# 16 - Page 110 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS...

This preview shows pages 1–4. Sign up to view the full content.

Page 110 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS * Introduction and Preview Now the origin of frequency fluctuations is expected to be interactions of our molecule (or more appropriately our electronic transitions) with its environment. This we can treat with our D.H.O. model, which is a general approach to coupling to nuclear vibrations. We found that iH t 0 µ e 0 µ e iH t t µ ( ) µ ( 0 ) = p n n n n i 2 − ω eg t H t iH t e g e e e = µ eg We can write this in terms of the Hamiltonian that describes the electronic energy gap’s dependence on Q (deviation relative to ω eg ): H eg = H e TOT H TOT = ω = H e H g (Energy Gap Hamiltonian) g eg 2 i ω t iH t e eg eg t C µµ ( ) = e µ eg Now if we believe there are interactions that lead to fluctuations in the energy group—variations in d or ω 0 , then our H eg is now time-dependent! t i t + C µµ ( ) = e ω eg t exp i d τ H eg ( ) τ = 0 Performing the cumulant expansion: i t exp = 0 τ − i t d τ H eg ( ) 1 eg τ 2 eg d τ H eg ( ) = exp = 0 τ +  − i 2 0 t d τ 2 0 τ 2 d τ H ( ) H ( τ 1 ) + + = * See Mukamel, Ch. 8 and Ch. 7

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Page 111 t t Defining δω eg ( ) H eg ( ) = F t g t ( ) ( ) = e ( ) = t d τ 2 d τ δω ( τ δω ( 0 ) g t 1 ) 1 eg eg 0 0 τ 2 So we have an expression for how the time-dependence of the energy gap Hamiltonian leads to the lineshape. Also note: H 0 = H e + E e + H g + E g 2 2 Q 2 H = p + 1 g 2 m 2 m ω D = = ω + H + 2 H eg eg g 2 2 1 2 2 H = H H = 1 m ω 0 ( Q d ) 2 m ω Q eg e g 2 0 2 2 2 = − ω 0 m d Q + 1 m ω 0 d ±²³² 2 ²³² ´ ± ´ linear in Q const Note that this looks very much like a Hamiltonian that describes the coupling of an electronic system to a bath [one degree of freedom here] of H.O. with a linear coupling between the two! H H S + H + H SB = B = e E + λ e + g E g g H S e 2 2 H B = p + 1 m ω 0 Q 2 2 m 2 2 H SB ( H ) = m ω 0 d 2 Q ²³² eg ± ´ coupling strength Fluctuations in coupling to bath could lead to line broadening! Equivalently, coupling to a bath of many harmonic oscillators should lead to line-broadening.
Page 112 Time-Dependent Energy Gap Hamiltonian Let’s work through this more carefully. Start by defining reduced coordinates E λ e H g H TOT TOT = ω 0 m p = p ~ 2 m ω 0 E D q = q ~ 2 = m ω 0 d = d ~ 2 = E A H = ω 0 p 2 + q + d 2 0 d Q = e ~ ~ ~ H eg 2 H g = ω 0

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern