Dealing with normalized quantum functions

[Jimmy25]

Homework Statement

If <∅| is normalized, show that:

<∅|∅>=1=<∅|n><n|∅>

(where ∅ is a non-eigenfunction wave function composed of Ʃc(n)ψ(n).

Homework Equations

The Attempt at a Solution

I can show that <∅|∅>=Ʃc*(n)c(n) (=1). But the next part of the question asks to use your proof to show Ʃc*(n)c(n)=1 so that's not the way it is supposed to be done. I feel like I'm missing something very simple.

 

[bbbeard]

Homework Statement

If <∅| is normalized, show that:

<∅|∅>=1=<∅|n><n|∅>

(where ∅ is a non-eigenfunction wave function composed of Ʃc(n)ψ(n).

It seems to me that the problem statement must be


<∅|∅>=1=Ʃ_n<∅|n><n|∅>.

This is just the usual expansion over basis states. What you wrote can't be true for arbitrary n; it only works when you sum over all the basis states.

In matrix language, Ʃ_n|n><n| is just the identity matrix.

BBB

 

[Jimmy25]

Aha that was what I suspected.

With the correction, is there a way to prove that it is equal to one without having to use that Ʃc*(n)c(n) (=1)?

 

[bbbeard]

Aha that was what I suspected.

With the correction, is there a way to prove that it is equal to one without having to use that Ʃc*(n)c(n) (=1)?

I would just say Ʃ_n|n><n| = I, the identity matrix, because that is the definition of a complete set of states {|n>}. Then

<θ|I|θ> = <θ|θ> =1

because |θ> is normalized. This is not the complicated part of quantum mechanics.

BBB