Linear combination of wave function of a Hydrogen Atom

[czng71]

Homework Statement

I am given a linear combination of wave function of HYDROGEN ATOM Ψ=1/2(Ψ200 +Ψ310+Ψ311+Ψ31-1), where the subscripts are n, l, m respectively.

I was asked to find all the possible outcomes when measuring Lx and their corresponding probabilities.

Homework Equations

The Attempt at a Solution

Should I construct matrices of the raising and lowering operator? How should I do so?
I don't really know how to deal with a linear combination of wave functions.

 

[vela]

Homework Statement

I am given a linear combination of wave function of HYDROGEN ATOM Ψ=1/2(Ψ200 +Ψ310+Ψ311+Ψ31-1), where the subscripts are n, l, m respectively.

I was asked to find all the possible outcomes when measuring Lx and their corresponding probabilities.

Homework Equations

The Attempt at a Solution

Should I construct matrices of the raising and lowering operator? How should I do so?

Why would you want to do this? I'm not suggesting you shouldn't, but do you have a reason for doing this? It seems like you're just guessing at this point.

Do you have some sort of strategy for solving the problem? Can you explain conceptually what you want to do even if you don't know how to do the actual math yet? That's where you need to start.

I don't really know how to deal with a linear combination of wave functions.

 

[czng71]

Why would you want to do this? I'm not suggesting you shouldn't, but do you have a reason for doing this? It seems like you're just guessing at this point.

Do you have some sort of strategy for solving the problem? Can you explain conceptually what you want to do even if you don't know how to do the actual math yet? That's where you need to start.

I think (I guess?) I can find the eigenfunction of Lx first by constructing the matrices of raising and lowering operator, by the relation of Lx=1/2(L+ + L_). Then I can find the eigenvalues, which are the possible outcome when I measure Lx.

 

[vela]

That's a good start. To find the columns of a matrix representing an operator, you calculate what the operator does to the basis states.

 

[czng71]

After some work, I worked out that Lx = h(bar) /2 (0 1 0) for l = 1
.　　　　　　　　　　　　　　　　　　　　　　　　　　1 0 1
.　　　　　　　　　　　　　　　　　　　　　　　　　　0 1 0

But now there is the problem: in the question there is a linear combination of states where l = 1 and l = 0. I don't know how to deal with it.

Thank you very much for your help!

 

[czng71]

I think I have worked out a better method:

Lx Ψ= 1/2 (L+ +L_)Ψ = 1/2 (L+Ψ + L_Ψ)
= 1/2 [(L+)(Ψ200 +Ψ310+Ψ311+Ψ31-1) + (L_)(Ψ200 +Ψ310+Ψ311+Ψ31-1)]

Plug in L+(Ym)=h(bar)√(l+m+1)(l-m) (Ym+1) and similarly for L_.

Then we can get the answer? I found that Lx = √2h(bar) or 0.
Is that correct?

Many Thanks!

I am actually still working on how the probabilities add up to 1. I am having trouble on this. There may be mistakes in my calculation or the entire method is incorrect.

 

[vela]

Don't equate making a measurement with applying the corresponding operator to the state.