(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

1. Consider a deuterium atom (composed of a nucleus of spin I=1 and an electron. The total angular momentum of the atom is [itex]\vec{F}=\vec{J}+\vec{I}[/itex], the eigenvalues of [itex]J^2[/itex] and [itex]F^2[/itex] are [itex]J(J+1)\hbar^2[/itex] and [itex]F(F+1)\hbar^2[/itex] respectively.

a) What are the possible values of the quantum numbers J and F for a deuterium atom in the 1s ground state?

b) Same as a for deuterium in the 2p excited state.

c) Same as a for hydrogen in the 2p state.

2. Consider particle A of spin-3/2 which can disintegrate into two particles, B of spin-1/2 and C of spin-0. We place ourselves in the rest frame of A and total angular momentum is conserved during the disintegration.

a) What values can be taken on by the relative orbital angular momentum of the two final particles.? Show that there is only one possible value if the parity of the relative orbital state is fixed. Would this result remain valid if particle A had spin greater than 3/2?

b) Assume particle A is initially in the spin state characterized by the eigenvalue [itex]m_A\hbar[/itex] of it's spin component along the z axis. We know the final orbital state has definite parity. Is it possible to determine the parity by measuring the probabilities of finding particle B either in state [itex]|+\rangle[/itex], or [itex]|-\rangle[/itex]?

2. Relevant equations

[itex]\vec{J}=\vec{L}+\vec{S}[/itex], for the electron.

The electron is a spin-1/2 particle.

3. The attempt at a solution

1.a) Since we are in the 1s state, the quantum number n=0, so l=0 and J=1/2 for the electron. This would lead to F=1/2, 3/2 for the atom.

b) In the 2p state n=2, so l=0, 1 and J=1/2, 3/2 for the electron. Then for the entire atom F=1/2, 3/2 (for J=1/2), and F=3/2, 5/2 (for J=3/2).

c) Again with quantum numbers, n=2 so l=0, 1. Thus J=1/2, 3/2. However this time since it is regular hydrogen, I=1/2, for the proton, so F=0, 1 (for J=1/2), and F=1, 2 (for J=3/2).

I'm pretty sure those 3 are correct, but I'm still confused with how I arrived at them. For example, in 1a, I used J=±1/2 (since the total angular momentum was equal to the spin and the electron could be spin up or down, it seemed logical at the time). A similar process was used for the rest. Is this the correct thinking, or was I simply lucky with my incorrect method?

2. a) In the rest frame of A the total angular momentum is J=3/2, so I have [itex]\frac{3}{2}=L_B+\frac{1}{2}+L_C[/itex], given the spin of the final particles, L being the orbital angular momentum of the respective particles. My first guess is the either [itex]L_B=1, L_C=0[/itex], or [itex]L_B=0, L_C=1[/itex]. However, I don't see how parity would come into play with this answer, which leads me to believe it is either wrong or incomplete.

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# Homework Help: Spin and angular momentum

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