Matrix of Hamiltonian, system's state - quantum

[Dassinia]

Hello
I'm reaaaaaally stuck here, if someone can explain me it'd be great because i just dn't know how to do all that

Homework Statement

Consider a system with the moment of inertia l=1
A base of the space of the states is constituted by the three eigenvectors of Lz |+1> , |0>, |-1> of eigenvalues +h, 0, -h (h=h/2π)
The Hamiltonian has the form : H=wo/h(Lu²-Lv²)

where Lu and Lv are the component of L in the directions Ou & Ov of the xOz plan, 45° from Ox and Oz. wo is a real constant.

a. Write the matrix representing H in the basis |+1> , |0> and |-1>
What are the stationnary states of the system and their energies ?
b. At t=0 the system is in the state

|ψ(0)>=1/√2(|+1>-|-1>)
What is |ψ(t)> at time t ? If we measure Lz what are the probabilities of the different results ?


c. Calculate <Lx>(t) <Ly>(t) & <Lz>(t)

Homework Equations

The Attempt at a Solution

a.I am soooo lost I know I have to find Lx and Lz first , then Lu & Lv to find H
But I just don't know how to do that

b. We have
|ψ(t)>=|ψ(0)>exp(-iHt/h)
|+1> and |-1> are not eigenstates of H so I have to express them in eigenstates of H but how ?
And I will have to convert the energies |En> found in term of |+1> AND |-1> , how ?

c.

Thanks

 

[Simon Bridge]

$\renewcommand{\br}[1]{| #1 \rangle} \renewcommand{\kt}[1]{\langle #1 |}$

Hello
I'm reaaaaaally stuck here, if someone can explain me it'd be great because i just dn't know how to do all that

Homework Statement

Consider a system with the moment of inertia l=1
A base of the space of the states is constituted by the three eigenvectors of Lz |+1> , |0>, |-1> of eigenvalues +h, 0, -h (h=h/2π)
The Hamiltonian has the form : H=wo/h(Lu²-Lv²)

where Lu and Lv are the component of L in the directions Ou & Ov of the xOz plan, 45° from Ox and Oz. wo is a real constant.

Lets see if I can translate:
System with moment of inertia $I=1$ in some units.
Use a basis of eigenvectors of z-angular-momentum $\{ \br{+1}, \br{0}, \br{-1} \}$

$L_z\br{+1}=\hbar\br{+1},\; L_z\br{0}=0,\; L_z\br{-1}=-\hbar\br{-1}$

The Hamiltonian has form: $$\hat{H}=\frac{\omega_0}{\hbar}\left( L_u^2-L_v^2 \right)$$

The u and v directions are what??
$L_z$ is the component of $\hat{L}$ in the z direction ...

a. Write the matrix representing H in the basis |+1> , |0> and |-1>
What are the stationnary states of the system and their energies ?
b. At t=0 the system is in the state

|ψ(0)>=1/√2(|+1>-|-1>)
What is |ψ(t)> at time t ? If we measure Lz what are the probabilities of the different results ?


c. Calculate <Lx>(t) <Ly>(t) & <Lz>(t)

Homework Equations

The Attempt at a Solution

a.I am soooo lost I know I have to find Lx and Lz first , then Lu & Lv to find H
But I just don't know how to do that

Yes you need to know how the u and v components relate to the x and z components. You were told - but I'm not sure I follow your notation so you'll have to go back to your notes to see what it means.

b. We have
|ψ(t)>=|ψ(0)>exp(-iHt/h)
|+1> and |-1> are not eigenstates of H so I have to express them in eigenstates of H but how ?
And I will have to convert the energies |En> found in term of |+1> AND |-1> , how ?

But they are eigenstates of $L_z$
You need to be able to expand one set of eigenstates in terms of the others - it's a change of basis.

In the previous you found the eigenstates of $H$ in terms of those of $L_z$ ... so you need to do the first one, well, first.