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Massachusetts Institute of Technology
Physics Department
Physics 8.321 Fall 2006
Quantum Theory I October 2, 2006
Assignment 5
Due October 13, 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.
Reading topics for this period
- Classical mechanics and canonical quantization; Schr¨odinger and Heisenberg “pic- tures” of time evolution; two state systems; simple harmonic oscillator.
Reading Recommendations 5
- 8.321 lecture notes on time evolution in quantum mechanics (posted on the website), classical Hamiltonian mechanics, and canonical quantization.
- Sakurai, §2.1 and 2.2 discussed the basics of time evolution including “pictures”.
- Review of classical mechanics (in addition to 8.321 posted lecture notes): Shankar, §, especially §2.5-2.7.
- The basics of motion in a magnetic field are presented in Gottfried & Yan, §4.3, which has been scanned and put on the 8.321 website.
- Two state systems are presented in Gottfried & Yan, §4.1, which has been scanned and put on the 8.321 website.
- The harmonic oscillator is discussed in almost every textbook. Sakurai §2.7; Shankar §7; and Gottfried & Yan §4.2.
Problem Set 5
Topics covered in the problems
- Motion of a charged particle in a magnetic field, and the importance of the vector potential in quantum mechanics.
- Motion in the Schr¨odinger and Heisenberg pictures.
- Time dependence of the density matrix.
- Canonical Quantization in the Presence of Static Magnetic and Electric Fields This is an important subject that we will return to from time to time in 8.321 and 8.322. It also illustrates the power as well as the shortcomings of the canonical quantization method. You may have studied the Hamiltonian formulation of this motion in classical mechanics. In that case the first few sections of the problem are review. The classical equation of motion for a particle in constant electric and magnetic fields is the Lorentz force law, m d^2 ~x dt^2
= F~ = e E~ + e c
~x˙ × B~
(using Gaussian units). Remember that static fields can be described by φ and A~, the electrostatic and magnetic vector potentials,
E^ ~ = −∇~φ and B~ = ∇ ×~ A~
(a) Consider the Lagrangian
L =
m~ x˙ 2 + e c
~x˙ · A~(~x) − eφ(~x) (1)
What is the canonical momentum, ~p = ∂L/∂ ~x˙? Note that it is not m ~x˙. Show that the Euler Lagrange equations, d~p/dt = ∂L/∂~x, give the Lorentz force law. You will have to remember that, although both A~ and φ have no explicit time dependence, they depend implicitly on time via the argument ~x(t). Thus (^) dtd A~ = ( ~x˙ · ∇~) A~. [You’ll also need some vector calculus identities, or the help of a text like Jackson’s.] (b) Find the Hamiltonian, H = ~x˙ · ~p − L. Combining the results of parts (a) and (b), it appears that the energy can be written as E = 12 m ~x˙^2 + eφ (an elementary result since the magnetic field does no work). What is the conceptual difference between H and E in classical mechanics? (c) Quantize this system canonically: [xj , pk] = iℏδjk, etc.. Then write the Schr¨odinger equation in coordinate space. (d) Show that A~ = − 12 ~x × B~ 0 is a vector potential corresponding to a constant field B^ ~ 0. Substitute this into the Schr¨odinger equation (with φ = 0) to obtain ( −
ℏ^2
2 m ∇~ 2 − e 2 mc L~ · B~ 0 + e
2 8 mc^2 ρ^2 B 02
ψ(~x) = Eψ(~x) (2)
Here L~ = ~x×p~ and ρ = Bˆ 0 ×~x is the radial coordinate in the plane perpendicular to B~ 0.
- Time Dependence of the Density Matrix
In lecture we discussed only the time dependence of pure states. Mixed states evolve in time too. We defined an arbitrary density matrix by ρ =
k pk|ψk〉〈ψk^ |. (a) The time dependence of states in the Schr¨odinger picture induces a natural definition of the time dependent density matrix in the Schr¨odinger picture, ρS (t). What is it? Write ρS (t) in terms of ρS (0) and the time evolution operator U (t, 0). Compare this result with the relation between a Heisenberg picture operator, QH (t) and the Schr¨odinger picture operator, QS. (b) How does the expectation value of an observable, Q, in the mixed state evolve with time? Remember at a fixed time we found 〈Q〉 = Tr[Qρ]. What is the analagous equation at a time t, in terms of ρS (t), or in terms of QH (t)? (c) What is the Schr¨odinger equation for ρS (t)? (d) Prove that a pure state cannot evolve into a mixed state or vica versa.
- Quantum Consequences of a Magnetic Field (Sakurai, §2, Problem 25)
Consider an electron confined to the interior of a hollow cylindrical shell whose axis coincides with the z-axis. The wave function is required to vanish on the inner and outer walls, ρ = ρa and ρ = ρb, and also at the top and bottom, z = 0, L.
(a) Find the energy eigenvalues and eigenfunctions (ignore normalization). Show that the eigenvalues are given by
Elmn =
ℏ^2
2 me
k^2 mn +
lπ L
where kmn is the nth^ root of the transcendental equation,
Jm(kmnρb)Nm(kmnρa) − Jm(kmnρa)Nm(kmnρb) = 0
(b) Repeat the same problem when there is a uniform magnetic field, B~ = B ˆz for 0 < ρ < ρa. Note that the energy eigenvalues are influenced by the magnetic field even though the electron never “touches” the magnetic field. (c) Compare, in particular, the ground state of the B = 0 problem with that of the B 6 = 0 problem. Show that if we require the ground state energy to be unchanged in the presence of B, we obtain the “flux quantization” condition,
πρ^2 aB =
2 πN ℏc e , for N = 0, ± 1 , ± 2 , ± 3 , ...
