schlegelii
- 8
- 0
- 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??
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??