Finding Expectation Values & Expressing Eigenstates

In summary: The two eigenvectors of a matrix are the vectors that represent the principal diagonal (or first column) of the matrix.
  • #1
broegger
257
0
Two quick ones :)

Hi, two questions:

1) How can I find the expectation value of the x-component of the angular momentum, [tex]\langle L_x \rangle[/tex], when I know [tex]\langle L^2 \rangle[/tex] and [tex]\langle L_z \rangle[/tex]?

2) Say, I have a state [tex]|\Psi \rangle[/tex] and two operators A and B represented as matrices. Now, [tex]|\Psi \rangle[/tex] is given as a linear combination of the eigenstates of A and I want to express them as a linear combination of the eigenstates of the operator B instead. How do I do that?

Thanks :)
 
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  • #2
broegger said:
1) How can I find the expectation value of the x-component of the angular momentum, [tex]\langle L_x \rangle[/tex], when I know [tex]\langle L^2 \rangle[/tex] and [tex]\langle L_z \rangle[/tex]?

I don't think you can ! I think you can if you are in an EIGENSTATE of L^2 and Lz, but not if you only know the expectation values, because I think I can make up two different wave functions with same expectation for L^2 and Lz and different expectation for Lx...

cheers,
Patrick.
 
  • #3
Oh, they ARE eigenstates of both L^2 and Lz. Sorry. What do I do? :)
 
  • #4
broegger said:
Oh, they ARE eigenstates of both L^2 and Lz. Sorry. What do I do? :)

I think you will always find 0, no ? Because of the cylindrical symmetry of these states ?

The reason is that you can write Jx as 1/2 (J_+ + J_-) and these, acting on a state |j,m> will give you |j,m+1> and |j,m-1>, so
<j,m| Jx |j,m> = 0, no ?

cheers,
patrick.
 
  • #5
I don't quite get it. What does the |J>'s represent?
 
  • #6
broegger said:
I don't quite get it. What does the |J>'s represent?

Sorry, I should have written L. J stands in general for an angular momentum, L for an orbital angular momentum and S for a spin angular momentum. One usually uses the notation J if it doesn't matter (as is the case here) whether it is orbital or spin angular momentum one talks about.

I would think you are aware of J+ and J- (or L+ and L-), the ladder operators of angular momentum ?

cheers,
Patrick.
 
  • #7
broegger said:
2) Say, I have a state [tex]|\Psi \rangle[/tex] and two operators A and B represented as matrices. Now, [tex]|\Psi \rangle[/tex] is given as a linear combination of the eigenstates of A and I want to express them as a linear combination of the eigenstates of the operator B instead. How do I do that?

I guess you mean that you work in a certain basis, in which you get to know the components of A (matrix) and B (matrix).
However, there is an ambiguity in the way you know psi: if you know its projections on the eigenvectors of A, you don't necessarily know its components in the basis in which A is given, because eigenvectors are only defined upto a complex multiplication factor ! So depending on how these eigenvectors were fixed your components will have an ambiguity (each of them individually) of a complex factor: just as well say that you don't know them (except if they happen to be 0).

Happily, you can hold the same reasoning for B :-) so I would say that a possible answer to the question will always be, that in a suitably scaled set of eigenvectors of B, psi will always take on the components {1,1,1,1,...1}
:redface:

cheers,
Patrick.
 
  • #8
Sorry, I'm wasting your time. The situation in 1) is this:

An electron moving in a Coulomb-field from a proton, is in the following state (at time t=0)

[tex]|\phi,t=0> = \tfrac4{5}|100> + \tfrac{3i}5|211>[/tex]​

where [tex]|nlm>[/tex] is the usual energy eigenstates of the hydrogenatom. They are also eigenstates of angularmomentum:


[tex]L^2|nlm>=l(l+1)\hslash^2|nlm>[/tex]

[tex]L_z|nlm>=m\hslash|nlm>[/tex]

The questions are now:

a) Calculate the expectation value <E> in the state [tex]|\phi,t=0>[/tex].

b) Calculate the expectation values for L^2 and L_z in the state [tex]|\phi,t=0>[/tex].

c) Calculate the expectation value for L_x in the state [tex]|\phi,t=0>[/tex].

It's c) I'm having trouble with. Hope this helps.
 
Last edited:
  • #9
[tex] \langle \hat{L}_{x}\rangle_{|\psi\rangle} =:\langle\psi|\hat{L}_{x}|\psi\rangle [/tex]

Okay?

Now,use the fact that

[tex] \hat{L}_{x}=\frac{1}{2}\left(\hat{L}_{+}+\hat{L}_{-}\right) [/tex]

and the action of [itex] \hat{\mbox{L}}_{+} [/itex] and [itex]\hat{\mbox{L}}_{-} [/itex] on an arbitrary hydrogenoid wavefunction [itex] |n,l,m\rangle [/itex] (see textbook) and the orthonormalization of the states

[tex] \langle n,l,m|n',l',m'\rangle =\delta_{nn'}\delta_{ll'}\delta_{mm'}[/tex]

Daniel.
 
  • #10
Yep, I with you now :) Thank you both.
 
  • #11
I assume that eigenvectors of both A and B provide the complete basis. (?)
well, about matrices

you have
|c>=|A><A|c>

you need to find

|c>=|B><B|c>

where |A> and |B> - eigenvectors

then

<b|c>=<B|A><A|c>;
 
  • #12
I don't follow.What and who are those A,B,a,b,,blah,blah,blah...?

Daniel.
 
  • #13
that is an old linear algebra in QM notation. I assume that eigenvectors of each matrix forms an ortogonal basis.

You have vector in the basis of the A matrix

[tex]\vec\psi=a_1 \vec{a_1}+a_2 \vec{a_2}+...[/tex] (1)

you need to find

[tex]\vec\psi=b_1 \vec{ b_1}+b_2 \vec{ b_2}+...[/tex] (2)

so multiply (1) by [tex]\vec{b_1}, \vec{b_2}...[/tex] and so on.

[tex]b_1 =a_1 (\vec {a_1}\vec{ b_1})+a_2 (\vec {a_1}\vec{ b_1})+...[/tex]
 
  • #14
Okay.What relevance does it have here?

Daniel.
 
  • #15
well,

Say, I have a state and two operators A and B represented as matrices. Now, [tex]\psi[/tex] is given as a linear combination of the eigenstates of A

that was (1)


and I want to express them as a linear combination of the eigenstates of the operator B instead.
that was (2)
 
  • #16
Are u hijacking the thread?This was the context

Two quick ones :)

--------------------------------------------------------------------------------

Hi, two questions:

1) How can I find the expectation value of the x-component of the angular momentum, , when I know and ?

2) Say, I have a state and two operators A and B represented as matrices. Now, is given as a linear combination of the eigenstates of A and I want to express them as a linear combination of the eigenstates of the operator B instead. How do I do that?

Thanks :)


[end quote]


Daniel.
 

Related to Finding Expectation Values & Expressing Eigenstates

1. What is an expectation value?

An expectation value is a measure of the average value that a physical quantity will take on in a given system. It is calculated by taking the sum of the products of all possible values of the quantity and their corresponding probabilities.

2. How is an expectation value calculated?

An expectation value is calculated by taking the integral of the product of a quantity and its probability distribution function over all possible values of the quantity. This is represented by the formula E = ∫x*P(x)dx, where E is the expectation value, x is the quantity, and P(x) is the probability distribution function.

3. What are eigenstates?

Eigenstates are the set of possible states that a quantum mechanical system can exist in, with each state having a unique energy and corresponding probability. These states are represented by eigenvectors, which are solutions to the Schrödinger equation.

4. How do you express eigenstates?

Eigenstates are typically expressed as linear combinations of basis states, which are states that have definite values for certain physical quantities. This is represented by the formula |ψ> = ∑c_i|b_i>, where |ψ> is the eigenstate, c_i is the coefficient for each basis state, and |b_i> is the basis state.

5. What is the significance of finding expectation values and expressing eigenstates?

Finding expectation values and expressing eigenstates allows us to make predictions about the behavior of quantum mechanical systems and understand how they evolve over time. These calculations and expressions are essential in the study of quantum mechanics and have many applications in fields such as chemistry, physics, and engineering.

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