Question

a hydrogen atom is in the 2p state. How much time must elapse for there to be a 1.3 % chance that this atom will undergo a quantum jump to the ground state?

Answer 1

Hydrogen in 2p state exited state (n=2, l=1, ml= -1, 0,1, mg = ± ½, max 6 electrons) and ground state ( n=1, l=0, ml= 0, 1, mg= = ± ½, max 4 electrons). The spontaneous transition from a 2p to a 1s state.

Wave function of a hydrozen

ᴪn,l,m (r, Ɵ, ҩ) = Rm,l (r) Yl,m (Ɵ, ҩ)

where αo is the Bohr radius specified. All of the other possible 2p to 1s matrix elements are zero because of the selection rules.

D² = 2 ¹⁵/3¹⁰ (e αo) ²

for m= -1,0,1 Clearly, the transition rate is independent of the quantum number m. It turns out that this is a general result.

Now the energy of hydrozen atom

E= Eo/n²

Eo = ground state energy

Hence energy of photon

Hw= Eo/4-Eo = -3/4 Eo = 10.2 ev

Wave length= 1.215 X 10^-7

W 2p to 1s = w3d2/3eohc3

Which reduces to

W 2p to 1s = 6.27 X 10^-8/s

The mean life time of a hydrozen 2p state is 1.6 ns

Incidentally, since the 2p state only has a finite life-time, it follows from the energy-time uncertainty relation that the energy of this state is uncertain by an amount

4 X 10^-7 eV