# Mean Square Fluctuation (Q.M.)

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
Homework Helper
Gold Member
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.

Last edited:
DrClaude
Mentor
On edit: The definition of mean square fluctuation is
##\left < H^2 - <H> \right >^2##.
Typo: ##\left < H - \langle H \rangle \right >^2##

kuruman
Homework Helper
Gold Member
Typo: ⟨H−⟨H⟩⟩2
Thanks, @DrClaude.

Orodruin
Staff Emeritus
Homework Helper
Gold Member
Typo: ##\left < H - \langle H \rangle \right >^2##
That would be identically zero. You mean
$$\left< (H - \langle H\rangle)^2\right>$$

DrClaude and BvU
kuruman
Homework Helper
Gold Member
That would be identically zero. You mean
##\left< (H - \langle H\rangle)^2\right>##
Yesss.

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

kuruman