130_notes.dvi

this but allow us to deal with single exponentials instead of real functions.) Our final result, the decay rate, will be indepen ...

29.5 Electric Dipole Approximation and Selection Rules We can now expand thee−i ~k·~r ≈ 1 −i~k·~r+...term to allow us to compute ...

= e^2 (Ei−En) 2 π ̄h^2 m^2 c^3 ∑ λ ∫ dΩγ ∣ ∣ ∣ ∣ im(En−Ei) ̄h 〈φn|ˆǫ·~r|φi〉 ∣ ∣ ∣ ∣ 2 = αω^3 in 2 πc^2 ∑ λ ∫ dΩγ|〈φn|ˆǫ·~r|φi〉|^ ...

∫ dΩYℓ∗nmnY 1 mYℓimi = √ 3(2ℓi+ 1) 4 π(2ℓn+ 1) 〈ℓn 0 |ℓi 100 〉〈ℓnmn|ℓi 1 mim〉 ∫ dΩYℓ∗nmn ( ǫzY 10 + −ǫx+iǫy √ 2 Y 11 + ǫx+iǫy √ ...

29.6 Explicit 2p to 1s Decay Rate Starting from the summary equation for electric dipole transitions,above, Γtot= αω^3 in 2 πc^2 ...

= 1 4 π 1 3 ∑ mi ( ǫ^2 zδmi 0 + 1 2 (ǫ^2 x+ǫ^2 y)(δmi(−1)+δmi 1 ) ) = 1 12 π ( ǫ^2 z+ 1 2 (ǫ^2 x+ǫ^2 y)(1 + 1) ) = 1 12 π ( ǫ^2 ...

Lets pick two transverse polarization vectors (to sum over) thatform a right handed system with the direction of photon propagat ...

Γtot = αωin^3 2 πc^2 |~rni|^22 π ∫^1 − 1 dx(1−x^2 ) Γtot = αωin^3 2 πc^2 |~rni|^22 π [ x− x^3 3 ] 1 − 1 Γtot = αωin^3 2 πc^2 |~r ...

We have performed that radial integration which will be unchanged. Assume that we start in a polarized state withmi= 1. We then ...

Hereαrepresents all the (other) quantum numbers of the state, not the angular momentum quantum numbers.jmrepresent the usual ang ...

29.11Lifetime and Line Width Now we have computed the lifetime of a state. For some atomic, nuclear, or particle states, this li ...

As Γ→0 this line shape approaches a delta function,δ(ω−ω 0 ). For the 2p to 1s transition in hydrogen, we’ve calculated a decay ...

Doppler broadening is a simple non-quantum effect. We know that thefrequency of photons is shifted if the source is moving – shi ...

Thus a large fraction of the radiation is emitted (and reabsorbed) without a large energy shift. (Remember that the crystal may ...

He Ne (2p) (2p) (3s) (2p) (4s) (2p) (5s) (2p) (3p) 2 S 1 S 1 0 2 S^3 1 1 0 5 5 5 5 6 collision collision visible laser IR laser ...

29.13Examples 29.13.1The 2P to 1S Decay Rate in Hydrogen 29.14Derivations and Computations 29.14.1Energy in Field for a Given Ve ...

29.14.3Estimate of Atomic Decay Rate We have the formula Γtot= e^2 (Ei−En) 2 π ̄h^2 m^2 c^3 ∫ dΩγ|〈φn|e−i( ~k·~r) ˆǫ·~pe|φi〉|^2 ...

29.16Sample Test Problems A hydrogen atom is in then= 5,^3 D 52 state. To which states is it allowed to decay via electric dipo ...

30 Scattering This material is covered inGasiorowicz Chapter 23. Scattering of one object from another is perhaps our best way o ...

Clearly there is more information available from scattering than whether a particle scatters or not. For example,Rutherford disc ...