# Variational Method / Proof using bra-ket notation [closed]

I want to prove that the expectation value of the Hamiltonian $$H$$ in a state $$\Psi$$ is an upper bound to the ground state energy.

The proof in my textbook makes sense but does it using integrals. So I tried to do it using bra-ket notation because it just seems a lot more aesthetic that way.

Consider a Hamiltonian $$H$$ with a complete set of orthonormal eigenfunctions $$\phi_1, \phi_2, \phi_3, ...,$$ with associated eigenvalues $$E_1, E_2, E_3, ...,$$ such that $$E_1 \leq E_2 \leq E_3 \leq ...$$.

Then any state of the system $$\Psi$$ can be expanded as a linearly combinations of the eigenfunctions, so $$\Psi = \sum_{n=1}^{\infty} c_n \phi_n$$.

Now consider the expectation value of the Hamiltonian in the state $$\Phi$$:

\begin{align} I &= \frac{<\Psi | H|\Psi>}{<\Psi | \Psi>} \\ & = \frac{<\sum_n c_n \phi_n | H | \sum_m c_m \phi_m >}{<\sum_n c_n \phi_n | \sum_m c_m \phi_m>} \\ & = \frac{\sum_{n,m} c_n^* c_m<\phi_n|H|\phi_m>}{\sum_{n, m} c_n^* c_m <\phi_n | \phi_m>}\end{align}

and then since the eigenfunctions are orthonormal, $$I = \frac{\sum_n |c_n|^2 E_n}{\sum_n |c_n|^2}$$. From here it is easy.

But is what I've done so far ok? Am I missing anything?

Thank you.

## closed as off-topic by Kyle Kanos, GiorgioP, Jon Custer, Dvij Mankad, ZeroTheHeroApr 15 at 3:56

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – Kyle Kanos, GiorgioP, Jon Custer, Dvij Mankad, ZeroTheHero
If this question can be reworded to fit the rules in the help center, please edit the question.

• Seems right to me. – Manvendra Somvanshi Apr 11 at 12:45
• @ManvendraSomvanshi yayyyy!! – PhysicsMathsLove Apr 11 at 12:45
• The expression you got is the definition of weighted average, which is expectation value in quantum mechanics. – Manvendra Somvanshi Apr 11 at 13:05
• Why the downvote? What’s wrong with this post? – PhysicsMathsLove Apr 11 at 13:13