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 ( ) ⟨ ⃗휇 ⟩ ∗ ⃗, ⃗̂휇 ⃗, 휏 ∗ ⃗, ⃗̂ ⃗, 휏 (t) = ∫ Ψ (r t) Ψ(r t)d = ∫ 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 ( ) ⟨ ⃗휇 ⟩ ∗ ⃗, ⃗̂ ⃗, 휏 (t) = ∫ Ψ (r t) −qer Ψ(r t)d V For a system in eigenstate of Hamiltonian, where wave function has the form, ℏ ⃗, 휓 ⃗ −iEnt∕ Ψn(r t) = n(r)e Electric dipole moment expectation value is ( ) ℏ ℏ ⟨ ⃗휇 ⟩ 휓∗ ⃗ iEnt∕ ⃗̂ 휓 ⃗ −iEnt∕ 휏 (t) = ∫ n (r)e −qer n(r)e d V Time dependent exponential terms cancel out leaving us with ( ) ⟨ ⃗휇 ⟩ 휓∗ ⃗ ⃗̂ 휓 ⃗ 휏 (t) = ∫ 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

⟨ ⃗휇 ⟩ ∗ ⃗, ⃗̂휇 ⃗, 휏 (t) = ∫ Ψ (r t) Ψ(r t)d V [ ] [ ] ∗ ∗ ⃗, ∗ ∗ ⃗, ⃗̂휇 ⃗, ⃗, 휏 = ∫ am(t)Ψm(r t) + an(t)Ψn(r t) am(t)Ψm(r t) + an(t)Ψn(r t) d V

⟨ ⃗휇 ⟩ ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 (t) = am(t)am(t) ∫ Ψm(r t) Ψm(r t)d + am(t)an(t) ∫ Ψm(r t) Ψn(r t)d V V ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 + an(t)am(t) ∫ Ψn(r t) Ψm(r t)d + an(t)an(t) ∫ 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

⟨ ⃗휇 ⟩ ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 (t) = am(t)am(t) ∫ Ψm(r t) Ψm(r t)d + am(t)an(t) ∫ Ψm(r t) Ψn(r t)d V ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ V Time Independent ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 + an(t)am(t) ∫ Ψn(r t) Ψm(r t)d + an(t)an(t) ∫ 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

⟨ ⃗휇 ⟩ ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 (t) = am(t)an(t) ∫ Ψm(r t) Ψn(r t)d + an(t)am(t) ∫ Ψn(r t) Ψm(r t)d V V Two integrals are complex conjugates of each other. Since ⟨ ⃗휇(t)⟩ must be real we simplify to { } ⟨ ⃗휇 ⟩ ℜ ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 (t) = am(t)an(t) ∫ 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 { } ⟨ ⃗휇 ⟩ ℜ ∗ ∗ ⃗, ⃗̂휇 ⃗, 휏 (t) = am(t)an(t) ∫ Ψm(r t) Ψn(r t)d V ℏ Inserting stationary state wave function, Ψ (⃗r, t) = 휓 (⃗r)e−iEnt∕ , gives { [n n ] } ℏ ⟨ ⃗휇 ⟩ ℜ ∗ 휓∗ ⃗ ⃗̂휇휓 ⃗ 휏 i(Em−En)t∕ (t) = am(t)an(t) ∫ m(r) n(r)d e ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟V ⟨ ⃗휇⟩ mn 휔 ℏ mn = (Em − En)∕ is angular frequency of emitted light and ⟨ ⃗휇⟩ mn is transition dipole moment—peak magnitude of dipole oscillation

⟨ ⃗휇⟩ 휓∗ ⃗ ⃗̂휇휓 ⃗ 휏 mn = ∫ m(r) n(r)d V Finally, write oscillating electric dipole moment vector as { } 휔 ⟨ ⃗휇 ⟩ ℜ ∗ ⟨ ⃗휇⟩ 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 휔 ℏ 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 { } ⟨ ⃗휇⟩ is transition dipole moment–amplitude of 휔 mn ⟨ ⃗휇 ⟩ ℜ ∗ ⟨ ⃗휇⟩ i mnt (t) = am(t)an(t) mn e dipole oscillation where

⟨ ⃗휇⟩ 휓∗ ⃗ ⃗̂휇휓 ⃗ 휏 mn = ∫ m(r) n(r)d V

Integrals give transition selection rules for various spectroscopies ( ) ( ) ( ) 휇 = 휓∗ ̂휇 휓 d휏, 휇 = 휓∗ ̂휇 휓 d휏, 휇 = 휓∗ ̂휇 휓 d휏 x mn ∫ m x n y mn ∫ m y n z mn ∫ 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, ⃗(⃗r, t) is

̂ ⃗̂휇 ⋅ ⃗ ⃗, ⃗̂휇 ⋅ ⃗ ⃗ 휔 V(t) = − (r t) = − 0(r) cos t ̂ ̂ ̂ 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 ⃗ and assume system is in spatially uniform ⃗(t) oscillating in time. Holds for atoms and molecules until x-ray wavelengths and shorter.

̂ 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 ̂ ̂ 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. ̂ ̂ ̂ ⃗, Putting (t) = 0 + V(t) and Ψ(r t) into time dependent Schrödinger Equation 휕Ψ(⃗r, t) ̂ (t)Ψ(⃗r, t) = iℏ 휕t and skipping many steps we eventually get da (t) i m ≈ − Ψ∗ (⃗r, t)V̂ (t)Ψ (⃗r, t)d휏 ℏ ∫ m n dt V

P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Rate of Light Absorption and Stimulated Emission ℏ ̂ ⃗̂휇 ⋅ ⃗ ⃗ 휔 휓 −iEnt∕ Using our expression for V(t) = − 0(r) cos t and Ψn = ne we get [ ] dam(t) i ∗ ̂ i ∗ ⃗ i휔 t ⃗ ≈ − Ψ (⃗r, t)V(t)Ψ (⃗r, t)d휏 = − 휓 ̂휇휓 d휏 e mn ⋅  cos 휔t dt ℏ ∫ m n ℏ ∫ 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 ⟨휇 ⟩2 | |2 mn t |a (t)| = u(휈 ) m 휖 ℏ2 mn 6 0 휈 휈 휔 휋 u( mn) is radiation density at mn = mn∕2 . Rate at which n → m transition from absorption of light energy occurs is

| |2 ⟨휇 ⟩2 d |am(t)| mn R → = = u(휈 ) n m 휖 ℏ2 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. 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 quantum electrodynamics (QED).

Beyond scope of course to give QED treatment.

Instead, we examine Einstein’s approach to absorption and stimulated emission of light and see what he learned about spontaneous emission.

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

P. J. Grandinetti Chapter 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 ( ) 휋휈2 휈 휈 8 h u( ) = 휈 3 eh ∕kBT − 1 c0 to 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, 휈 mn, that excites Nn atoms from level n to m, and is proportional to light intensity shining on atom 휈 Rn→m = NnBnmu( mn)

Bnm is Einstein’s proportionality constant for light absorption. For Light Emission, when atom drops from m to n level, he proposed 2 processes: Spontaneous Emission and Stimulated Emission

Absorption Stimulated Spontaneous Emission Emission

P. J. Grandinetti Chapter 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 away energy at rate proportional only to number of atoms in mth level,

spont Rm→n = NmAmn

▶ 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 by external oscillating electric fields—light in cavity—and that stimulated emission rate is proportional to light intensity and number of atoms in mth level.

stimul 휈 Rm→n = NmBmnu( mn)

Bmn is Einstein’s proportional constant for stimulated emission.

P. J. Grandinetti Chapter 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 at equilibrium spont stimul Rn→m = Rm→n + Rm→n we obtain 휈 휈 NnBnmu( mn) = Nm[Amn + Bmnu( mn)] Einstein knew from Boltzmann’s statistical mechanics that n and m populations at equilibrium depends on temperature Nm ℏ휔 = e−(Em−En)∕kBT = e− ∕kBT Nn We can rearrange the rate expression and substitute for Nm∕Nn to get

Nn ℏ휔 휈 휈 ∕kBT 휈 Amn + Bmnu( mn) = Bmnu( mn) = e Bnmu( mn) Nm and then get A u(휈 ) = mn mn ℏ휔∕k T Bnme B − Bmn P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Light Absorption and Emission (Meanwhile, back in 1916) Einstein compared his expression to Planck’s ( ) A 8휋휈2 h휈 u(휈 ) = mn = mn mn mn ℏ휔∕k T 3 h휈∕k T Bnme B − Bmn c e B − 1 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟0 Einstein Planck

Bnm = Bmn, i.e., stimulated emission and absorption rates must be equal to agree with Planck. ( ) ( ) 2 A 1 8휋휈 h휈 u(휈 ) = mn = mn mn mn ℏ휔∕k T 3 h휈∕k T Bnm e B − 1 c e B − 1 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟0 Einstein Planck

Relationship between spontaneous and stimulated emission rates:

A 8휋휈2 8휋h휈3 mn = mn h휈 = mn Amazing Einstein got this far without full quantum theory. B 3 mn 3 nm c0 c0

P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Light Absorption and Emission Fast forward to Schrödinger’s discovery of quantum wave equation, which gives light absorption rate as ⟨휇 ⟩2 mn R → = u(휈 ) = B u(휈 ) n m 휖 ℏ2 mn mn mn 6 0 setting this equal to Einstein’s rate for absorption gives ⟨휇 ⟩2 B = mn mn 휖 ℏ2 6 0 from which we can calculate the spontaneous emission rate 8휋h휈3 ⟨휇 ⟩2 A = mn mn mn 3 6휖 ℏ2 c0 0 For H atom, spontaneous emission rate from 1st excited state to ground state is ∼ 108/s in agreement with what Amn expression above would give. QED tells us that quantized electromagnetic field has zero point energy. It is these “vacuum fluctuations” that “stimulate” charge oscillations that lead to spontaneous emission process.

P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Selection Rules for Transitions

P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Transition Selection Rules

In all spectroscopies you find that transition rate between certain levels will be nearly zero.

This is because corresponding transition moment integral is zero.

For electric dipole transitions we found that transition rate depends on

⟨ ⃗휇⟩ 휓∗ ⃗ ⃗̂휇휓 ⃗ 휏 mn = ∫ m(r) n(r)d V 휓 휓 m and n are stationary states in absence of electric field.

P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Harmonic Oscillator Selection Rules Consider quantum harmonic oscillator transitions.

Imagine vibration of diatomic molecule with electric dipole moment.

This is 1D problem so we write electric dipole moment operator of harmonic oscillator in series expansion about its value at equilibrium

d휇(r ) 1 d2휇(r ) 휇(r̂) = 휇(r ) + e (r̂ − r ) + e (r̂ − r )2 + ⋯ e dr e 2 dr2 e

휇 1st term in expansion, (re), is permanent electric dipole moment of harmonic oscillator associated with oscillator at rest.

2nd term describes linear variation in electric dipole moment with changing r.

We will ignore 3rd and higher-order terms in expansion.

P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Harmonic Oscillator Plug 1st two terms of expansion into transition moment integral ( ) ∞ d휇(r ) ⟨휇 ⟩ 휓∗ 휇 e ̂ 휓 , nm = ∫ m(r) (re) + (r − re) n(r)dr −∞ dr :0 ( ) ∞  d휇(r ) ∞ 휇 휓∗ 휓 e 휓∗ ̂ 휓 = (re)∫ m(r) n(r)dr + ∫ m(r)(r − re) n(r)dr  −∞ dr −∞ Since m ≠ n we know that 1st integral is zero as stationary state wave functions are orthogonal leaving us with ( ) d휇(r ) ∞ ⟨휇 ⟩ e 휓∗ ̂ 휓 nm = ∫ m(r)(r − re) n(r)dr dr −∞ 휉 In quantum harmonic oscillator it is convenient to transform into coordinate using x = r − re and 휉 = 훼x to obtain ( ) d휇(r ) ∞ ⟨휇 ⟩ e 휒∗ 휉 휉휒̂ 휉 휉 nm = ∫ m( ) n( )d dr −∞

P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Harmonic Oscillator With harmonic oscillator wave function, 휒 (휉), we obtain ( n) ∞ d휇(r ) 2 ⟨휇 ⟩ e 휉 −휉 휉 nm = AmAn ∫ Hm Hne d dr −∞ 휉 1 Using recursive relation, Hn = Hn+1 + nHn−1, we obtain ( ) 2 [ ] ∞ ∞ d휇(r ) 2 2 ⟨휇 ⟩ e 1 −휉 휉 −휉 휉 nm = AmAn ∫ HmHn+1e d + n ∫ HmHn−1e d dr 2 −∞ −∞ To simplify expression we rearrange ∞ ∞ 훿 휉 휉 −휉2 휉 훿 휉 휉 −휉2 휉 m,n AmAn ∫ Hm( )Hn( )e d = m,n to ∫ Hm( )Hn( )e d = −∞ −∞ AmAn Substitute into expression for ⟨휇 ⟩ gives (nm )[ ] d휇(r ) A A ⟨휇 ⟩ e 1 n 훿 n 훿 nm = m,n+1 + n m,n−1 dr 2 An+1 An−1

P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Harmonic Oscillator Recalling ≡ 1 An √ 2nn! 휋1∕2 we finally obtain transition dipole moment for harmonic oscillator ( ) [√ √ ] 휇 d (re) n + 1 n ⟨휇 ⟩ = 훿 , + 훿 , nm dr 2 m n+1 2 m n−1

→ ⟨휇 ⟩2 For absorption, m = n + 1, transition is n n + 1 and mn gives ( ) 휈 휈 휇 2 u( mn) 2 u( mn) d (re) n + 1 R → = ⟨휇 ⟩ = n n+1 휖 ℏ2 mn 휖 ℏ2 6 0 6 0 dr 2

→ ⟨휇 ⟩2 For emission, m = n − 1, transition is n n − 1 and mn gives ( ) 휈 휈 휇 2 u( mn) 2 u( mn) d (re) n R → = ⟨휇 ⟩ = n n−1 휖 ℏ2 mn 휖 ℏ2 6 0 6 0 dr 2 P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Harmonic Oscillator

( ) ( ) 휈 휇 2 휈 휇 2 u( mn) d (re) n + 1 u( mn) d (re) n R → = and R → = n n+1 휖 ℏ2 n n−1 휖 ℏ2 6 0 dr 2 6 0 dr 2 Selection rule for harmonic oscillator is Δn = ±1. ( ) 휇 Also, for allowed transitions d (re)∕dr must be non-zero.

For allowed transition it is not important whether a molecule has permanent dipole moment but rather that dipole moment of molecule varies as molecule vibrates.

In later lectures we will examine transition selection rules for other types of quantized motion, such as quantized rigid rotor and orbital motion of electrons in atoms and molecules.

P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics Web Apps by Paul Falstad

Quantum transitions in one dimension

P. J. Grandinetti Chapter 14: Radiating Dipoles in Quantum Mechanics