please answer all the parts of the question with complete solution Suppose the e

Question

please answer all the parts of the question with complete solution

Suppose the electron in a hydrogen atom occupies the combined position and spin state Determine the normalization constant A. What values of total and z-component orbital angular momentum might be measured and with what probabilities? What values of total and z-component spin angular momentum might be measured and with what probabilities? What values of z-component total angular momentum (combined spin and orbital) might be measured? What energies might you measure and with what probabilities?

Explanation / Answer

a) since states are orthogonal
<psi|psi> = A^2*(5^2 + 12^2) = 1

A = 1/sqrt(169) = 1/13

b) for first term total = sqrt(l (l+1)) hbar = sqrt( 2*3) hbar = sqrt(6) hbar
and mz = 1 hbar

with prob (5/13)^2 = 25/169

for second term total = sqrt( 3*4)hbar = sqrt(12) hbar

mz = -2 hbar
prob = (12/13)^2 = 144/169

c) total is always 1/2 hbar prob 1

first term z is -1/2 hbar prob 25/169

second term z is 1/2 hbar prob 144/169

d) J = L + S

max j = l + s = 2.5 or 3.5

so jz can vary from -3.5 to 3.5 so -3.5, -2.5, - 1.5 , -0.5, 0.5, 1.5, 2.5 ,35

e) E = -13.6/n^2
first term E = -13.6/5^2 with prob 25/169

second erm E = -13.6/4^2 with prob 144/169

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