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Chapter 14 Radiating Dipoles in Quantum Mechanics

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Chapter 14 Radiating Dipoles in Quantum Mechanics Chapter 14 Radiating Dipoles in Quantum Mechanics P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Electric dipole moment vector operator Electric dipole moment vector operator for collection of charges is ∑N ⃗̂휇 ⃗̂ = qkr k=1 Single charged quantum particle bound in some potential well, e.g., a negatively charged electron bound to a positively charged nucleus, would be [ ] ⃗̂휇 ⃗̂ ̂⃗ ̂⃗ ̂⃗ = *qer = *qe xex + yey + zez Expectation value for electric dipole moment vector in Ψ(⃗r; t/ state is ( ) ê ⃗휇 ë < ⃗; ⃗̂휇 ⃗; 휏 < ⃗; ⃗̂ ⃗; 휏 .t/ = Ê Ψ .r t/ Ψ(r t/d = Ê Ψ .r t/ *qer Ψ(r t/d V V Here, d휏 = dx dy dz P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Time dependence of electric dipole moment Energy Eigenstate Starting with ( ) ê ⃗휇 ë < ⃗; ⃗̂ ⃗; 휏 .t/ = Ê Ψ .r t/ *qer Ψ(r t/d V For a system in eigenstate of Hamiltonian, where wave function has the form, ` ⃗; ⃗ *iEnt_ Ψn.r t/ = n.r/e Electric dipole moment expectation value is ( ) ` ` ê ⃗휇 ë < ⃗ iEnt_ ⃗̂ ⃗ *iEnt_ 휏 .t/ = Ê n .r/e *qer n.r/e d V Time dependent exponential terms cancel out leaving us with ( ) ê ⃗휇 ë < ⃗ ⃗̂ ⃗ 휏 .t/ = Ê n .r/ *qer n.r/d No time dependence!! V No bound charged quantum particle in energy eigenstate can radiate away energy as light or at least it appears that way – Good news for Rutherford’s atomic model. P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics But then how does a bound charged quantum particle in an excited energy eigenstate radiate light and fall to lower energy eigenstate? P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Time dependence of electric dipole moment - A Transition During transition wave function must change from Ψm to Ψn During transition wave function must be linear combination of Ψm and Ψn ⃗; ⃗; ⃗; Ψ(r t/ = am.t)Ψm.r t/ + an.t)Ψn.r t/ Before transition we have am.0/ = 1 and an.0/ = 0 After transition am(∞) = 0 and an(∞) = 1 P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Time dependence of electric dipole moment - A Transition To maintain normalization during transition we require | |2 | |2 : am.t/ + an.t/ = 1 Electric dipole moment expectation value for Ψ(⃗r; t/ is ê ⃗휇 ë < ⃗; ⃗̂휇 ⃗; 휏 .t/ = Ê Ψ .r t/ Ψ(r t/d V [ ] [ ] < < ⃗; < < ⃗; ⃗̂휇 ⃗; ⃗; 휏 = Ê am.t)Ψm.r t/ + an.t)Ψn.r t/ am.t)Ψm.r t/ + an.t)Ψn.r t/ d V ê ⃗휇 ë < < ⃗; ⃗̂휇 ⃗; 휏 < < ⃗; ⃗̂휇 ⃗; 휏 .t/ = am.t/am.t/ Ê Ψm.r t/ Ψm.r t/d + am.t/an.t/ Ê Ψm.r t/ Ψn.r t/d V V < < ⃗; ⃗̂휇 ⃗; 휏 < < ⃗; ⃗̂휇 ⃗; 휏 + an.t/am.t/ Ê Ψn.r t/ Ψm.r t/d + an.t/an.t/ Ê Ψn.r t/ Ψn.r t/d V V P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Time dependence of electric dipole moment - A Transition ê ⃗휇 ë < < ⃗; ⃗̂휇 ⃗; 휏 < < ⃗; ⃗̂휇 ⃗; 휏 .t/ = am.t/am.t/ Ê Ψm.r t/ Ψm.r t/d + am.t/an.t/ Ê Ψm.r t/ Ψn.r t/d V «­­­­­­­­­­­¯­­­­­­­­­­­¬ V Time Independent < < ⃗; ⃗̂휇 ⃗; 휏 < < ⃗; ⃗̂휇 ⃗; 휏 + an.t/am.t/ Ê Ψn.r t/ Ψm.r t/d + an.t/an.t/ Ê Ψn.r t/ Ψn.r t/d V V «­­­­­­­­­­¯­­­­­­­­­­¬ Time Independent 1st and 4th terms still have slower time dependence due to an.t/ and am.t/ but this electric dipole variation will not lead to appreciable energy radiation. Drop these terms and focus on faster oscillating 2nd and 3rd terms ê ⃗휇 ë < < ⃗; ⃗̂휇 ⃗; 휏 < < ⃗; ⃗̂휇 ⃗; 휏 .t/ = am.t/an.t/ Ê Ψm.r t/ Ψn.r t/d + an.t/am.t/ Ê Ψn.r t/ Ψm.r t/d V V Two integrals are complex conjugates of each other. Since ê ⃗휇.t/ë must be real we simplify to < = ê ⃗휇 ë R < < ⃗; ⃗̂휇 ⃗; 휏 .t/ = am.t/an.t/ Ê Ψm.r t/ Ψn.r t/d V P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Time dependence of electric dipole moment - A Transition < = ê ⃗휇 ë R < < ⃗; ⃗̂휇 ⃗; 휏 .t/ = am.t/an.t/ Ê Ψm.r t/ Ψn.r t/d V ` Inserting stationary state wave function, Ψ .⃗r; t/ = .⃗r/e*iEnt_ , gives < 4n n 5 = ` ê ⃗휇 ë R < < ⃗ ⃗̂휇 ⃗ 휏 i.Em*En/t_ .t/ = am.t/an.t/ Ê m.r/ n.r/d e «­­­­­­­­­­¯­­­­­­­­­­¬V ê ⃗휇ë mn ! ` mn = .Em * En/_ is angular frequency of emitted light and ê ⃗휇ë mn is transition dipole moment—peak magnitude of dipole oscillation ê ⃗휇ë < ⃗ ⃗̂휇 ⃗ 휏 mn = Ê m.r/ n.r/d V Finally, write oscillating electric dipole moment vector as { } ! ê ⃗휇 ë R < ê ⃗휇ë i mnt .t/ = am.t/an.t/ mn e P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Transition dipole moment In summary, the superposition state ! ` mn = .Em * En/_ is angular frequency of emitted ⃗; ⃗; ⃗; light Ψ(r t/ = am.t)Ψm.r t/ + an.t)Ψn.r t/ a< .t/a .t/ gives time scale of transition. has oscillating electric dipole moment vector m n { } ê ⃗휇ë is transition dipole moment–amplitude of ! mn ê ⃗휇 ë R < ê ⃗휇ë i mnt .t/ = am.t/an.t/ mn e dipole oscillation where ê ⃗휇ë < ⃗ ⃗̂휇 ⃗ 휏 mn = Ê m.r/ n.r/d V Integrals give transition selection rules for various spectroscopies ( ) ( ) ( ) 휇 = < ̂휇 d휏; 휇 = < ̂휇 d휏; 휇 = < ̂휇 d휏 x mn Ê m x n y mn Ê m y n z mn Ê m z n V V V |( ) |2 |( ) |2 |( ) |2 where 휇2 = | 휇 | + | 휇 | + | 휇 | mn | x mn| | y mn| | z mn| P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Okay, so... Wave function in superposition of energy eigenstates can have oscillating electric dipole moment 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 and final eigenstates in first place? One way to shine light onto the atom. The interaction of the atom and light leads to absorption and stimulated emission of light. Absorption Stimulated Spontaneous Emission Emission P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Rate of Light Absorption and Stimulated Emission Potential energy of ⃗̂휇 of quantum system interacting with a time dependent electric field, E⃗.⃗r; t/ is ̂ ⃗̂휇 ⋅ E⃗ ⃗; ⃗̂휇 ⋅ E⃗ ⃗ ! V.t/ = * .r t/ = * 0.r/ cos t Ĥ Ĥ ̂ and Hamiltonian becomes .t/ = 0 + V.t/. If light wavelength is long compared to system size (atom or molecule) we can ignore ⃗r dependence of E⃗ and assume system is in spatially uniform E⃗.t/ oscillating in time. Holds for atoms and molecules until x-ray wavelengths and shorter. Ĥ When time dependent perturbation is present old stationary states of 0 (before light was turned on) are no longer stationary states. P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Rate of Light Absorption and Stimulated Emission To describe time dependence of electric dipole moment write wave function as linear Ĥ ̂ combination of stationary state eigenfunction of 0, i.e., when V.t/ is absent. ∑n ⃗; ⃗; Ψ(r t/ = am.t)Ψm.r t/ m=1 Need to determine time dependence of am.t/ coefficients. | |2 | |2 As initial condition take an.0/ = 1 and am≠n.0/ = 0. Ĥ Ĥ ̂ ⃗; Putting .t/ = 0 + V.t/ and Ψ(r t/ into time dependent Schrödinger Equation )Ψ(⃗r; t/ Ĥ .t)Ψ(⃗r; t/ = i` )t and skipping many steps we eventually get da .t/ i m ù * Ψ< .⃗r; t/V̂ .t)Ψ .⃗r; t/d휏 ` Ê m n dt V P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Rate of Light Absorption and Stimulated Emission ` ̂ ⃗̂휇 ⋅ E⃗ ⃗ ! *iEnt_ Using our expression for V.t/ = * 0.r/ cos t and Ψn = ne we get 4 5 dam.t/ i < ̂ i < ⃗ i! t ⃗ ù * Ψ .⃗r; t/V.t/Ψ .⃗r; t/d휏 = * ̂휇 d휏 e mn ⋅ E cos !t dt ` Ê m n ` Ê m n 0 V «­­­­­­­¯­­­­­­­¬V transition moment integral | |2 | |2 ≠ Setting an.0/ = 1 and am.0/ = 0 for n m (after many steps) we find ê휇 ë2 | |2 mn t |a .t/| = u.휈 / m 휖 `2 mn 6 0 휈 휈 ! 휋 u. mn/ is radiation density at mn = mn_2 . Rate at which n → m transition from absorption of light energy occurs is | |2 ê휇 ë2 d |am.t/| mn R → = = u.휈 / n m 휖 `2 mn dt 6 0 P. J. Grandinetti Chapter 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 its ground state in the absence of any electromagnetic radiation, i.e., V̂ .t/ = 0 How does spontaneous emission happen? In this lecture’s derivations we treat light as classical E&M wave.
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