Chapter 14 - Radiating Dipoles In Quantum Mechanics - Grandinetti

Chapter 14Radiating Dipoles in Quantum MechanicsP. J. GrandinettiChem. 4300P. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

P. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Electric dipole moment vector operatorElectric dipole moment vector operator for collection of charges is𝜇⃗̂ N qk ⃗r̂k 1Single charged quantum particle bound in some potential well, e.g., a negatively chargedelectron bound to a positively charged nucleus, would be[]𝜇⃗̂ qe ⃗r̂ qe x̂ e⃗x ŷ e⃗y ẑ e⃗zExpectation value for electric dipole moment vector in Ψ(⃗r, t) state is⟨𝜇(t)⟩⃗ V⃗̂ r, t)d𝜏 Ψ (⃗r, t)𝜇Ψ(⃗ V()Ψ (⃗r, t) qe ⃗r̂ Ψ(⃗r, t)d𝜏Here, d𝜏 dx dy dzP. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Time dependence of electric dipole momentEnergy EigenstateStarting with⟨𝜇(t)⟩⃗ ()Ψ (⃗r, t) qe ⃗r̂ Ψ(⃗r, t)d𝜏 VFor a system in eigenstate of Hamiltonian, where wave function has the form,Ψn (⃗r, t) 𝜓n (⃗r)e iEn t ℏElectric dipole moment expectation value is⟨𝜇(t)⟩⃗ V()𝜓n (⃗r)eiEn t ℏ qe ⃗r̂ 𝜓n (⃗r)e iEn t ℏ d𝜏Time dependent exponential terms cancel out leaving us with() ⃗𝜓 (⃗r) qe r̂ 𝜓n (⃗r)d𝜏⟨𝜇(t)⟩⃗ No time dependence!! V nNo bound charged quantum particle in energy eigenstate can radiate away energy as lightor at least it appears that way – Good news for Rutherford’s atomic model.P. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

But then how does a bound charged quantum particle in anexcited energy eigenstate radiate light and fall to lower energy eigenstate?P. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Time dependence of electric dipole moment - A TransitionDuring transition wave function must change from Ψm to ΨnDuring transition wave function must be linear combination of Ψm and ΨnΨ(⃗r, t) am (t)Ψm (⃗r, t) an (t)Ψn (⃗r, t)Before transition we have am (0) 1 and an (0) 0After transition am ( ) 0 and an ( ) 1P. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Time dependence of electric dipole moment - A TransitionTo maintain normalization during transition we require am (t) 2 an (t) 2 1.Electric dipole moment expectation value for Ψ(⃗r, t) is⟨𝜇(t)⟩⃗ V V⃗̂ r, t)d𝜏Ψ (⃗r, t)𝜇Ψ(⃗[ ] []am (t)Ψ m (⃗r, t) a n (t)Ψ n (⃗r, t) 𝜇⃗̂ am (t)Ψm (⃗r, t) an (t)Ψn (⃗r, t) d𝜏⟨𝜇(t)⟩⃗ a m (t)am (t) a n (t)am (t)P. J. Grandinetti V V⃗̂ m (⃗r, t)d𝜏Ψ m (⃗r, t)𝜇Ψ a m (t)an (t)⃗̂ m (⃗r, t)d𝜏Ψ n (⃗r, t)𝜇Ψ a n (t)an (t) V V⃗̂ n (⃗r, t)d𝜏Ψ m (⃗r, t)𝜇Ψ⃗̂ n (⃗r, t)d𝜏Ψ n (⃗r, t)𝜇ΨChapter 14: Radiating Dipoles in Quantum Mechanics

Time dependence of electric dipole moment - A Transition⟨𝜇(t)⟩⃗ a m (t)am (t)⃗̂ m (⃗r, t)d𝜏Ψ m (⃗r, t)𝜇Ψ V a m (t)an (t) a n (t)an (t) V⃗̂ n (⃗r, t)d𝜏Ψ m (⃗r, t)𝜇ΨTime Independent a n (t)am (t) V⃗̂ m (⃗r, t)d𝜏Ψ n (⃗r, t)𝜇Ψ⃗̂ n (⃗r, t)d𝜏Ψ n (⃗r, t)𝜇Ψ V Time Independent1st and 4th terms still have slower time dependence due to an (t) and am (t) but this electric dipolevariation will not lead to appreciable energy radiation.Drop these terms and focus on faster oscillating 2nd and 3rd terms⟨𝜇(t)⟩⃗ a m (t)an (t) V⃗̂ n (⃗r, t)d𝜏Ψ m (⃗r, t)𝜇Ψ a n (t)am (t) V⃗̂ m (⃗r, t)d𝜏Ψ n (⃗r, t)𝜇ΨTwo integrals are complex conjugates of each other.Since ⟨𝜇(t)⟩⃗must be real we simplify to{}⃗̂ n (⃗r, t)d𝜏⟨𝜇(t)⟩⃗ ℜ a m (t)an (t) Ψ m (⃗r, t)𝜇Ψ VP. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Time dependence of electric dipole moment - A Transition{⟨𝜇(t)⟩⃗ ℜa m (t)an (t) V⃗̂ n (⃗r, t)d𝜏Ψ m (⃗r, t)𝜇Ψ}Inserting stationary state wave function, Ψn (⃗r, t) 𝜓n (⃗r)e iEn t ℏ , gives{[]} i(E E)t ℏ⃗̂ n (⃗r)d𝜏 e m n⟨𝜇(t)⟩⃗ ℜ am (t)an (t)𝜓 (⃗r)𝜇𝜓 V m ⟨𝜇⟩⃗ mn𝜔mn (Em En ) ℏ is angular frequency of emitted light and⟨𝜇⟩⃗ mn is transition dipole moment—peak magnitude of dipole oscillation⟨𝜇⟩⃗ mn V⃗̂ n (⃗r)d𝜏𝜓m (⃗r)𝜇𝜓Finally, write oscillating electric dipole moment vector as}{⃗ mn ei𝜔mn t⟨𝜇(t)⟩⃗ ℜ a m (t)an (t) ⟨𝜇⟩P. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Transition dipole momentIn summary, the superposition state𝜔mn (Em En ) ℏ is angular frequency of emittedlightΨ(⃗r, t) am (t)Ψm (⃗r, t) an (t)Ψn (⃗r, t)has oscillating electric dipole moment vector{}⟨𝜇(t)⟩⃗ ℜ a m (t)an (t) ⟨𝜇⟩⃗ mn ei𝜔mn ta m (t)an (t) gives time scale of transition.⟨𝜇⟩⃗ mn is transition dipole moment–amplitude ofdipole oscillationwhere⟨𝜇⟩⃗ mn V⃗̂ n (⃗r)d𝜏𝜓m (⃗r)𝜇𝜓Integrals give transition selection rules for various spectroscopies( )( )( )𝜇x mn 𝜓m 𝜇̂ x 𝜓n d𝜏, 𝜇y mn 𝜓m 𝜇̂ y 𝜓n d𝜏, 𝜇z mn 𝜓 𝜇̂ 𝜓 d𝜏 V V V m z n( ) 2 ( ) 2 ( ) 22 𝜇where 𝜇mn x mn 𝜇y mn 𝜇z mn P. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Okay, so.Wave function in superposition of energy eigenstates can have oscillating electric dipolemoment which will emit light until system is entirely in lower energy state.But how does atom in higher energy eigenstate get into this superposition of initial andfinal eigenstates in first place?One way to shine light onto the atom. The interaction of the atom and light leads toabsorption and stimulated emission of light.AbsorptionP. J. apter 14: Radiating Dipoles in Quantum Mechanics

Rate of Light Absorption and Stimulated Emission⃗ r, t) isPotential energy of 𝜇⃗̂ of quantum system interacting with a time dependent electric field, (⃗⃗ r, t) 𝜇⃗̂ ⃗0 (⃗r) cos 𝜔t̂ 𝜇⃗̂ (⃗V(t)̂ ̂ 0 V(t).̂and Hamiltonian becomes (t)If light wavelength is long compared to system size (atom or molecule) we can ignore ⃗r⃗ oscillating in time. Holds fordependence of ⃗ and assume system is in spatially uniform (t)atoms and molecules until x-ray wavelengths and shorter.When time dependent perturbation is present old stationary states of ̂ 0 (before light was turnedon) are no longer stationary states.P. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Rate of Light Absorption and Stimulated EmissionTo describe time dependence of electric dipole moment write wave function as linear̂ is absent.combination of stationary state eigenfunction of ̂ 0 , i.e., when V(t)Ψ(⃗r, t) n am (t)Ψm (⃗r, t)m 1Need to determine time dependence of am (t) coefficients.As initial condition take an (0) 2 1 and am n (0) 2 0.̂ ̂ 0 V(t)̂ and Ψ(⃗r, t) into time dependent Schrödinger EquationPutting (t) Ψ(⃗r, t)̂ (t)Ψ(⃗r, t) iℏ tand skipping many steps we eventually getdam (t)îr, t)d𝜏 Ψ (⃗r, t)V(t)Ψn (⃗dtℏ V mP. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Rate of Light Absorption and Stimulated Emission̂ 𝜇⃗̂ ⃗0 (⃗r) cos 𝜔t and Ψn 𝜓n e iEn t ℏ we getUsing our expression for V(t)[]dam (t)ii ⃗̂ Ψ (⃗r, t)V(t)Ψn (⃗r, t)d𝜏 𝜓 𝜇𝜓̂ d𝜏ei𝜔mn t ⃗0 cos 𝜔t V m ndtℏ V mℏ transition moment integralSetting an (0) 2 1 and am (0) 2 0 for n m (after many steps) we find2 am (t) 2 ⟨𝜇mn ⟩ t u(𝜈mn ) 6𝜖0 ℏ2u(𝜈mn ) is radiation density at 𝜈mn 𝜔mn 2𝜋.Rate at which n m transition from absorption of light energy occurs isRn mP. J. Grandinetti2d am (t) ⟨𝜇 ⟩2 mn 2 u(𝜈mn )dt6𝜖0 ℏChapter 14: Radiating Dipoles in Quantum Mechanics

Okay, but what about spontaneous emission?An atom or molecule in an excited energy eigenstate can spontaneously emit light and return to itŝ 0ground state in the absence of any electromagnetic radiation, i.e., V(t)How does spontaneous emission happen?In this lecture’s derivations we treat light as classical E&M wave. No mention of photons.Treating light classically gives no explanation for how superposition gets formed.To explain spontaneous emission we need quantum field theory, which for light is called quantumelectrodynamics (QED).Beyond scope of course to give QED treatment.Instead, we examine Einstein’s approach to absorption and stimulated emission of light and seewhat he learned about spontaneous emission.P. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Light Absorption and Emission (Meanwhile, back in 1916)P. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Light Absorption and Emission (Meanwhile, back in 1916)Before any of this quantum theory was worked out. At this time Einstein used Planck’s distribution()h𝜈8𝜋𝜈 2u(𝜈) 3eh𝜈 kB T 1c0to examine how atom interacts with light inside cavity full of radiation.For Light Absorption he said atom’s light absorption rate, Rn m , depends on light frequency,𝜈mn , that excites Nn atoms from level n to m, and is proportional to light intensity shining on atomRn m Nn Bnm u(𝜈mn )Bnm is Einstein’s proportionality constant for light absorption.For Light Emission, when atom drops from m to n level, he proposed 2 processes: SpontaneousEmission and Stimulated EmissionAbsorptionP. J. apter 14: Radiating Dipoles in Quantum Mechanics

Light Absorption and Emission (Meanwhile, back in 1916)Spontaneous Emission: When no light is in cavity, atom spontaneously radiates awayenergy at rate proportional only to number of atoms in mth level,Rspontm n Nm Amn Amn is Einsteins proportional constant for spontaneous emission.Einstein knew oscillating dipoles radiate and he assumed the same for atoms.Remember, in 1916, he wouldn’t know about stationary states and Schrödinger Eq.incorrectly predicting atoms do not spontaneously radiate.Stimulated Emission: From E&M Einstein guessed that emission was also generated byexternal oscillating electric fields—light in cavity—and that stimulated emission rate isproportional to light intensity and number of atoms in mth level.Rstimulm n Nm Bmn u(𝜈mn )Bmn is Einstein’s proportional constant for stimulated emission.P. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Light Absorption and Emission (Meanwhile, back in 1916)Taking 3 processes together and assuming that absorption and emission rates are equal atequilibriumstimulRn m Rspontm n Rm nwe obtainNn Bnm u(𝜈mn ) Nm [Amn Bmn u(𝜈mn )]Einstein knew from Boltzmann’s statistical mechanics that n and m populations at equilibriumdepends on temperatureNm e (Em En ) kB T e ℏ𝜔 kB TNnWe can rearrange the rate expression and substitute for Nm Nn to getNAmn Bmn u(𝜈mn ) n Bmn u(𝜈mn ) eℏ𝜔 kB T Bnm u(𝜈mn )Nmand then getAmnu(𝜈mn ) ℏ𝜔 kBT BBnm emnP. J. GrandinettiChapter 14: Radiating Dipoles in Quantum Mechanics

Light Absorption and Emission (Meanwhile, back in 1916)Einstein compared his expression to Planck’su(𝜈mn ) AmnBnm eℏ𝜔 kB T Bmn Einstein 28𝜋𝜈mn(h𝜈mn)eh𝜈 kB T 1c30 PlanckBnm Bmn , i.e., stimulated emission and absorption rates must be equal to agree with Planck.))(2 (8𝜋𝜈mnAmnh𝜈mn1u(𝜈mn ) Bnm eℏ𝜔 kB T 1eh𝜈 kB T 1c30 EinsteinPlanckRelationship between spontaneous and stimulated emission rates:328𝜋h𝜈mn8𝜋𝜈mnAmnh𝜈 mnBnmc30c30P. J. GrandinettiAmazing Einstein got this far without full quantum theory.Chapter 14: Radiating Dipoles in Quantum Mechanics