5.72 Problem set #4 (spring 2012)
Cao
I. Harmonic Oscillator
1) For a harmonic oscillator, show that 𝐶(𝑡) = ⟨𝑥(𝑡)𝑥(0)⟩ satisfies 𝐶̈ + 𝜔02 𝐶 = 0.
2) Solve for 𝐶(𝑡) and its Fourier transform 𝐶̃ (𝜔) = ∫ 𝐶(𝑡)𝑒 𝑖𝜔𝑡 𝑑𝑡.
3) The forced oscillator obeys the equation of motion
𝑚𝑥̈ + 𝑚𝜔02 𝑥 = 𝑓(𝜔)𝑒 −𝑖𝜔𝑡 .
Derive the expression for 𝜒(𝜔) from the above equation.
4) Write the formula for 𝐾(𝑡).
𝛽𝜔
̇
5) Verify 𝐾(𝑡) = −𝛽𝐶(𝑡)
[i.e. 𝜒 ′′ =
𝐶̃ (𝜔)].
2
6) *Verify the Kramers-Kronig relations.
II. The relaxation of rotational motions can be described by the rotational diffusion
equation
𝜕𝑝
𝜕𝑡
= 𝐷𝑅 ∇2 𝑝, where ∇2 is the angular part of the Laplacian operator
1 𝜕
𝜕
1 𝜕2
∇ =
sin 𝜃
+ 2
.
sin 𝜃 𝜕𝜃
𝜕𝜃
sin 𝜃 𝜕𝜙 2
1) Show that the average orientation 𝑢(𝑡) = ⟨cos 𝜃(𝑡)⟩ satisfies
𝑢̇ (𝑡) = −2𝐷𝑅 𝑢(𝑡).
2) Show that the orientational correlation function is given by
1
𝐶(𝑡) = ⟨cos 𝜃(𝑡) cos 𝜃(0)⟩ = 𝑒 −2𝐷𝑅𝑡 .
3
3) Write down 𝐾(𝑡), 𝜒′, and 𝜒′′.
4) Calculate the response to a monochromatic force 𝐹(𝑡) = 𝐹0 cos 𝜔0 𝑡, which
couples to the system according to 𝐻 ′ = −𝐹(𝑡) cos 𝜃(𝑡).
5) Calculate the average absorption rate.
(Ref: McQuarrie, p. 398, prob. 17-19)
2
III. Repeat the steps in Problem I for a damped oscillator described by
𝑚𝑥̈ + 𝑚𝛾𝑥̇ + 𝑚𝜔02 𝑥 = 𝑓(𝑡) + 𝐹(𝑡),
where 𝑓(𝑡) is the random force and 𝐹(𝑡) is the external driving force.
1) Show the position correlation function satisfies
𝑚𝐶̈ + 𝑚𝛾𝐶̇ + 𝑚𝜔02 𝐶 = 0
with the initial condition 𝐶(0) = (𝛽𝑚𝜔02 )−1 .
2) Derive explicit expressions for 𝐶(𝑡) and its Fourier transform 𝐶̃ (𝜔).
3) Show that under the external force the average position 𝑥̅ satisfies
𝐹(𝑡)
𝑥̈̅ + 𝛾𝑥̇̅ + 𝜔02 𝑥̅ =
.
𝑚
4) Solve 𝜒(𝜔) from the above equation.
5) Derive 𝐾(𝑡) using the expression for 𝜒(𝜔) found above.
1
6) Verify 𝐾(𝑡) = −𝛽𝐶̇ (𝑡).
IV. *Derivation of quantum response theory (Ref: McQuarrie, Berne, Kubo, Reichl).
1
1) Define the quantum Liouville operator as ℒ𝐴 = [𝐻, 𝐴]. Show that 𝐴̇(𝑡) =
ℏ
𝑖ℒ(𝑡)𝐴(𝑡) and 𝜌̇ = −𝑖ℒ𝜌, where 𝜌 is the density matrix and 𝐴(𝑡) is a Heisenberg
operator.
2) Given 𝐻 = 𝐻0 − 𝐴𝐹(𝑡), use perturbation theory to show
where 𝐾(𝑡 − 𝑡
′)
𝑖
⟨𝐴(𝑡)⟩ =
′
𝑡
−∞
= ⟨[𝐴(𝑡), 𝐴(𝑡 )]⟩.
ℏ
𝐾(𝑡 − 𝑡 ′ )𝐹(𝑡 ′ )𝑑𝑡′ ,
3) Show that the classical limit of the quantum commutator is:
where { , } is the Poisson bracket.
4) Show that the classical limit of 𝐾(𝑡) is −𝛽𝐶̇ (𝑡).
1
𝑖ℏ
[𝐴, 𝐵] = {𝐴, 𝐵},
V. Spectroscopic measurements are expressed as polarization responses. We calculate
the response function of a linear harmonic oscillator as an example.
1) The linear response function is defined as
𝑖
𝑖
𝑅(𝑡) = ⟨0|[𝛼(𝑡), 𝛼(0)]|0⟩ = [⟨0|𝛼(𝑡)𝛼(0)|0⟩ − ⟨0|𝛼(0)𝛼(𝑡)|0⟩],
ℏ
ℏ
where the transition dipole is assumed to be linear in coordinate 𝛼(𝑡) = 𝛼0 𝑥(𝑡).
Show 𝑅(𝑡) =
sin 𝜔𝑡
𝑚𝜔
Useful expressions:
𝑥(𝑡) =
ℏ
2𝑚𝜔
𝛼02 .
𝑎𝑒 −𝑖𝜔𝑡 + 𝑎† 𝑒 𝑖𝜔𝑡 , 0 𝑎𝑎† 0 = 1, 0 𝑎† 𝑎 0 = 0.
2) Show that the same result can be obtained classically by replacing the quantum
commutation with the Poisson bracket,
1
𝜕𝛼(𝑡) 𝜕𝛼(0) 𝜕𝛼(𝑡) 𝜕𝛼(0)
[𝛼(𝑡), 𝛼(0)] → {𝛼(𝑡), 𝛼(0)} =
−
𝑖ℏ
𝜕𝑥(0) 𝜕𝑝(0) 𝜕𝑝(0) 𝜕𝑥(0)
Useful expression: 𝑥(𝑡) = 𝑥(0) cos 𝜔𝑡 +
𝑝(0)
𝑚𝜔
sin 𝜔𝑡
3) The non-linear response function is defined as
𝑖 2
𝑅(𝑡1 , 𝑡2 ) =
0 [𝛼(𝑡1 + 𝑡2 ), 𝛼(𝑡1 )], 𝛼(0) 0 ,
ℏ
If 𝛼 = 𝛼0 𝑥, show 𝑅(𝑡1 , 𝑡2 ) = 0 from the balance of 𝑎 and 𝑎† operators.
4) *To obtain non-vanishing non-linear response for the harmonic oscillator, we
introduce a non-linear coordinate dependence,
𝛼 = 𝛼0 𝑥 + 𝛼 ′ 𝑥 2 + ⋯
Show that the leading order term in 𝛼′ is proportional to 𝛼02 𝛼′.
2
MIT OpenCourseWare
http://ocw.mit.edu
5.72 Statistical Mechanics
Spring 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.