b / i / Solution

{c}

For
$$
H=-\frac{d^2}{dx^2}-V_0e^{-x^2}
$$
and the trial state $\psi_a(x)=e^{-ax^2/2}$,
$$
E_0\leq E(a)
=\frac a2-V_0\sqrt{\frac a{1+a}}.
$$
This is negative for some $a>0$ for every $V_0>0$, proving that the attractive well has a bound state. For small $V_0$, taking $a=V_0^2$ gives
$$
-V_0<E_0\leq-\frac12V_0^2+O(V_0^4).
$$

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 Incoming links: Gaussian variational bound for an attractive Gaussian well