Question

2. (2.5 pts) What is the lifetime (in seconds) for the hydrogen atom to decay from then 3,1 = 0, mı = 0 state to the n = 2,1 = 1, ml = 0 state? Hint: Recall that the lifetime is the inverse of the Einstein coefficient A32. Also that r = r sin θ cos φ ex + r sin sin ф ey + r cos ez.

Answer 1

We know the ratio of the Einstein a co-efficient and B co - efficient is given by A_m rightarrow n/B_m rightarrow n = 8 pi h r_mn^3/c^3 or A_m rightarrow n/B_m rightarrow n = 2hw_mn^3/pi c^3 or A_m + n = b_m rightarrow n. 2hw_mn^3/pi c^3 where b_m rightarrow n = 2 pi/3h^2 Vertical-bar mu_mn Vertical-bar ^2 B_m rightarrow n = 2 pi e^2/3h^2 Vertical-bar < m Vertical-bar r Vertical-bar n > Vertical-bar ^2 and w_mn = 2 pi r_mn mu_mn = er m rightarrow initial state of transition n rightarrow final state of transition or A_m + n = 16 pi e_0 alpha w_mn^3/3e^2 Vertical-bar < m Vertical-bar r Vertical-bar n > Vertical-bar ^2 alpha = fine structure constant = e^2/4 pi e_0 ch = 1/137 now for hydrogen atom according to the problem we have decay from (n = 3, l = 0 m_2 = 0) state to the (n = 2, l = 1, m_1 = 0) state. So at first we write the wave function of two state psi_300 = 1/81 squareroot 3 pi .1/(a_0)^3/2 (27 - 18^2/a_0 + 2r^2
here I have calculated the lifetime for the hydrogen atom for the decay ...

All the calculations are given above ...Step by step..Hope you understand....