Problem 4. (25 points) A hydrogen atom is in a P/2state with the z-component of

Question

Problem 4. (25 points) A hydrogen atom is in a P/2state with the z-component of the total angular momentum up along the z-axis. (a) What are the values of I, s, j, and m of the total angular momentum state? (b) Express the total angular momentum state in terms of the product states of |lm)and |sm,). (c) With what probability will the electron be found with spin down? (d) Compute the probability per unit solid angle PC.) that the electron will be found at spherical angles , (independent of radial distance and spin). (e) What is the expectation value of the effective magnetic moment of the state [hint: useful formula: jim

Explanation / Answer

(a) answer: As Hydrogen atom is in a state 2P1/2,

Comparing with (2s+1)Lj ,

2s+1=2 which implies s=1/2,

And corresponding to P, value of l is 1. l=1

j has value j=1/2.

Hence, mj has values - j to +j.

mj=-1/2,+1/2.

(b) answer: Total angular momentum states in terms of the product of |l ml> & |s ms> as follows

|j mj>=|l ml> |s ms>.................mj=ml+ms

s=1/2 , ms=+1/2,-1/2

j=1/2 , mj=+1/2,-1/2

l=1, ml=+1, - 1

Hence,

|1/2, +1/2>= |1,+1> |1, - 1/2>

|1/2, - 1/2>= |1,-1> |1, +1/2>

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