Average value in superposition question

1. Jan 15, 2012

Alexitron

Hi there.
A quick question: When calculating the average energy (or any value) of a particle which is in superposition, why the sums have different summation index (n,m)?

Thanks!

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2. Jan 15, 2012

kith

If you multiply a number a=a1+a2 with itself, you get a1² + a2² + 2a1a2. If you take the same summation index for both sums in your problem, you end up only with terms an² and don't get the mixed terms.

/edit: You may be wondering why in the end, you actually don't get the mixed terms. But remember that this is due to the orthogonality ψnψmnm of the eigenfunctions of the (hermitian) Hamiltonian. This is not true for arbitrary sums of wavefunctions.

Last edited: Jan 15, 2012
3. Jan 15, 2012

Alexitron

Thanks a lot for your reply.
it bothers me for some time

So for example,(ψ=c1ψ1 + c2ψ2) when calculating the average position (or any other Item A) of a particle, if we had the same summation index we wouldn't have the terms X_12 and X_21. But when we calculate the energy we don't have these mixed (E_12,E_21) terms but eigenvalues (E_1,E_2) due to orthogonality.

4. Jan 15, 2012

kith

Yes. But you do understand why it is mathematically wrong to use the same summation index, don't you?

5. Jan 15, 2012

Alexitron

Well, i know how the values of <x> and <E> are calculated and that energy is quantized but the position for example is not , but to be honest i don't understand why the conjugate ψ* has different index. The only thing i can think is that we lose the factors A12 and A21 when calculating <A> (what you said in your previous reply). Is that the reason? And what's the physical meaning of A12 and A21 that energy doesn't have (I know why energy doesn't have them)?
Sorry for asking such trivial things, but I'm stuck here.

I mean what's wrong with the following?

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6. Jan 15, 2012

kith

Forget about QM for a moment and think purely mathematical. ψ is a complex number which can be written as a sum Ʃnψn, and so is ψ*. If you multiply two sums, you can't use the same index because this doesn't yield any mixed terms.

Actually, there's nothing specific to energy here. ψ can be decomposed in an orthogonal set of eigenvectors to any observable, so <A> = Ʃn|cn|²an is the result for the average value of an arbitrary observable A.

Terms c1c2 do play a role when you calculate probabilities instead of average values. They are responsible for interference (see the double slit for example).

/edit: Your second calculation only yields the same result because ψn is an eigenvector to H. Else you would get Ʃn|cn|² * I, where I stands for the numerical value of the integral. In general, this is not the same as Ʃn,mcn*cm.

Last edited: Jan 15, 2012
7. Jan 15, 2012

Alexitron

Thanks for your time. You helped me a lot to figure it out.

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