What Is the Mean Square Fluctuation in Energy?

[NikBreslin]

Homework Statement

A state of a particle in the potential box of width a with infinitely high walls is described by the wave function:
Ψ(x)=Ax(x-a)
Find the probability distribution of various value of particle energy, mean value and mean square fluctuation of energy.

Homework Equations

Energy Operator H: -hbar² / 2m * d²/dx²
Expectation Value of H is Integral of Ψ*HΨ with respect to x
ΔC²=(<H²>-<H>²)

The Attempt at a Solution

I'm not sure if by mean fluctuation they mean ΔC or ΔC² I have solved the first 2 parts and know the expectation value is 5 hbar²/(m*a²). Because of the wave equation I know expectation value of H² is 0. So is my answer ΔC or ΔC² and if it is the prior, what does an imaginary value mean?

 

[kuruman]

I'm not sure if by mean fluctuation they mean ΔC or ΔC2

The question asked is "mean square fluctuation of energy.

Because of the wave equation I know expectation value of H² is 0.

Can you explain this?

On edit: The definition of mean square fluctuation is
$\left < H^2 - <H> \right >^2$. This cannot be negative. Derive the expression you quoted for the mean square fluctuation from this definition and you will see where and why you got confused.

 

[DrClaude]

On edit: The definition of mean square fluctuation is
$\left < H^2 - <H> \right >^2$.

Typo: $\left < H - \langle H \rangle \right >^2$

 

[kuruman]

Typo: ⟨H−⟨H⟩⟩²

Thanks, @DrClaude.

 

[Orodruin]

Typo: $\left < H - \langle H \rangle \right >^2$

That would be identically zero. You mean
$$
\left< (H - \langle H\rangle)^2\right>
$$

 

[kuruman]

That would be identically zero. You mean
$\left< (H - \langle H\rangle)^2\right>$

Yesss.

 

[DrClaude]

Does this thread break the record number of typos? (At least by different people )

 

[kuruman]

Does this thread break the record number of typos? (At least by different people)

I don't think it's the typos as much as the "reados", if I may coin the word, because of the notation involving nested angular brackets and parentheses. This, $<\psi |(H-<H>)^2|\psi>$ is more legible and easier to proofread, at least to people familiar with Dirac notation.