# 5.74 Introductory Quantum Mechanics II

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 Andrei Tokmakoff, MIT Department of Chemistry, 3/15/ QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS Introduction In describing fluctuations in a quantum mechanical system, we will now address how they manifest themselves in an electronic absorption spectrum by returning to the Displaced Harmonic Oscillator model. As previously discussed, we can also interpret the DHO model in terms of an electronic energy gap which is modulated as a result of interactions with nuclear motion. While this motion is periodic for the case coupling to a single harmonic oscillator, we will look this more carefully for a continuous distribution of oscillators, and show the correspondence to classical stochastic equations of motion. Energy Gap Hamiltonian Now let s work through the description of the Energy Gap Hamiltonian more carefully. Remember that the Hamiltonian for coupling of an electronic transition to a harmonic dree of freedom is written as H = H e + E e + H g + E g (7.48) H = ω + H + H g (7.49) where the Energy Gap Hamiltonian is H = H e H g. (7.5) Note how eq. (7.49) can be thought of as an electronic system interacting with a harmonic bath, where H plays the role of the system-bath interaction: H = H S + H SB + H B (7.51) We will express the energy gap Hamiltonian through reduced coordinates for the momentum, coordinate, and displacement of the oscillator p = ω m pˆ. (7.5) q = mω qˆ (7.53) See Mukamel, Ch. 8 and Ch. 7.

3 7-1 From (7.5) we have d = H e = ω ( mω d (7.54) p + (q d ) ) H g = ω ( p + q ) (7.55) H = ω dq + ω d (7.56) = ω dq + λ So, we see that the energy gap Hamiltonian describes a linear coupling of the electronic system to the coordinate q. The slope of H versus q is the coupling strength, and the average value of H in the ground state, H (q=), is offset by the reorganization energy λ. To obtain the absorption lineshape from the dipole correlation function we must evaluate the dephasing function. C μμ () t = μ e iωt F () t (7.57) e ih g t e ih e F()= t t = U U (7.58) g e We now want to rewrite the dephasing function in terms of the time dependence to the energy gap H ; that is, if F()= t U, then what is U? This involves a transformation of the dynamics to a new frame of reference and a new Hamiltonian. The transformation from the DHO Hamiltonian to the EG Hamiltonian is similar to our derivation of the interaction picture. Note the mapping H e = H g + H H = H +V (7.59) Then we see that we can represent the time dependence of H by evolution under H g. The time-propagators are

4 7-13 ih t e e = e ih g t i t exp+ d τ H τ ( ) (7.6) U e = UgU and ih t H ()= t e g H e = Ug H U g ih gt. (7.61) Remembering the equivalence between H g and the bath mode(s) H B indicates that the time dependence of the EG Hamiltonian reflects how the electronic energy gap is modulated as a result of the interactions with the bath. That is U g = U B. Equation (7.6) immediately implies that U ( τ )= exp + i t dτ H ( ) τ (7.6) F()= t ih t t ih t e g e e = exp + i d τ (7.63) τ H ( ) Note: Transformation of time-propagators to a new Hamiltonian If we have e ih A t Ae ih B t and we want to express this in terms of ( B A )t ih BA t Ae ih H = Ae, we will now be evolving the system under a different Hamiltonian H BA. We must perform a transformation into this new frame of reference, which involves a unitary transformation under the reference Hamiltonian: H new = H ref + H diff ih t i e ih new t ref = e exp + H τ = U H U diff ( ) ref diff ref t dτ H diff ( ) τ

5 7-14 This is what we did for the interaction picture. Now, proceeding a bit differently, we can express the time evolution under the Hamiltonian of H B relative to H A as H B = H A + H BA e ih t = e ih t i exp + t dτ H BA ( ) B A τ ( ) +ih A t ih A t where H BA τ = e H BA e. This implies: e +ih t e ih t = exp i dτ H τ A B + t BA ( ) Using the second-order cumulant expansion allows the dephasing function to be written as F t ()= exp i t dτ H ( τ ) i t τ ( τ ) H ( ) H ( ) H ( ) H τ dτ dτ 1 τ τ (7.64) Note that the cumulant expansion is here written as a time-ordered expansion here. The first exponential term depends on the mean value of H H = ω d = λ (7.65) This is a result of how we defined H. Alternatively, the EG Hamiltonian could also be defined relative to the energy gap at Q = : H = H e H g λ. In fact this is a more common definition. In this case the leading term in (7.64) would be zero, and the mean energy gap that describes the high frequency (system) oscillation in the dipole correlation function is ω + λ. The second exponential term in (7.64) is a correlation function that describes the time dependence of the energy gap H ( ) τ H (τ 1 ) 1 = δ H τ δ H τ 1 H (τ ) H (τ ) ( ) ( ) (7.66) where δ H = H H. (7.67) Defining the time-dependent energy gap frequency in terms of the EG Hamiltonian as

6 7-15 we obtain C (τ τ δ H δω (7.68), ) = δω (τ τ ) δω ( ) (7.69) 1 1 F()= t exp t τ i λt dτ dτ1 C (τ τ 1 ) (7.7) So, the dipole correlation function can be expressed as g()= t i E E +λ t/ g t C μμ () t = μ e ( ) e g e ( ) (τ ) (7.71) t dτ τ dτ δω τ δω ( ). (7.7) 1 1 This is the correlation function expression that determines the absorption lineshape for a timedependent energy gap. It is a perfectly general expression at this point. The only approximation made for the bath is the second cumulant expansion. Now, let s look specifically at the case where the bath we are coupled to is a single harmonic mode. Evaluating the energy gap correlation function C n n 1 ih t g e g n ()= t p n δω (t)δω () n = p n n δ H e ih t δ H n (7.73) = ω ( +1)e iω t + n e +i ω D n t Here, as before, D= d, and n is the thermally averaged occupation number for the oscillator Note that C is a complex quantity with n = n p n n a a n = (e β ω 1) 1. (7.74) C t = C + ic (7.75) ( ) C ()= t ω D coth ( βω cos C () t = ω Dsin ω t ( ) ) (ω t) (7.76) x x ( ) ( ) (e e ). As the temperature is raised well beyond the frequency of Here coth x = e x + e x the oscillator, C becomes real, C >> C, and C t ( ) limit in which the energy gap is modulated at the frequency of the oscillator. ~cosω t. This is the simple classical

7 7-16 Evaluating (7.7) gives the lineshape function g()= t D coth (βω /)( 1 cosω t) + i (sinω t ω t) = g + ig (7.77) We also have real ( g ) and imaginary ( g ) contributions to F( t). Alternatively, we can write this in a form that more closely parallels our earlier DHO expressions g()= t D n e i t +iω t ω 1 e 1 + e 1 id t ( ω + ) ( i t ) ω ω ) + n e + ω D n ( 1 1 ) id ω t = ( +1) e i t ( i t The leading term gives us a vibrational progression, the second term leads to hot bands, and the final term is the reorganization energy. (7.78) Looking at the low temperature limit for this expression, coth (β ω / ) 1 and n, we have g()= t D[1 cosω t + i sin ω t iω t ] i 1 e ω t = D i ω t. (7.79) Combining with we have our old result: i t/ g( t) id ω t g t) ()= e λ = e ( F t (7.8) F t ω ()= exp D (e i t 1). (7.81) In the high temperature limit coth (βω ) β ω and g >> g. From eq. (7.77) we obtain F t ()= exp D (1 cos(ω t)) β ω DkT /ω 1 DkT j = e cos j (ω t) j= j! ω (7.8) which leads to an absorption spectrum which is a series of sidebands equally spaced on either side of ω.

8 7-17 Spectral representation of energy gap correlation function Since time- and frequency domain representations are complementary, and one form may be preferable over another, it is possible to express the frequency correlation function in terms of its spectrum. We define a Fourier transform pair that relates the time and frequency domain representations: C ( ) C t ω = + e i t ω t dt = Re + e i t C () t dt. (7.83) ω C () ( )= 1 + e i t ( * The second equality in eq. (7.83) follows from C ( ) = C Where C C t π ω C ω )dt (7.84) t (t). Also it implies that C (ω) = C (ω) + C (ω) (7.85) ω and C ω are the Fourier transforms of the real and imaginary components of ( ) ( ), respectively. Note that C ω is an entirely real quantity. ( ) ( ) With these definitions in hand, we can the spectrum of the energy gap correlation function for coupling to a single harmonic mode spectrum (eq. (7.73)): C ( ω ) = ω D ( ) (n +1)δ ( ω ω ) + n ( + ). (7.86) ω δ ω ω This spectrum characterizes the thermally averaged balance between upward energy transition of the system and downward in the bath δ (ω ω ) and vice versa in δ (ω+ ω ). This is given by the detailed balance expression C ( ω) =e β ω C ( ω ). (7.87) The balance of rates tends toward equal with increasing temperature. Fourier transforms of eqs. (7.76) gives two other representations of the energy gap spectrum C ( ω ) = ω D ( ω ) coth ( βω ) δ ( ω ω ) +δ ω ω ( ) ( ) δ ( ω ω ) δ ( ω ω ) C ( + ) (7.88) ω = ω D ω + +. (7.89) The representations in eqs. (7.86), (7.88), and (7.89) are not independent, but can be related to one another through the detailed balance expression:

9 7-18 C (ω ) = coth ( β ω )C (ω ) (7.9) C ω = 1+ coth βω ))C ( ) ( ( (ω ) (7.91) Due to its independence on temperature, C (ω ) is a commonly used representation. Also. from eqs. (7.7) and (7.84) we obtain the lineshape function as g t + 1 C (ω) ()= dω π ω exp( iωt) + iωt 1. (7.9)

10 7-19 Distribution of States: Coupling to a Harmonic Bath More generally for condensed phase problems, the system coordinates that we observe in an experiment will interact with a continuum of nuclear motions that may reflect molecular vibrations, phonons, or intermolecular interactions. We conceive of this continuum as continuous distribution of harmonic oscillators of varying mode frequency. The Energy Gap Hamiltonian is readily generalized to the case of a continuous distribution of motions if we statistically characterize the density of states and the strength of interaction between the system and this bath. This method is also referred to as the Spin-Boson Model used for treating a spin two-level system interacting with a quantum harmonic bath. Following our earlier discussion of the DHO model, the generalization of the EG Hamiltonian to the multimode case is H = ω + H + H B (7.93) H B = ω ( p + q ) (7.94) H = ω d q + λ (7.95) λ = ω d (7.96) Note that the time-dependence to H results from the interaction with the bath: B ih Bt H (t) = e ih t H e (7.97) Also, since the harmonic modes are normal to one another, the dephasing function and lineshape function are readily obtained from ()= F (t) g( t ) = g (t) (7.98) F t For a continuum, we assume that the number of modes are so numerous as to be continuous, and that the sums in the equations above can be replaced by intrals over a continuous distribution of states characterized by a density of states W ( ω ). Also the interaction with modes of a particular frequency are equal so that we can simply average over a frequency dependent coupling constant D ω = d ω. For instance, eq. (7.98) becomes ( ) ( )

11 7- g()= t dω W (ω ) g( t,ω ) (7.99) Coupling to a continuum leads to dephasing that results from interactions of modes of varying frequency. This will be characterized by damping of the energy gap frequency correlation function C ( t ) C ( t )= dω C (ω,t) W (ω ). (7.1) Here C (ω,t) = δω ( ω,t) δω ( ω,) refers to the energy gap frequency correlation function for a single harmonic mode given in eq. (7.73). While eq. (7.1) expresses the modulation of the energy gap in the time domain, we can alternatively express the continuous distribution of coupled bath modes in the frequency domain: ( )= dω W ( ) + e i t C ω, ) C ω ω ω ( t dt. (7.11) ω = dω W ( ω ) C ( ) An intral of a single harmonic mode spectrum over a continuous density of states provides a coupling weighted density of states that reflects the action spectrum for the system-bath interaction. We evaluate this with the single harmonic mode spectrum, eq. (7.86). We see that the spectrum of the correlation function for positive frequencies is related to the product of the density of states and the frequency dependent coupling C ( ) ω = ω D ω ω n +1) (7.1) This is an action spectrum that reflects the coupling weighted density of states of the bath that contributes to the spectrum. ( )W ( )( More commonly, the frequency domain representation of the coupled density of states in eq. (7.1) is expressed as a spectral density ρ ω C (ω) ( ) πω ω D ω δ ω ω ) π (7.13) 1 = W ( ω) D ( ω) π = 1 dω W ( ) ( ) ( From eqs. (7.7) and (7.11) we obtain the lineshape function in two forms

12 7-1 g t ()= + dω 1 C ( ω) exp( iωt ) + iωt 1 π ω ωρω 1 = d ( ) coth β ω ( cosωt ) + i (sinωt ωt ). (7.14) In this expression the temperature dependence implies that in the high temperature limit, the real part of g(t) will dominate, as expected for a classical system. The reorganization energy is obtained from the first moment of the spectral density λ = dωωρ( ω ). (7.15) This is a perfectly general expression for the lineshape function in terms of an arbitrary spectral distribution describing the time-scale and amplitude of energy gap fluctuations. Given a spectral density ρ(ω), you can calculate spectroscopy and other time-dependent processes in a fluctuating environment. Now, let s evaluate the lineshape function for two special cases of the spectral density. To keep things simple, we will look specifically at the high temperature limit, kt >> ω. Here coth (βω ) β ω and we can nlect the imaginary part of the frequency correlation function and lineshape function: 1) What happens when C ( ω ) grows linearly with frequency? This represents a system that is coupled with equal strength to a continuum of modes. Setting C (ω) = Γω and evaluating + 1 C (ω) ()= dω πβ ω ω (1 cos ωt) g t =Γt. (7.16) A linearly increasing spectral density leads to a homogeneous Lorentzian lineshape with width Γ. This case corresponds to a spectral density that linearly decreases with frequency, and is also referred to as the white noise spectrum. ) Now take the case that we choose a Lorentzian spectral density centered at ω=. Specifically, let s write the imaginary part of the Lorentzian lineshape in the form

13 7- Λω ( ω)= λ. (7.17) ω +Λ C Here, in the high temperature (classical) limit kt >> Λ, nlecting the imaginary part, we find: g t λkt () Λ This expression looks familiar. If we equate exp( Λt) + Λ t 1 (7.18) λkt Δ = (7.19) and τ c = 1, (7.11) Λ we obtain the same lineshape function as the classical Gaussian-stochastic model: g t ()=Δτ c exp( t /τ c ) + t /τ c 1 (7.111) So, the interaction of an electronic transition with a harmonic bath leads to line broadening that is equivalent to random fluctuations of the energy gap.

14 7-3 Coupling to a Harmonic Bath: Correspondence to Stochastic Equation ** So, why does coupling to a quantum harmonic bath give the same results as the classical stochastic equations for fluctuations? Why does coupling to a continuum of bath states have the same physical meaning as random fluctuations? The answer is that in both cases, we really have imperfect knowledge of the particles of the bath, and observing a subset of those particles will have a random character that can alternatively be viewed as a correlation function or a spectral density for the time-scales of motion of the bath. To take this discussion further, let s again consider the electronic absorption spectrum from a classical perspective. It s quite common to think that the electronic transition of interest is coupled to a particular nuclear coordinate Q which we will call a local coordinate. This local coordinate could be an intramolecular normal vibrational mode, a intermolecular rattling in a solvent shell, a lattice vibration, or another motion that influences the electronic transition. The idea is that we take the observed electronic transition to be linearly dependent on one or more local coordinates. Therefore describing Q allows us to describe the spectroscopy. However, since this local mode has further drees of freedom that it may be interacting with, we are extracting a particular coordinate out or a continuum of other motions, the local mode will appear to feel a fluctuating environment a friction. Classically, we would describe the fluctuations in Q as Brownian motion, described by a Langevin equation. In the simplest sense this is an equation that restates Newton s equation of motion F=ma for a fluctuating force acting on a harmonic coordinate Q. ( ) + mω Q + m Q = f t) (7.11) mq t γ ( Here the terms on the left side represent a damped harmonic oscillator. The first term is ma, the second term is the restoring force of the harmonic potential F res = V Q, and the third term allows friction γ to damp the motion of the coordinate. The motion of Q is driven by f(t), a random fluctuating force. We take f(t) to follow Gaussian statistics and obey the classical fluctuation-dissipation theorem: f ( t) = (7.113) f (t) f () = m γ kt δ (t) (7.114) ** See: Nitzan, Ch. 8; Mukamel, Ch. 8; M. Cho and G.R. Fleming, Chromophore-solvent dynamics, Annu. Rev. Phys. Chem. 47 (1996) 19.

15 7-4 Here the delta function indicates that we have a Markovian system the fluctuations immediately loose all correlation on the time scale of the evolution of Q. A more general description is the Generalized Langevin Equation, which accounts for the possibility that the damping may be time-dependent and carry memory of earlier configurations t () ( ) ( ) () mq t + mω Q + m dτ γ t τ Q τ = f t. (7.115) γ (t τ ), the memory kernel, is a correlation function that describes the time-scales of the fluctuating force and obeys f ( t) f (τ ) = mkt γ (t τ ). (7.116) The GLE reduces to the Markovian limit eq. (7.11) for the case that γ (t τ ) = γδ (t τ ). The Langevin equation can be used to describe the correlation function for the time dependence of Q. For the Markovian case, eq. (7.11), ω cos Ω + γω sin Ωt γ t/ t t e (7.117) C QQ ()= m kt where the reduced frequency Ω= ω γ 4. The frequency domain expression is C ( ) QQ ω = γ kt 1. (7.118) mπ ( ω ω ) + ω γ In the case of the GLE, similar expression are obtained, although now the damping constant is replaced by γ ( ω ), which is the frequency spectrum of the correlation function for the fluctuating force on the oscillator. This coordinate correlation function is just what we need for describing lineshapes. Note the quantum mechanical energy gap correlation function was C ()= t δ H t δ H = ω d q t q (7.119) ( ) ( ) () ( ) We can get obtain exactly the same behavior as the classical GLE by solving the quantum mechanical problem by coupling to a bath of N harmonic oscillators, specified by coordinates q. N H nuc = ω ( p + q ) (7.1) =1 With this Hamiltonian, we can construct N harmonic coordinates any way we like with the appropriate unitary transformation. Specifically, we want to transform to a frame of reference

16 7-5 that includes our local mode Q and N 1 other linearly coupled normal modes, X i. Given the transformation: Q X 1 Ux = X (7.11) X n 1 we can write N 1 H = ω p + Q + ω p + X nuc ( ) ( ) + Q c X (7.1) =1 Here we have expressed the Hamiltonian as a primary local mode Q linearly coupled to the remaining drees of freedom with a strength c. In the following section, we describe how the correlation function for the coordinate Q in a Hamiltonian of this form is the same as the classical GLE, and reflects the fluctuating force acting on Q. Electronic transition, ω Primary coordinate, Q Bath of H.O.s, X Therefore, a harmonic bath can be used to construct the behavior corresponding to random fluctuations. The important thing to remember when using a harmonic bath is that it is an abstract entity and does not have a clear physical interpretation in and of itself. If the spectral density has a peak at a frequency that corresponds to a known vibration of the molecule, it is reasonable to assume that the electronic transition is coupled to this motion. On the other hand if the spectral density is broad and featureless, as is common for low frequency intermolecular motions in condensed phases, then it is difficult to ascribe a clear microscopic origin to the motion. It is challenging to evaluate and understand both the frequency dependent density of states and the frequency dependent coupling, making is that much more challenging to assign the spectral density. Straties that are meant to decompose and assign these effects remain an active area of research.

17 7-6 The Brownian Oscillator Now we do back to our energy gap Hamiltonian and express it in a form that describes the energy gap dependence on one primary vibration which is linearly coupled to the remaining modes of a quantum bath. This formulation is known as the Brownian oscillator model. We bin by writing H = H + H + H S B SB (7.13) where the system Hamiltonian is the full Hamiltonian for a displaced harmonic oscillator Hamiltonian which described the coupling of the electronic energy gap to a local mode, q. H S = E H E E + G H G G (7.14) The remaining terms describe the interaction of the primary oscillator q with the remaining coordinates of the bath x a H B + H SB = p + m ω x c q (7.15) m m ω Note here each of the bath oscillators is expressed as a displaced harmonic oscillator to the primary mode. Here c is the coupling strength. This can be expressed in a somewhat more familiar form by separating H B = ω ( p + q ) H = q c x + λ SB (7.16) The Brownian Oscillator Hamiltonian can now be used to solve for the modulation of the electronic energy gap induced by the bath. We start with C t = δ H t δ H = ξ q t q (7.17) ξ = ω d is the measure of the coupling of our primary oscillator to the electronic transition. The correlation functions for q are complicated to solve for, but can be done analytically: ( ) () ( ) () ( ) ( ) ωγ ω C ( ω ) = ξ m (ω ω ) + ω γ ( ) ω. (7.18) Here γ ( ω ) is the spectral distribution of couplings between our primary vibration and the bath:

18 7-7 ( ) m c ω ( ) γ ω = π δ ω ω (7.19) Here we see that the correlation function for the motion of the Brownian oscillator primary coordinate is equivalent to the randomly fluctuation coordinate described by the GLE, where the friction spectrum is described the magnitude of couplings between the primary and bath oscillators. For the case that we can replace γ (ω) with a constantγ, the energy gap time correlation function can be obtained as C 1 () t = ξ exp ( γ t / ) sin Ωt (7.13) m Ω where Ω= ω γ /4 is the reduced frequency. Using this model to describe the energy gap correlation function allows one to vary the parameters to interpolates smoothly between the coherent undamped limit and the overdamped Gaussian stochastic limit. Consider the following: 1) If we set γ, we recover our earlier result for C (t) and g(t) for coupling to a single undamped nuclear coordinates and leads to fine structure on the electronic spectrum ) For weak damping γ << ω, eq. (7.13) becomes C ()= t ξ exp( γ / ) sinω t. (7.131) mω 3) For strong damping γ >> ω i, Ω is imaginary and we can re-write the expression in an overdamped form C () t ξ Λ exp ( Λt) (7.13) m ω ω where Λ=. (7.133) γ This is the correlation function for the Gaussian-stochastic model. Absorption lineshapes are calculated as before, by calculating the lineshape function from the spectral density above. This model allows a bath to be constructed with all possible time scales,

19 7-8 by summing over many nuclear drees of freedom, each of which may be under- or overdamped. In the frequency domain: ωγ ω C ω = C ( ), i ( ω ) = ξ i. (7.134) i i ω ω ω γ ω. m ( i ) + ( ) i ( )

### Light and Matter. Thursday, 8/31/2006 Physics 158 Peter Beyersdorf. Document info

Light and Matter Thursday, 8/31/2006 Physics 158 Peter Beyersdorf Document info 3. 1 1 Class Outline Common materials used in optics Index of refraction absorption Classical model of light absorption Light

### Damped harmonic motion

Damped harmonic motion March 3, 016 Harmonic motion is studied in the presence of a damping force proportional to the velocity. The complex method is introduced, and the different cases of under-damping,

### 10.1 VIBRATIONAL RELAXATION *

Andrei Tokmakoff, MIT Department of Cemistry, 3//009 p. 0-0. VIRATIONAL RELAXATION * Here we want to address ow a quantum mecanical vibration undergoes irreversible energy dissipation as a result of interactions

### Optical coherence spectroscopy in solution: Determining the system-bath correlation function

Optical coherence spectroscopy in solution: Determining the system-bath correlation function Lewis D. Book a, David C. Arnett b and Norbert F. Scherer a a Department of Chemistry and The James Franck Institute,

### Brownian Motion and Langevin Equations

1 Brownian Motion and Langevin Equations 1.1 Langevin Equation and the Fluctuation- Dissipation Theorem The theory of Brownian motion is perhaps the simplest approximate way to treat the dynamics of nonequilibrium

### Time dependent perturbation theory 1 D. E. Soper 2 University of Oregon 11 May 2012

Time dependent perturbation theory D. E. Soper University of Oregon May 0 offer here some background for Chapter 5 of J. J. Sakurai, Modern Quantum Mechanics. The problem Let the hamiltonian for a system

### Work sheet / Things to know. Chapter 3

MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients

### Oscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is

Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 1-8. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring

### Simple Explanation of Fermi Arcs in Cuprate Pseudogaps: A Motional Narrowing Phenomenon

Simple Explanation of Fermi Arcs in Cuprate Pseudogaps: A Motional Narrowing Phenomenon ABSTRACT: ARPES measurements on underdoped cuprates above the superconducting transition temperature exhibit the

### 6. Molecular structure and spectroscopy I

6. Molecular structure and spectroscopy I 1 6. Molecular structure and spectroscopy I 1 molecular spectroscopy introduction 2 light-matter interaction 6.1 molecular spectroscopy introduction 2 Molecular

### 2.3 Damping, phases and all that

2.3. DAMPING, PHASES AND ALL THAT 107 2.3 Damping, phases and all that If we imagine taking our idealized mass on a spring and dunking it in water or, more dramatically, in molasses), then there will be

### Wave Phenomena Physics 15c. Lecture 11 Dispersion

Wave Phenomena Physics 15c Lecture 11 Dispersion What We Did Last Time Defined Fourier transform f (t) = F(ω)e iωt dω F(ω) = 1 2π f(t) and F(w) represent a function in time and frequency domains Analyzed

### Chapter 3. Periodic functions

Chapter 3. Periodic functions Why do lights flicker? For that matter, why do they give off light at all? They are fed by an alternating current which turns into heat because of the electrical resistance

### 22.51 Quantum Theory of Radiation Interactions

.5 Quantum Theory of Radiation Interactions Mid-Term Exam October 3, Name:.................. In this mid-term we will study the dynamics of an atomic clock, which is one of the applications of David Wineland

### 2. Determine whether the following pair of functions are linearly dependent, or linearly independent:

Topics to be covered on the exam include: Recognizing, and verifying solutions to homogeneous second-order linear differential equations, and their corresponding Initial Value Problems Recognizing and

### We have seen that the total magnetic moment or magnetization, M, of a sample of nuclear spins is the sum of the nuclear moments and is given by:

Bloch Equations We have seen that the total magnetic moment or magnetization, M, of a sample of nuclear spins is the sum of the nuclear moments and is given by: M = [] µ i i In terms of the total spin

### Vibrations: Second Order Systems with One Degree of Freedom, Free Response

Single Degree of Freedom System 1.003J/1.053J Dynamics and Control I, Spring 007 Professor Thomas Peacock 5//007 Lecture 0 Vibrations: Second Order Systems with One Degree of Freedom, Free Response Single

### Energy during a burst of deceleration

Problem 1. Energy during a burst of deceleration A particle of charge e moves at constant velocity, βc, for t < 0. During the short time interval, 0 < t < t its velocity remains in the same direction but

### Exercises Lecture 15

AM1 Mathematical Analysis 1 Oct. 011 Feb. 01 Date: January 7 Exercises Lecture 15 Harmonic Oscillators In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium

### Assignment 8. [η j, η k ] = J jk

Assignment 8 Goldstein 9.8 Prove directly that the transformation is canonical and find a generating function. Q 1 = q 1, P 1 = p 1 p Q = p, P = q 1 q We can establish that the transformation is canonical

### Lecture 1: The Equilibrium Green Function Method

Lecture 1: The Equilibrium Green Function Method Mark Jarrell April 27, 2011 Contents 1 Why Green functions? 2 2 Different types of Green functions 4 2.1 Retarded, advanced, time ordered and Matsubara

### Wave Phenomena Physics 15c

Wave Phenomena Physics 5c Lecture Fourier Analysis (H&L Sections 3. 4) (Georgi Chapter ) What We Did Last ime Studied reflection of mechanical waves Similar to reflection of electromagnetic waves Mechanical

### 4.2 Homogeneous Linear Equations

4.2 Homogeneous Linear Equations Homogeneous Linear Equations with Constant Coefficients Consider the first-order linear differential equation with constant coefficients a 0 and b. If f(t) = 0 then this

### Lecture #4: The Classical Wave Equation and Separation of Variables

5.61 Fall 013 Lecture #4 page 1 Lecture #4: The Classical Wave Equation and Separation of Variables Last time: Two-slit experiment paths to same point on screen paths differ by nλ-constructive interference

### Physics 505 Homework No. 8 Solutions S Spinor rotations. Somewhat based on a problem in Schwabl.

Physics 505 Homework No 8 s S8- Spinor rotations Somewhat based on a problem in Schwabl a) Suppose n is a unit vector We are interested in n σ Show that n σ) = I where I is the identity matrix We will

### 7.3. QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS: THE ENERGY GAP HAMILTONIAN

Andrei Tokmakoff, MIT Deparmen of Cemisry, 3/5/8 7-5 7.3. QUANTUM MECANICAL TREATMENT OF FLUCTUATIONS: TE ENERGY GAP AMILTONIAN Inroducion In describing flucuaions in a quanum mecanical sysem, we will

### Section Mass Spring Systems

Asst. Prof. Hottovy SM212-Section 3.1. Section 5.1-2 Mass Spring Systems Name: Purpose: To investigate the mass spring systems in Chapter 5. Procedure: Work on the following activity with 2-3 other students

### Coupled quantum oscillators within independent quantum reservoirs

Coupled quantum oscillators within independent quantum reservoirs Illarion Dorofeyev * Institute for Physics of Microstructures, Russian Academy of Sciences, 6395, GSP-5 Nizhny Novgorod, Russia Abstract

### Linear Second Order ODEs

Chapter 3 Linear Second Order ODEs In this chapter we study ODEs of the form (3.1) y + p(t)y + q(t)y = f(t), where p, q, and f are given functions. Since there are two derivatives, we might expect that

### Paper B2: Radiation and Matter - Basic Laser Physics

Paper B2: Radiation and Matter - Basic Laser Physics Dr Simon Hooker Michaelmas Term 2006 ii Contents 1 The Interaction of Radiation and Matter 1 1.1 Introduction.............................. 1 1.2 Radiation...............................

### Undamped Free Vibrations (Simple Harmonic Motion; SHM also called Simple Harmonic Oscillator)

Section 3. 7 Mass-Spring Systems (no damping) Key Terms/ Ideas: Hooke s Law of Springs Undamped Free Vibrations (Simple Harmonic Motion; SHM also called Simple Harmonic Oscillator) Amplitude Natural Frequency

### Forced Response - Particular Solution x p (t)

Governing Equation 1.003J/1.053J Dynamics and Control I, Spring 007 Proessor Peacoc 5/7/007 Lecture 1 Vibrations: Second Order Systems - Forced Response Governing Equation Figure 1: Cart attached to spring

### Vibrations and Waves Physics Year 1. Handout 1: Course Details

Vibrations and Waves Jan-Feb 2011 Handout 1: Course Details Office Hours Vibrations and Waves Physics Year 1 Handout 1: Course Details Dr Carl Paterson (Blackett 621, carl.paterson@imperial.ac.uk Office

### obtained in Chapter 14 to this case requires that the E1 approximation

Chapter 15 The tools of time-dependent perturbation theory can be applied to transitions among electronic, vibrational, and rotational states of molecules. I. Rotational Transitions Within the approximation

### Fourier series. Complex Fourier series. Positive and negative frequencies. Fourier series demonstration. Notes. Notes. Notes.

Fourier series Fourier series of a periodic function f (t) with period T and corresponding angular frequency ω /T : f (t) a 0 + (a n cos(nωt) + b n sin(nωt)), n1 Fourier series is a linear sum of cosine

### PHY217: Vibrations and Waves

Assessed Problem set 1 Issued: 5 November 01 PHY17: Vibrations and Waves Deadline for submission: 5 pm Thursday 15th November, to the V&W pigeon hole in the Physics reception on the 1st floor of the GO

### Chapter 11: Dielectric Properties of Materials

Chapter 11: Dielectric Properties of Materials Lindhardt January 30, 2017 Contents 1 Classical Dielectric Response of Materials 2 1.1 Conditions on ɛ............................. 4 1.2 Kramer s Kronig

### arxiv:cond-mat/ v3 [cond-mat.stat-mech] 21 May 2007

A generalized Chudley-Elliott vibration-jump model in activated atom surface diffusion arxiv:cond-mat/0609249v3 [cond-mat.stat-mech] 21 May 2007 R. Martínez-Casado a,b, J. L. Vega b, A. S. Sanz b, and

### Solutions for homework 5

1 Section 4.3 Solutions for homework 5 17. The following equation has repeated, real, characteristic roots. Find the general solution. y 4y + 4y = 0. The characteristic equation is λ 4λ + 4 = 0 which has

### NMR Dynamics and Relaxation

NMR Dynamics and Relaxation Günter Hempel MLU Halle, Institut für Physik, FG Festkörper-NMR 1 Introduction: Relaxation Two basic magnetic relaxation processes: Longitudinal relaxation: T 1 Relaxation Return

### 5.62 Physical Chemistry II Spring 2008

MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.62 Spring 2008 Lecture

### Preface. Preface to the Third Edition. Preface to the Second Edition. Preface to the First Edition. 1 Introduction 1

xi Contents Preface Preface to the Third Edition Preface to the Second Edition Preface to the First Edition v vii viii ix 1 Introduction 1 I GENERAL THEORY OF OPEN QUANTUM SYSTEMS 5 Diverse limited approaches:

### Thermodynamical cost of accuracy and stability of information processing

Thermodynamical cost of accuracy and stability of information processing Robert Alicki Instytut Fizyki Teoretycznej i Astrofizyki Uniwersytet Gdański, Poland e-mail: fizra@univ.gda.pl Fields Institute,

### Vibrations Qualifying Exam Study Material

Vibrations Qualifying Exam Study Material The candidate is expected to have a thorough understanding of engineering vibrations topics. These topics are listed below for clarification. Not all instructors

### Quantum Reservoir Engineering

Departments of Physics and Applied Physics, Yale University Quantum Reservoir Engineering Towards Quantum Simulators with Superconducting Qubits SMG Claudia De Grandi (Yale University) Siddiqi Group (Berkeley)

### Second Order Linear ODEs, Part II

Craig J. Sutton craig.j.sutton@dartmouth.edu Department of Mathematics Dartmouth College Math 23 Differential Equations Winter 2013 Outline Non-homogeneous Linear Equations 1 Non-homogeneous Linear Equations

### Unforced Mechanical Vibrations

Unforced Mechanical Vibrations Today we begin to consider applications of second order ordinary differential equations. 1. Spring-Mass Systems 2. Unforced Systems: Damped Motion 1 Spring-Mass Systems We

### Lab 11 - Free, Damped, and Forced Oscillations

Lab 11 Free, Damped, and Forced Oscillations L11-1 Name Date Partners Lab 11 - Free, Damped, and Forced Oscillations OBJECTIVES To understand the free oscillations of a mass and spring. To understand how

### Non-Markovian stochastic Schrödinger equation

Journal of Chemical Physics 111 (1999) 5676-569 Non-Markovian stochastic Schrödinger equation P. Gaspard Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems, Faculté des Sciences,

### Laplace Transform of Spherical Bessel Functions

Laplace Transform of Spherical Bessel Functions arxiv:math-ph/01000v 18 Jan 00 A. Ludu Department of Chemistry and Physics, Northwestern State University, Natchitoches, LA 71497 R. F. O Connell Department

### 4. Sinusoidal solutions

16 4. Sinusoidal solutions Many things in nature are periodic, even sinusoidal. We will begin by reviewing terms surrounding periodic functions. If an LTI system is fed a periodic input signal, we have

### Consider a system of n ODEs. parameter t, periodic with period T, Let Φ t to be the fundamental matrix of this system, satisfying the following:

Consider a system of n ODEs d ψ t = A t ψ t, dt ψ: R M n 1 C where A: R M n n C is a continuous, (n n) matrix-valued function of real parameter t, periodic with period T, t R k Z A t + kt = A t. Let Φ

### Chemistry 431. NC State University. Lecture 17. Vibrational Spectroscopy

Chemistry 43 Lecture 7 Vibrational Spectroscopy NC State University The Dipole Moment Expansion The permanent dipole moment of a molecule oscillates about an equilibrium value as the molecule vibrates.

### A new class of sum rules for products of Bessel functions. G. Bevilacqua, 1, a) V. Biancalana, 1 Y. Dancheva, 1 L. Moi, 1 2, b) and T.

A new class of sum rules for products of Bessel functions G. Bevilacqua,, a V. Biancalana, Y. Dancheva, L. Moi,, b and T. Mansour CNISM and Dipartimento di Fisica, Università di Siena, Via Roma 56, 5300

### VIII.B Equilibrium Dynamics of a Field

VIII.B Equilibrium Dynamics of a Field The next step is to generalize the Langevin formalism to a collection of degrees of freedom, most conveniently described by a continuous field. Let us consider the

### Oscillatory Motion SHM

Chapter 15 Oscillatory Motion SHM Dr. Armen Kocharian Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A

### 12.1. Exponential shift. The calculation (10.1)

62 12. Resonance and the exponential shift law 12.1. Exponential shift. The calculation (10.1) (1) p(d)e = p(r)e extends to a formula for the effect of the operator p(d) on a product of the form e u, where

### Outline of parts 1 and 2

to Harmonic Loading http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March, 6 Outline of parts and of an Oscillator

### Lecture #6 Chemical Exchange

Lecture #6 Chemical Exchange Topics Introduction Effects on longitudinal magnetization Effects on transverse magnetization Examples Handouts and Reading assignments Kowalewski, Chapter 13 Levitt, sections

### Optical Lattices. Chapter Polarization

Chapter Optical Lattices Abstract In this chapter we give details of the atomic physics that underlies the Bose- Hubbard model used to describe ultracold atoms in optical lattices. We show how the AC-Stark

### Driven Harmonic Oscillator

Driven Harmonic Oscillator Physics 6B Lab Experiment 1 APPARATUS Computer and interface Mechanical vibrator and spring holder Stands, etc. to hold vibrator Motion sensor C-209 spring Weight holder and

### KEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY OSCILLATIONS AND WAVES PRACTICE EXAM

KEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY-10012 OSCILLATIONS AND WAVES PRACTICE EXAM Candidates should attempt ALL of PARTS A and B, and TWO questions from PART C. PARTS A and B should be answered

### A.1 Introduction & Rant

Scott Hughes 1 March 004 A.1 Introduction & Rant Massachusetts Institute of Technology Department of Physics 8.0 Spring 004 Supplementary notes: Complex numbers Complex numbers are numbers that consist

### Math 240: Spring-mass Systems

Math 240: Spring-mass Systems Ryan Blair University of Pennsylvania Tuesday March 1, 2011 Ryan Blair (U Penn) Math 240: Spring-mass Systems Tuesday March 1, 2011 1 / 15 Outline 1 Review 2 Today s Goals

### Andrei V. Pisliakov, Tomáš Manal, and Graham R. Fleming

Two-dimensional optical three-pulse photon echo spectroscopy. II. Signatures of coherent electronic motion and exciton population transfer in dimer two-dimensional spectra Andrei V. Pisliakov, Tomáš Manal,

### Spectral Resolution. Spectral resolution is a measure of the ability to separate nearby features in wavelength space.

Spectral Resolution Spectral resolution is a measure of the ability to separate nearby features in wavelength space. R, minimum wavelength separation of two resolved features. Delta lambda often set to

### The Einstein A and B Coefficients

The Einstein A and B Coefficients Austen Groener Department of Physics - Drexel University, Philadelphia, Pennsylvania 19104, USA Quantum Mechanics III December 10, 010 Abstract In this paper, the Einstein

### Lecture 4 Fiber Optical Communication Lecture 4, Slide 1

ecture 4 Dispersion in single-mode fibers Material dispersion Waveguide dispersion imitations from dispersion Propagation equations Gaussian pulse broadening Bit-rate limitations Fiber losses Fiber Optical

### Atomic cross sections

Chapter 12 Atomic cross sections The probability that an absorber (atom of a given species in a given excitation state and ionziation level) will interact with an incident photon of wavelength λ is quantified

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.044 Statistical Physics I Spring Term 2013 Notes on the Microcanonical Ensemble The object of this endeavor is to impose a simple probability

### Generalized Nyquist theorem

Non-Equilibrium Statistical Physics Prof. Dr. Sergey Denisov WS 215/16 Generalized Nyquist theorem by Stefan Gorol & Dominikus Zielke Agenda Introduction to the Generalized Nyquist theorem Derivation of

### Likewise, any operator, including the most generic Hamiltonian, can be written in this basis as H11 H

Finite Dimensional systems/ilbert space Finite dimensional systems form an important sub-class of degrees of freedom in the physical world To begin with, they describe angular momenta with fixed modulus

### MA 266 Review Topics - Exam # 2 (updated)

MA 66 Reiew Topics - Exam # updated Spring First Order Differential Equations Separable, st Order Linear, Homogeneous, Exact Second Order Linear Homogeneous with Equations Constant Coefficients The differential

### Third-order nonlinear optical response of energy transfer systems

hird-order nonlinear optical response of energy transfer systems Mino Yang and Graham R. Fleming Citation: he Journal of Chemical Physics 111, 27 (1999); doi: 1.163/1.479359 View online: http://dx.doi.org/1.163/1.479359

### Principles of Molecular Spectroscopy

Principles of Molecular Spectroscopy What variables do we need to characterize a molecule? Nuclear and electronic configurations: What is the structure of the molecule? What are the bond lengths? How strong

### 11. Recursion Method: Concepts

University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 16 11. Recursion Method: Concepts Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative

### Chapter 1. Harmonic Oscillator. 1.1 Energy Analysis

Chapter 1 Harmonic Oscillator Figure 1.1 illustrates the prototypical harmonic oscillator, the mass-spring system. A mass is attached to one end of a spring. The other end of the spring is attached to

### Coherent states, beam splitters and photons

Coherent states, beam splitters and photons S.J. van Enk 1. Each mode of the electromagnetic (radiation) field with frequency ω is described mathematically by a 1D harmonic oscillator with frequency ω.

### 9 Atomic Coherence in Three-Level Atoms

9 Atomic Coherence in Three-Level Atoms 9.1 Coherent trapping - dark states In multi-level systems coherent superpositions between different states (atomic coherence) may lead to dramatic changes of light

### NONLINEAR OPTICS. Ch. 1 INTRODUCTION TO NONLINEAR OPTICS

NONLINEAR OPTICS Ch. 1 INTRODUCTION TO NONLINEAR OPTICS Nonlinear regime - Order of magnitude Origin of the nonlinearities - Induced Dipole and Polarization - Description of the classical anharmonic oscillator

### Electron spins in nonmagnetic semiconductors

Electron spins in nonmagnetic semiconductors Yuichiro K. Kato Institute of Engineering Innovation, The University of Tokyo Physics of non-interacting spins Optical spin injection and detection Spin manipulation

### ( ) /, so that we can ignore all

Physics 531: Atomic Physics Problem Set #5 Due Wednesday, November 2, 2011 Problem 1: The ac-stark effect Suppose an atom is perturbed by a monochromatic electric field oscillating at frequency ω L E(t)

### PH575 Spring Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5

PH575 Spring 2009 Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5 PH575 Spring 2009 POP QUIZ Phonons are: A. Fermions B. Bosons C. Lattice vibrations D. Light/matter interactions PH575 Spring 2009 POP QUIZ

### Lecture #8: Quantum Mechanical Harmonic Oscillator

5.61 Fall, 013 Lecture #8 Page 1 Last time Lecture #8: Quantum Mechanical Harmonic Oscillator Classical Mechanical Harmonic Oscillator * V(x) = 1 kx (leading term in power series expansion of most V(x)

### Interaction of Molecules with Radiation

3 Interaction of Molecules with Radiation Atoms and molecules can exist in many states that are different with respect to the electron configuration, angular momentum, parity, and energy. Transitions between

### Fourier Series & The Fourier Transform

Fourier Series & The Fourier Transform What is the Fourier Transform? Anharmonic Waves Fourier Cosine Series for even functions Fourier Sine Series for odd functions The continuous limit: the Fourier transform

### Oscillations Simple Harmonic Motion

Oscillations Simple Harmonic Motion Lana Sheridan De Anza College Dec 1, 2017 Overview oscillations simple harmonic motion (SHM) spring systems energy in SHM pendula damped oscillations Oscillations and

### Chapter 2: Linear Constant Coefficient Higher Order Equations

Chapter 2: Linear Constant Coefficient Higher Order Equations The wave equation is a linear partial differential equation, which means that sums of solutions are still solutions, just as for linear ordinary

### arxiv:hep-th/ v1 1 Sep 2003

Classical brackets for dissipative systems Giuseppe Bimonte, Giampiero Esposito, Giuseppe Marmo and Cosimo Stornaiolo arxiv:hep-th/0309020v1 1 Sep 2003 Dipartimento di Scienze Fisiche, Università di Napoli,

### Computational Physics (6810): Session 8

Computational Physics (6810): Session 8 Dick Furnstahl Nuclear Theory Group OSU Physics Department February 24, 2014 Differential equation solving Session 7 Preview Session 8 Stuff Solving differential

### Quantum Chemistry. NC State University. Lecture 5. The electronic structure of molecules Absorption spectroscopy Fluorescence spectroscopy

Quantum Chemistry Lecture 5 The electronic structure of molecules Absorption spectroscopy Fluorescence spectroscopy NC State University 3.5 Selective absorption and emission by atmospheric gases (source:

### Math Assignment 5

Math 2280 - Assignment 5 Dylan Zwick Fall 2013 Section 3.4-1, 5, 18, 21 Section 3.5-1, 11, 23, 28, 35, 47, 56 Section 3.6-1, 2, 9, 17, 24 1 Section 3.4 - Mechanical Vibrations 3.4.1 - Determine the period

### EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS

EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS CONTENTS 3 Be able to understand how to manipulate trigonometric expressions and apply

### ODEs. September 7, Consider the following system of two coupled first-order ordinary differential equations (ODEs): A =

ODEs September 7, 2017 In [1]: using Interact, PyPlot 1 Exponential growth and decay Consider the following system of two coupled first-order ordinary differential equations (ODEs): d x/dt = A x for the

### BASIC WAVE CONCEPTS. Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, Giancoli?

1 BASIC WAVE CONCEPTS Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, 9.1.2 Giancoli? REVIEW SINGLE OSCILLATOR: The oscillation functions you re used to describe how one quantity (position, charge, electric field,

### MOLECULAR SPECTROSCOPY

MOLECULAR SPECTROSCOPY First Edition Jeanne L. McHale University of Idaho PRENTICE HALL, Upper Saddle River, New Jersey 07458 CONTENTS PREFACE xiii 1 INTRODUCTION AND REVIEW 1 1.1 Historical Perspective