A silly question about quantum mechanics?

Question

Let ψ₁, ψ₂, ... , ψᵥ be an orthonormal basis set.

And,
ψ = c₁ψ₁ + c₂ψ₂ + ... + cᵥψᵥ

It is given in the lecture note I am provided with that since 〈ψᵤ|ψᵥ〉 = δᵤᵥ we can find

cᵣ = 〈ψᵣ|ψ〉

The proof is given and I understand that. The problem is, how can one apply this to find cᵣ without really knowing c₁, c₂, ..., cᵣ, ..., cᵥ? (ψ = c₁ψ₁ + c₂ψ₂ + ... + cᵥψᵥ)

PleaseStop. · Accepted Answer

This is silly. I do not even understand your problem here. What do you mean without really knowing c₁, c₂, ..., cᵣ, ..., cᵥ ?

You know c₁, c₂, cᵥ because you defined ψ already as a superposition of an orthonormal basis ψ₁, ψ₂, ... , ψᵥ .... 
ψ = c₁ψ₁ + c₂ψ₂ + ... + cᵥψᵥ

So by 〈ψᵤ|ψᵥ〉 = δᵤᵥ  any  〈ψᵣ|ψ〉 = cᵣ

ie.. 〈ψ₁|ψ〉 = c₁

EDIT:  
After I was done I figured that was what you meant. 
Even though you can easily see that is true, you still have to justify that is true. Thats how you do it. Simple but necessary.

This is important in the generalized statistical interpretation. Say ψᵣ is an eigenfunction corresponding to a discrete eigenvalue qn of a hermitian operator,  
then  |cᵣ|^2 is the probability that measurement on ψ will yield the eigenvalue qn.

You probably already know that.. and that seems trivial, but you still have to define the formalism of the theory properly using mathematics.  
 
Also how do you make all those nice symbols?

Anonymous · Answer

yeah, that's just silly.

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A silly question about quantum mechanics?

2 Answers