If someone could walk me through this example I'd be extremely grateful.(adsbygoogle = window.adsbygoogle || []).push({});

A Hamiltonian acts in a 2 j + 1 dimensional space through a set of three operators Jz , J± that obey angular momentum commutation relations. We wish to determine the evolution of some particular state [itex]|j,m_j \rangle[/itex] . The Hamiltonian is;

[itex]H = \epsilon(t)J_z + \alpha (t) J_+ + \alpha * (t) J_- \overset{j \rightarrow \frac{1}{2}}{\rightarrow}

\left( \begin{array}{ccc}

\frac{1}{2} \epsilon(t) & \alpha (t) \\

\alpha * (t) & -\frac{1}{2}\epsilon (t) \end{array} \right)[/itex]

I'm having trouble on this step for a few reasons

1. why are we letting j go to [itex]\frac{1}{2}[/itex]

2. That isn't necessarily unitary

The unitary operator acting in the 2 j + 1 dimensional space is a unitary representation of some operation in the group SU (2). It is simpler to determine how g(t) ∈ SU (2) evolves, and then construct its unitary representation, than it is to determine the time evolution of the (2 j + 1) × (2 j + 1) unitary matrix. Specifically, the equation of motion in the group is;

[itex]

\frac{d}{dt}

\left( \begin{array}{ccc}

a(t) & b (t) \\

-b * (t) & a* (t) \end{array} \right)

=

-\frac{i}{\hbar}

\left( \begin{array}{ccc}

\frac{1}{2} \epsilon(t) & \alpha (t) \\

\alpha * (t) & -\frac{1}{2}\epsilon (t) \end{array} \right)

\left( \begin{array}{ccc}

a(t) & b (t) \\

-b * (t) & a* (t) \end{array} \right)

[/itex]

I understand this part, it's just Schrodinger but in group form and the g(t) ∈ SU(2) is just [itex]\psi[/itex] (I think). I still have the problem with

1. No explicit Det=1 requirment

After some algebraic manipulation this matrix equation reduces to two equations for the complex coefficients a(t) and b(t) or three equations for the real coefficients of the Pauli spin matrices σ1 , σ2 , σ3 . These are first order equations and can be solved by standard integration methods (e.g., RK4). The initial conditions are [itex]a(t_i ) =

1[/itex], [itex]b(t_i ) = 0[/itex]. The final 2 × 2 unitary matrix is determined by [itex]a(t_f )[/itex], [itex]b(t_f )[/itex]. This is a group operation in SU (2) that can subsequently be mapped into the (2 j + 1) × (2 j + 1) unitary irreducible representation of this group. At this point the problem is solved, independent of the initial state [itex]|\psi(t_i )\rangle[/itex]

And here I'm not sure how exactly you map back into the original space

Thanks in advance!

(Any book recommendations that would help with this are also appreciated!)

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# Help understanding this Lie Algebra/Group stuff

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