What Is the State of a 1/2-Spin Particle at Time t?

[19matthew89]

Homework Statement

Consider a 1/2-spin particle. Its time evolution is ruled by operator $U(t)=e^{-i\Omega t}$ with $\Omega=A({\vec{\sigma}}\cdot {\vec{L}})^{2}$. A is a constant. If the state at t=0 is described by quantum number of ${\vec{L}}^2$, $L_{z}$ and $S_{z}$, $l=0$, $m=0$ and $s_{z}={1/2}$, determinate the state at a generic time t.

Homework Equations

$({\vec{\sigma}}\cdot {\vec{L}})^{2}={{\vec{L}}^{2}}-{{\vec{\sigma}}\cdot{\vec{L}}}$
and
$(\vec{\sigma}\cdot\vec{u})^{2n}=1$ with $2n$ even number and $\vec{u}$ a unitary vector

The Attempt at a Solution

I've used the relations I've written above to write the propagator as $e^{-iA{\vec{L}}^{2}t}$$e^{iA{{{\vec{\sigma}}\cdot{\vec{L}}}}t}$ and I've found out

$e^{-iA{\vec{L}}^{2}t}[{cos(At)+i{({\vec{\sigma}}\cdot\vec{L})}sin(At)]}$. But I don't think it is correct because L is not a versor.

Thanks

 

[mark726]

for your post. It looks like you have made some progress in your attempt to solve this problem. However, there are a few points that I would like to clarify and expand on.

Firstly, the operator U(t)=e^{-i\Omega t} is known as a time evolution operator. It describes how a system evolves in time, and in this case, it is dependent on the operator \Omega. This operator is given by \Omega=A({\vec{\sigma}}\cdot {\vec{L}})^{2}, where A is a constant. The first thing to note is that this operator is Hermitian, meaning that its eigenvalues are real. This is important because it means that the evolution of the system is unitary, which is a desirable property for a quantum system.

Next, you mention the operators {\vec{L}}^2 and {\vec{\sigma}}\cdot{\vec{L}}, which are both related to the angular momentum of the system. The first operator, {\vec{L}}^2, represents the total angular momentum squared, while the second operator, {\vec{\sigma}}\cdot{\vec{L}}, represents the spin-orbit coupling. These operators commute with each other, meaning that they can be diagonalized simultaneously. This allows us to use the quantum numbers l, m, and s_z to describe the state of the system.

To determine the state of the system at a generic time t, we need to apply the time evolution operator U(t) to the initial state at t=0. In this case, the initial state is described by quantum numbers l=0, m=0, and s_z=1/2. This state can be written as |l=0, m=0, s_z=1/2>, where the angular momentum quantum numbers are implicit.

Using the time evolution operator, we can write the state at a generic time t as:

U(t)|l=0, m=0, s_z=1/2>=e^{-i\Omega t}|l=0, m=0, s_z=1/2>

Substituting in the expression for \Omega, we get:

U(t)|l=0, m=0, s_z=1/2>=e^{-iA({\vec{\sigma}}\cdot {\vec{L}})^{2}t}|l=0, m=0, s_z=1/2