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Massachusetts Institute of Technology
Physics Department
Physics 8.321 Fall 2006
Quantum Theory I October 16, 2006
Assignment 7
Due October 27, 2006
Announcements
- The 8.321 midterm exam will take place in class on October 31 (Hallowe’en). It will be an hour and a half exam.
- The present expectation is that the midterm will cover all the material in the first two sections of the course: 1) Kinematics and 2) Dynamics. As of Monday, October 16, we have yet to cover the following sections in “dynamics”: a) finish up coherent states; b) energy-time uncertainty relation; c) the propagator; d) the path integral formulation of quantum mechanics. I hope to finish all of these by the end of lecture on Tuesday, Oct 24, one week before the midterm.
- Here’s some information about the schedule for the rest of the term: As previously announced, there will be additional makeup lectures on Nov. 3 and 17. There will be no lecture on Tuesday, Nov. 21 (the Tuesday before Thanksgiving). There will be no lecture on Thursday, Nov. 30. Problem set 11 will be distributed on Nov. 13 and due on Dec. 1. The last problem set, # 12, will be distributed on Nov. 27, and due Dec. 8.
Reading topics for this period
- Energy-time uncertainty relation; the propagator; the path integral approach to quan- tum mechanics.
Reading Recommendations 7
- The propagator in quantum mechanics is discussed in Sakurai §2.5 (which continues on to derive the path integral — next week’s subject in 8.321); similar material is covered in Gottfried & Yan §2.6.
- Propagators, Green’s functions, and path integrals, Sakurai §2.5, G & Y §2.6 (a–c); 2.7.
- Shankar, §8, takes a different approach to path integrals from the one presented in lecture.
Problem Set 7
Topics covered in the problems
- Forced harmonic oscillator
- Exponential decay of a state with a Lorentzian energy distribution.
- The propagator in one dimension.
- The density of states and the propagator.
- Forced Quantum Harmonic Oscillator Consider a quantum oscillator subject to a time dependent external force
H(t) =
P 2 +
ω^2 X^2 + f (t)X
where f (t) vanishes for t ≤ 0. Assume that at t = 0, the oscillator is in its ground state. Show that the state at a later time is a normalized coherent state N (t)
φ(t)
and find N (t) and φ(t) in terms of integrals of f (t). Hint: Assume a solution of the form |ψ, t〉 = N (t)ea†^ φ(t)| 0 〉 and derive differential equations for φ(t) and N (t). The eigenmodes of the electromagnetic field are “driven” by the coupling to time dependent currents and charge densities. Thus the radiation that is produced is naturally described in terms of coherent states.
- Exponential Decay When we study time dependent perturbation theory, we will see that unstable states decay with a exponential time dependence, P (t) = e−t/τ^ (although this result is not exact, especially at very short and very long times, it is extremely accurate for most purposes). The purpose of this problem is to find the energy distribution associated with this kind of time dependence and the relation between the uncertainty in energy and the lifetime. Suppose a state is given as a superposition of energy eigenstates at t = 0,
|Φ 0 〉 =
−∞
dEg(E)|E〉
and suppose the function g(E) is a simple pole at a complex energy,
g(E) = N
E − E 0 + iΓ/ 2 , where Γ > 0
and show (E − H) K(x, x′; E) = iℏδ(x − x′) (2) and show that K(x, x′; E) is analytic in the upper half complex plane. [K(x, x′; E) is the Green’s function for the time independent Schr¨odinger equation.] What would happen to the time dependence of K(x, t; x′, t′) if, instead, K was an- alytic in the lower half plane and had singularities in the u.h.p.? If you use completeness, state what you mean by it. (c) Show that K(x, x′, E) = iℏ
dE′^
ψ(x, E′)ψ∗(x′, E′) E − E′^ + i 0 +^
satisfies the equations of part (b). The strange notation, Σ
, means a sum over countable states and an integral over the continuum. The notation i 0 +^ means an arbitrarily small positive imaginary number that moves the poles in E into the l.h.p. (d) Another way to construct the Green’s function is by matching functions of x to the left and right of x′^ so they satisfy a jump condition at x = x′^ required by the δ-function in eq. (2). First, suppose V (x) = 0. Let ψ±(k, x) = e±ikx^ where k =
2 mE/ℏ^2 is defined to be positive. Show that the following form for K 0 is an acceptable Green’s function for a free particle. K 0 (+) (x, x′; E) = N
θ(x − x′)ψ+(k, x)ψ∗ +(k, x′) + θ(x′^ − x)ψ−(k, x)ψ∗−(k, x′)
To do this you will need to verify a) that K 0 (+) satisfies the free Schr¨odinger equation for x 6 = x′; and b) that it satisfies the appropriate jump condition (that follows from eq. (2)) at x = x′. This will determine the coefficient N. Is the resulting Green’s function analytic in the upper half plane?
The superscript (+)^ on K(+)^ signifies that this Green’s function contributes outgoing waves as x → ±∞ to the propagator. That is, the fourier component in K(x, t; x′, t′) with wave number k will go like eikx−iEt/ℏ^ as x → ∞ and as e−ikx−iEt/ℏ^ as x → −∞. Other choices are possible. The Green’s function can always be changed by adding a solution to the homogeneous Schr¨odinger equation (eq. (2) without the δ-function on the right hand side).
(e) Now return to the case where V (x) 6 = 0. Suppose you have been able to construct the outgoing wave Green’s function K(+)(x, x′; E) in this case. Show that the following wavefunction satisfies the Schr¨odinger equation with an incoming plane wave (from the left).
ψ(+)(x) = eikx^ +
iℏ
−∞
dx′K(+)(x, x′, E)V (x′)eikx
′ (5)
The transmission and reflection coefficients T (k) and R(k) for scattering in one dimension are defined by the limit as x → ±∞ of ψ(+)(x), lim x→−∞ ψ(+)(x) = eikx^ + R(k)e−ikx
lim x→∞ ψ(+)(x) = T (k)eikx^ (6)
Write expressions for T (k) and R(k) in terms of K(+)^ and V (x).
- The Density of States in Three Dimensions
The density of states of a quantum system is defined as the sum over all energies of a δ-function at each energy,
ρ(E) =
dN dE
dE′δ(E′^ − E) (7)
Obviously ρ(E) is a very singular function, but it is also very useful. [In scattering and decay we have to integrate over the density of states available to the reaction or decay products. In statistical mechanics we have to integrate the Boltzmann factor over the density of states to obtain the partition function, and so on.] Introductory courses in quantum mechanics usually “derive” an expression for the density of states of photons in a cavity (when studying black-body radiation),
dN =
d^3 xd^3 p h^3
d^3 xd^3 k (2π)^3
where, of course, p = ℏk. This has a lovely interpretation: one state per volume h in phase space. The object of this exercise is to derive this result from eq. (7).
(a) Use the definition of the Green’s function, K(~x, ~x′, E), and eq. (7) to show
dN dk
ℏk mπ
Im
d^3 x iK(~x, ~x, E) (9)
[Use the spectral representation for K (eq. (3)).] This enables us to define a density of states local in coordinate space,
dN dkd^3 x
ℏk mπ
Im [iK(~x, ~x, E)]
(b) Now consider the case V (~x) = 0. The free Green’s function, K 0 (~x, ~x′, E) has a particularly simple form in three dimensions,
K 0 (~x, ~x′, E) = N
eikr 4 πr
where r = |~x−~x′|. Show that this satisfies the defining equation, eq. (1), and find the constant N. [It will help to remember the equation obeyed by the Coulomb potential, −∇2 1 r = 4πδ^3 (~r).] (c) Finally, derive eq. (8) by substituting this into eq. (10) keeping |~x − ~x′| 6 = 0 until you study the expression carefully.
