Is the Ion Bound with this Energy Minimization?

[schlegelii]

Homework Statement

There exists some ion with a hamiltonian H = H0 + H1. The expectation value of H0 is $\frac{\hbar ^2}{m} a^2 - 2 e^2 a$ and the expectation value of H1 is $\frac{5}{8} e^2 a$, where $a$ is some parameter with units of inverse length. Is the ion bound?

Relevant Equations

H = H0 + H1
expectation value of H0 is $\frac{\hbar ^2}{m} a^2 - 2 e^2 a$
expectation value of H1 is $\frac{5}{8} e^2 a$

From what we did in class, I think we need to minimize the energy with respect to a.

Like $E = \frac{\hbar ^2}{m} a^2 - 2 e^2 a + \frac{5}{8} e^2 a = \frac{\hbar ^2}{m} a^2 - \frac{11}{8} e^2 a$, then minimize it

Finding the minimum value: $- (\frac{11}{16})^2 \frac{m e^4}{\hbar^2}$ when $a = \frac{11 m}{16 \hbar ^2} e^2$

I am not sure how to tell if this means the ion is bound or not?? The energy is negative, but does that help??

 

[Abhishek11235]

Yes. According to variational principle $$E_g\leq <\text H>$$. Since tighetst upper bound you have founded is negative,it is bounded.