Orbital angular momentum Hamiltonian

In summary, the conversation discusses different methods for computing the wave function at t=0 and applying the time-evolution operator U(t) in a system with quantum numbers l=1 and ml=0. The options include using the explicit form of the initial wave function, working with the state in its abstract form, or expressing the initial state as a vector in the z-basis and using the AM operators as 3x3 matrices.
  • #1
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Homework Statement
Consider a a system described by the following Hamiltonian:

H=(L^2)/2I -gBLy

where I is a momentum of Inertia, B is the y-component of a uniform magnetic field while finally g is a constant.
At t=0, a measurement of L^2 and Lz gives, respectively 2h^2 and 0 as results.
Under these hypotheses determine:
1) The state of the system at a generic time t;
2) The mean values of the energy and of Lx;
3) The minimal time at which the state of the system is an eigenstate of Lx.
(Hint: Remember that Lx and Ly are a combination of Ladder Operators L+ and L-).
Relevant Equations
Lx = 1/2(L+ + L-)
Ly=-i/2(L+ - L-)
I think that the quantum numbers are l=1 and ml=0, so I write the spherical harmonic Y=Squareroot(3/4pi)*cos(theta).
I would like to know how to compute the wave function at t=0, then I know it evolves with the time-evolution operator U(t), to answer the first request.
 
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  • #2
Sounds good. Now you don't even need the explicit form of the initial wave function. Just write down the time-evolution operator for the ##\ell=1## subspace!
 
  • #3
Do you mean last row?
IMG_20221031_141856.jpg
 
  • #4
... Replacing H with the Hamiltonian describing the system and Y with the spherical harmonic !?
 
  • #5
anakin said:
... Replacing H with the Hamiltonian describing the system and Y with the spherical harmonic !?
The initial state is ##\ket{1, 0}##. That can be represented as the spherical harmonic, ##Y_0^1##. Then you have to work out how to apply the time evolution operator to that function. Is that going to be easy?

Alternatively, you may continue to work with the state in its abstract form. Then you need to apply the time-evolution operator to that state. Is that possible?

Or, you could express the initial state as a vector in the z-basis. It would be ##(0, 1, 0)## in the usual convention. If you have seen the AM operators as 3x3 matrices in the case of ##l = 1##, then you may be able to express the time-evolution operator as a 3x3 matrix.

Lots of options!
 
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Likes vanhees71
  • #6
This latter method was what I had in mind in my previous posting.
 

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