- #1

- 10

- 0

## Homework Statement

For a wavefunction ##\psi##, the variance of the Hamiltonian operator ##\hat{H}## is defined as:

$$\sigma^2 = \big \langle \psi \mid (\hat{H} - \langle\hat{H}\rangle)^2 \psi \big\rangle$$

I want to prove that if ##\sigma^2 = 0##, then ##\psi## is a solution to the Schrödinger equation (##\hat{H}\psi = E\psi##).

##\psi## is a normalized wavefunction.

## Homework Equations

[/B]

Let ##\phi## be a wavefunction. The norm ##\parallel \phi \parallel = \sqrt{\langle \phi \mid \phi \rangle} ## is equal to zero if and only if the wavefunction ##\phi## is equal to zero (##\phi = 0 \Leftrightarrow \parallel \phi \parallel = 0) ##

Given hint: ##\hat{H} - \langle\hat{H}\rangle## is a hermitian operator.

## The Attempt at a Solution

[/B]

I think maybe the equation can be written as:

$$ \sigma^2 = (\hat{H}\psi - E\psi)\psi $$

I know that ##\sigma^2 = 0##.

##\psi## is a normalized wavefunction ##\Rightarrow \sqrt{\langle \psi\mid\psi \rangle} = 1##, and ##\psi \neq 0##. Then, ##\hat{H}\psi = E\psi = 0##, and ##\psi## is a solution to the Schrödinger equation.