Finding coefficients of superposition of states

[bznm]

I have some troubles in finding coefficients of superposition of states.

I have 2 particles, their spins are s1=3/2 and s2=1/2.
At t=0, the system is described by |a(0)>=|3/2, 1/2, 1/2, 1/2>

I have to find |a(t)>.

I have thought to proceed in the following way:

1) use the basis |s, s_z> where s=s1+s2 and s_z= s_1z+s_2z and find the expressions of these vectors in function of the "old" basis (old basis: |s1, s_1z, s_2, s_2z> )
2) find the expression of |a(0)> in this new basis and then find its expression in function of t.

But something went wrong... For example, if i want to find |s=2, s_z=1>, I have:
|s=2, s_z=1>=a1 |3/2, 3/2, 1/2, -1/2>+a2 |3/2, 1/2, 1/2, 1/2>

If I apply the operator J_, I obtain
$$0= \sqrt 3 a_1 |3/2, 1/2, 1/2, -1/2>+ 2 a_2 |3/2, -1/2, 1/2, 1/2>+a_2 |3/2, 1/2, 1/2, -1/2>$$.. is it wrong?

And now, how can I find a1 and a2 (normalized)?

 

[blue_leaf77]

Time evolution requires the knowledge of the system's Hamiltonian.

 

[bznm]

you're right. It is $$H=\alpha s_1 \cdot s_2$$
where alpha is constant. Could you help me?

 

[atyy]

Find the eigenstates of the Hamiltonian.

 

[blue_leaf77]

Step 1) and 2) are correct.

But something went wrong... For example, if i want to find |s=2, s_z=1>, I have:
|s=2, s_z=1>=a1 |3/2, 3/2, 1/2, -1/2>+a2 |3/2, 1/2, 1/2, 1/2>

I don't see anything wrong, you just haven't founnd the coefficients yet.

how can I find a1 and a2 (normalized)?

They are called Clebsch-Gordan coefficients. There is a way to calculate them so that the states are normalized, but you can also make use of your internet connection to find a related table.

 

[bznm]

They are called Clebsch-Gordan coefficients. There is a way to calculate them so that the states are normalized, but you can also make use of your internet connection to find a related table.

I have to calculate them ;)

 

[blue_leaf77]

There is one convention in calculating Clebsch-Gordan coefficient, namely the coefficient corresponding to $|s_{max},s_{max}\rangle$ is agreed to be unity. In other words
$$ |s_{max},s_{max}\rangle = |s_1,s_1\rangle |s_2,s_2\rangle $$
So, to get the coefficients for $|2,1\rangle$, you should start by applying lowering operator, $S_- = S_{1-}+S_{2-}$, to the state $|2,2\rangle$.

 

[bznm]

well, I have tried.

$$j-|2,2>=|2,1|=k_1 |3/2, 1/2, 1/2, 1/2>+k_2 |3/2, 3/2, 1/2, -1/2>$$ where |$$k_1>=\sqrt 3, k_2=1$$
if I want to normalize them,$$|k_1>=\sqrt 3 /2, k_2=1/2$$

If I apply j_ again,

$$j- |2,1>=|2,0>=k_1 *k_3 |3/2, -1/2, 1/2, 1/2>+k_1*k_4|3/2, 1/2, 1/2, -1/2>+k_2*k_5|3/2, 1/2, 1/2, -1/2>$$

where k_3=2, k_4=1, k_5=sqrt 3...

And so, I haven't obtained the correct result... what's wrong?

 

[blue_leaf77]

$$j-|2,2>=|2,1>$$$$j- |2,1>=|2,0>$$

Remember that $J_-|j,m\rangle = C_-|j,m-1\rangle$, regardless of whether the lowering operator is a total or a single operator

if I want to normalize them,$$|k_1>=\sqrt 3 /2, k_2=1/2$$

Calculate the coefficient which is supposed to be in front of $|2,1 \rangle$ on the left hand side, and you will find the same answer as you did here. Do the same for your second calculation.