Mean Square Fluctuation (Q.M.)

  • #1
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?
 

Answers and Replies

  • #2
kuruman
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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.
 
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  • #3
DrClaude
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On edit: The definition of mean square fluctuation is
##\left < H^2 - <H> \right >^2##.
Typo: ##\left < H - \langle H \rangle \right >^2##
 
  • #5
Orodruin
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Typo: ##\left < H - \langle H \rangle \right >^2##
That would be identically zero. You mean
$$
\left< (H - \langle H\rangle)^2\right>
$$
 
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  • #6
kuruman
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That would be identically zero. You mean
##\left< (H - \langle H\rangle)^2\right>##
Yesss.
 
  • #7
DrClaude
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Does this thread break the record number of typos? (At least by different people :rolleyes:)
 
  • #8
kuruman
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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.
 

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