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According to Fermi's golden rule, the spontaneous emission rate I for an atomic transition between two levels with energy difference E2 – E = ħw is given by г. |(2\+|1)(², where (2\4|1) = e ||| d°r ¢;(r)rv (r) is the matriæ element of the dipole operator (note that this is a three-component vector; e in front of the integral is the elementary charge). (a) Determine (2|µ|1) for the transition 2p → 1s in the hydrogen atom, using the wave functions V1 (r) = 1 çe=r/ao_ b2(r) = 1 -e=r/(2ao). V 32raž ao where r = Vr? + y? + 2². (b) OPTIONAL Use your result from part (a) and Fermi's golden rule to calculate the spontaneous emission rate r, where you can use w = 1.55 x 1016s-1 and ao = 5.29 x 10-11 m. From this, determine the decay time 7 = 1/T, and estimate the energy uncertainty AE z ħ/T of the spontaneously emitted photons.

According to Fermi's golden rule, the spontaneous emission rate I for an atomic transition between two levels with energy difference E2 – E = ħw is given by г. |(2\+|1)(², where (2\4|1) = e ||| d°r ¢;(r)rv (r) is the matriæ element of the dipole operator (note that this is a three-component vector; e in front of the integral is the elementary charge). (a) Determine (2|µ|1) for the transition 2p → 1s in the hydrogen atom, using the wave functions V1 (r) = 1 çe=r/ao_ b2(r) = 1 -e=r/(2ao). V 32raž ao where r = Vr? + y? + 2². (b) OPTIONAL Use your result from part (a) and Fermi's golden rule to calculate the spontaneous emission rate r, where you can use w = 1.55 x 1016s-1 and ao = 5.29 x 10-11 m. From this, determine the decay time 7 = 1/T, and estimate the energy uncertainty AE z ħ/T of the spontaneously emitted photons.

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According to Fermi's golden rule, the spontaneous emission rate [gamma] for an atomic transition between two levels with energy difference E2-E1 = [hbar][omega] is given by:

{Equation shown in in attached image}

a) Determine <2|[mu]|1> for the transition 2p -> 1s in the hydrogen atoms using the wave functions:

{Equations shown in in attached image}

Equations shown in image, help with part b would be appreciated too!

According to Fermi's golden rule, the spontaneous emission rate I for an atomic transition between two levels with energy difference E2 – E1 = w is given by T = 3uheses (2||1)|*, where (2||1) = e ||| dr;(r)rv1(r) is the matrix element of the dipole operator (note that this is a three-component vector; e in front of the integral is the elementary charge). (a) Determine (2|µ|1) for the transition 2p → 1s in the hydrogen atom, using the wave functions 1 e-r/ao. 1 1 (r) b2(r) e-r/(2a0), 32na, ao where r = Vx2 + y² + z². (b) OPTIONAL Use your result from part (a) and Fermi's golden rule to calculate the spontaneous emission rate r, where you can use w = 1.55 x 1016s-1 and ao = 5.29 x 10-11 m. From this, determine the decay time T = 1/T, and estimate the energy uncertainty AE z ħ/T of the spontaneously emitted photons.
Transcribed Image Text:According to Fermi's golden rule, the spontaneous emission rate I for an atomic transition between two levels with energy difference E2 – E1 = w is given by T = 3uheses (2||1)|*, where (2||1) = e ||| dr;(r)rv1(r) is the matrix element of the dipole operator (note that this is a three-component vector; e in front of the integral is the elementary charge). (a) Determine (2|µ|1) for the transition 2p → 1s in the hydrogen atom, using the wave functions 1 e-r/ao. 1 1 (r) b2(r) e-r/(2a0), 32na, ao where r = Vx2 + y² + z². (b) OPTIONAL Use your result from part (a) and Fermi's golden rule to calculate the spontaneous emission rate r, where you can use w = 1.55 x 1016s-1 and ao = 5.29 x 10-11 m. From this, determine the decay time T = 1/T, and estimate the energy uncertainty AE z ħ/T of the spontaneously emitted photons.
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