Average value in superposition question

[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!

 

[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 a_n² 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 ψ_nψ_m=δ_nm of the eigenfunctions of the (hermitian) Hamiltonian. This is not true for arbitrary sums of wavefunctions.

 

[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.

 

[kith]

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.

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

 

[Alexitron]

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

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?

 

[kith]

Well, i know how the values of <x> and <E> are calculated and that energy is quantized but the position for example is not (of course for t≠0), but to be honest i don't understand why the conjugate ψ* has different index.

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

And what's the physical meaning of A12 and A21 that energy doesn't have (I know why energy doesn't have them)?

Actually, there's nothing specific to energy here. ψ can be decomposed in an orthogonal set of eigenvectors to any observable, so <A> = Ʃ_n|c_n|²a_n 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 ψ_n is an eigenvector to H. Else you would get Ʃ_n|c_n|² * I, where I stands for the numerical value of the integral. In general, this is not the same as Ʃ_n,mc_n^*c_m.

 

[Alexitron]

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