Physics 551, Problem Set 2 due: Wednesday, September 26 at 10am grader: M. York Please give your completed problem sets to me at the beginning of class or place them in the “Physics 551” box in the physics department mailroom (Rutherford 103b) before the due date. You are encouraged to discuss these problems with your colleagues, but you must write up your own solutions; the solutions you hand in should reflect your own work and understanding. Late problem sets will be penalized 10% per day late, unless an extension has been obtained from me or a TA before the due date. Late problem sets will not be accepted after solutions are handed out.
Reading: Chapters 1 & 2 of Sakurai. Chapters 1 – 4 of Dirac’s “Principles of Quantum Mechanics.”
1. Chapter 1, problem 29a.
1 2 2. Consider a free particle of mass m moving in one dimension, where H = 2m p . Let x(t) and p(t) be the position and momentum operators in Heisenberg picture.
(a) Using the Heisenberg equation of motion show that p(t) = p(0) is independent of time. (b) Show that x(t) = x(0) + (p(0)/m)t. Conclude that the expectation value of the position hx(t)i takes the exact same form as in the classical mechanics of a free particle. (c) Compute [x(t), x(0)]. (d) Show that h(∆x(t))2ih(∆x(0))2i ≥ ct2 where c is a constant you should compute. Thus even though the average position of the particle obeys the classical equation of motion, the uncertainty in the position grows without bound at late times.
1 2 3. Consider a particle of mass m moving in one dimension with Hamiltonian H = 2m p +V (x) where the potential V (x) is non-zero. Let x(t) and p(t) be the position and momentum operators in Heisenberg picture.
1 (a) Write down the Heisenberg equations of motion and show that they imply that the expectation value hx(t)i obeys the classical equation of motion
d2hxi m = −hV 0(x)i dt2 This is known as Ehrenfest’s Theorem.
n (b) Consider the case V = V0x . Recall that classical dynamical systems obey the “virial theorem” which states that the average value of the kinetic energy is proportional to the average value of the potential energy: 1 n h p2i = hV i 2m 2 where hi denotes the long time average of a classical variable. Show that this is not quite true in quantum mechanics, and instead quantum expectation values obey 1 n 1 dhxpi h p2i = hV i + 2m 2 2 dt
d(xp) (Hint: Compute dt .)
4. Consider two operators A and B whose commutator is a constant, say [A, B] = c.
(a) Show that eABe−A = B + c (b) Compute eABne−A.(Hint: 1 = e−AeA). (c) Show that eAeBe−Ae−B = ec. This is a simple version of the Campbell-Baker- Hausdorf forumla.
5. Consider a one dimensional particle with Schrodinger picture observables x and p.
(a) Show that the operators U = eip and V = eix are unitary. Let |xi and |pi denote position and momentum eigenstates. Show that U n|xi is an eigenstate of x for any n and compute it’s eigenvalue. Show that V n|pi is an eigenstate of p and compute it’s eigenvalue. We often say that the operator U is a translation in position space, and V is a translation in momentum space. (b) Show that the operator W = ei¯h is unitary. What happens to a physical state when we act on it with the operator W n? (c) Show that U nV mU −nV −m = W p for some value of p which you should compute.
2 (d) Consider an operator of the form U nV mW p. Explain why it is that any product of the U’s, V ’s and W ’s can be written in this form, with the U’s on the left, the V ’s in the middle and the W ’s on the right. Show that if we multiply and two operators of this form we obtain another operator of this form. That is to say, show that
(U n1 V m1 W p1 )(U n2 V m2 W p2 ) = U n3 V m3 W p3
where n3, m3, p3 are numbers you should compute in terms of n1, n2, m1, m2, p1 and n m p p2. This means that operators of the form U V W form what is known as a group. The formula you have just derived is called the group multiplication law. (e) The group you have discovered is called the Heisenberg group. In fact, this group has another very simple form. Consider the set of 3 × 3 matrices of the form
1 a c 0 1 b 0 0 1
A matrix of this form is called unitriangular. The Heisenberg group can be thought of as the set of 3 × 3 unitriangular matrices. To see this, show that
1 a1 c1 1 a2 c2 1 a3 c3 0 1 b1 0 1 b2 = 0 1 b3 0 0 1 0 0 1 0 0 1
where a3, b3, c3 are numbers you should compute in terms of a1, a2, b1, b2, c1 and c2. Show that this multiplication law is the same as the multiplication law you derived in part (d), provided a, b and c are identified with n, m and p in some way (which you should determine).
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