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 transitionbetween two levels with energy difference E2 – E1 = w is given byT =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; ein front of the integral is the elementary charge).(a) Determine (2|µ|1) for the transition 2p → 1s in the hydrogen atom, using the wavefunctions1e-r/ao.11 (r)b2(r)e-r/(2a0),32na, aowhere r = Vx2 + y² + z².(b) OPTIONAL Use your result from part (a) and Fermi's golden rule to calculate thespontaneous emission rate r, where you can use w = 1.55 x 1016s-1 and ao = 5.29 x10-11 m.From this, determine the decay time T = 1/T, and estimate the energy uncertaintyAE z ħ/T of the spontaneously emitted photons.

Part a: Use the definition of expectation value and solve using spherical coordinate system.

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.