Question

Two Particle State problem, measurement of total angular momentum, finding probabilities, action of lowering operator

a) A particle of spin \(3 / 2\) and a particle of spin 1 are found in the state \(\left|\frac{3}{2}-\frac{1}{2}\right\rangle\) of total spin. Assume that the particles are in a state of zero orbital angular momentum.

Find which values of the z-component of the spin of the two particles can be measured and with what probabilities.

The following information applies to parts \(\mathbf{b}\) ) and \(\mathbf{c}\) ).

An electron in the hydrogen atom occupies the following position and spin state:

$$ \Psi=\frac{1}{5}\left(3 \psi_{322} \chi_{-}+4 \psi_{321} \chi_{+}\right) $$

It can be also written as:

$$ \begin{aligned} &\Psi=R_{32} \frac{1}{5}\left(3 Y_{2}^{2} \chi_{-}+4 Y_{2}^{1} \chi_{+}\right) \\ &\left.\left.|\Psi\rangle=R_{32} \frac{1}{5}(3(122\rangle \otimes|1 / 2-1 / 2\rangle)+4(121\rangle \otimes|1 / 21 / 2\rangle\right)\right) \end{aligned} $$

 b) If you measure the total angular momentum, what values would you get and with what probabilities?

 c) Find the action of the lowering operator \(J_{-}=L_{-}+S_{-}\)on the wave function.

Answer 1