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12.11: 2P-1S Transitions in Hydrogen

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Let us calculate the rate of spontaneous emission between the first excited state (i.e., n=2) and the ground-state (i.e., n=1) of a hydrogen atom. Now, the ground-state is characterized by l=m=0. Hence, in order to satisfy the selection rules ([e13.133]) and ([e13.134]), the excited state must have the quantum numbers l=1 and m=0,±1. Thus, we are dealing with a spontaneous transition from a 2P to a 1S state. Note, incidentally, that a spontaneous transition from a 2S to a 1S state is forbidden by our selection rules.

According to Section [s10.4], the wavefunction of a hydrogen atom takes the form ψn,l,m(r,θ,ϕ)=Rn,l(r)Yl,m(θ,ϕ),

where the radial functions Rn,l are given in Section [s10.4], and the spherical harmonics Yl,m are given in Section [sharm]. Some straightforward, but tedious, integration reveals that 1,0,0|x|2,1,±1=±2735a0,1,0,0|y|2,1,±1=i2735a0,1,0,0|z|2,1,0=22735a0,
where a0 is the Bohr radius specified in Equation ([e9.57]). All of the other possible 2P1S matrix elements are zero because of the selection rules. It follows from Equation ([e13.128]) that the modulus squared of the dipole moment for the 2P1S transition takes the same value d2=215310(ea0)2
for m=0, 1, or 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 the eigenstate of the hydrogen atom characterized by the quantum numbers n, l, m is E=E0/n2, where the ground-state energy E0 is specified in Equation ([e9.56]). Hence, the energy of the photon emitted during a 2P1S transition is ω=E0/4E0=34E0=10.2eV.

This corresponds to a wavelength of 1.215×107 m.

Finally, according to Equation ([e3.115]), the 2P1S transition rate is written w2P1S=ω3d23πϵ0c3,

which reduces to w2P1S=(23)8α5mec2=6.27×108s1
with the aid of Equations ([e13.139]) and ([e13.140]). Here, α=1/137 is the fine-structure constant. Hence, the mean life-time of a hydrogen 2P state is τ2P=(w2P1S)1=1.6ns.
Incidentally, because 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 ΔE2Pτ2P4×107eV.
This uncertainty gives rise to a finite width of the spectral line associated with the 2P1S transition. This natural line-width is of order

ΔλλΔE2Pω4×108

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 12.11: 2P-1S Transitions in Hydrogen is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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