Quantum Physics II (8.05) Fall 2002 Assignment 7

Transcription

1 Quantum Physics II (8.05) Fall 2002 Assignment 7 Readings for the next two weeks The ammonia molecule and the ammonia maser are presented in The Feynman Letures, Volume 3, Chapters 8 and 9. They are also discussed in Cohen-Tannoudji, p Although this is a very nice example of quantum dynamics, I have decided not to present the maser in lecture, given that I want to finish Section III of the course before the midterm. Your section instructors may choose to discuss it. And, regardless, you should read at the minimum the two chapters from Feynman. Spin precession is discussed in Sakurai, 2.1, and in Cohen-Tannoudji p C-T also discusses nuclear magnetic resonance. By this point in the course all of Cohen-Tannoudji s Chapter 4, including all the complements except E and H, warrants your attention. If you are curious to learn more about NMR than we will cover in 8.05, you can look at Principles of Magnetic Resonance by C. P. Schlichter, which is available in the physics reading room. For the midterm and final, you are only responsible for NMR as I present it in lecture; the book is for those who want more. Supplementary notes on neutrino oscillations and kaon physics, posted on the web page. These cover the same material which I will address in lecture. You are responsible for this material for the midterm. However, for the midterm and final you are not responsible for memorizing physics special to neutrinos or kaons. If I pose a problem in this area, I will tell you whatever you need to know about neutrinos or kaons, and ask you to solve a problem which demonstrates your understanding of quantum mechanical two-state systems. Although memorization of the special features of these systems is NOT called for, if you do face a problem of this nature on an exam you will be in much better shape if you have worked through these notes. In addition, I am handing out a packet of extra reading on neutrino oscillations. This contains four different items: (i) an article by Boris Kayser; (ii) notes on the space-time evolution of a neutrino beam; (iii) a paper published in Physical Review Letters describing the discovery of evidence for the oscillation of atmospheric neutrinos obtained using the Super-Kamiokande 1

2 detector. (iv) the paper published in Physical Review Letters which describes the 2002 discovery of direct evidence for solar neutrino oscillations. In addition, you can consult superk/ for a very elementary description of the SuperKamiokande experiment. And, from you can find links to the web pages of all the ongoing, completed, and proposed experiments studying neutrino oscillations, including many that I did not have time to mention in lecture. All the reading material mentioned in this bullet is optional. Problem Set 7 1. The Ammonia Molecule (16 points) Consider an ammonia molecule placed in an electric field ε. The Hamiltonian is given by ( ) E 0 + µε Δ Ĥ = Δ E 0 µε where µ is the electric dipole moment. [Point of notation: Call the basis states in the basis in which the Hamiltonian takes the form above + and, because they are the states in which the nitrogen atom is either above or below the plane of the hydrogens. These states are evidently not energy eigenstates.] In this problem, you will find and study the energy eigenstates and energy eigenvalues of the Hamiltonian H. (a) Show that the Hamiltonian may be written as H = E a ( sin(2θ)σ 1 + cos(2θ)σ 3 ) for some a and θ. Specify a and tan(2θ) in terms of variables in the original Hamiltonian. [General point of notation: the Pauli matrices are sometimes referred to as σ x,y,z and sometimes as σ 1,2,3.] (b) Use an identity derived on a past problem set to show that the Hamiltonian can be rewritten as H = E a (exp(iθσ 2 )σ 3 exp( iθσ 2 )). (c) Define U = exp(iθσ 2 ). Show that 1 = U + and 2 = U are the eigenstates of the Hamiltonian. What are the corresponding energy eigenvalues? Evaluate Û ĤU. [Notational clarification: U is unitary that s why I called it U but it is not the time evolution operator for a system with Hamiltonian H. In other words, Û exp( i Ht/ h).] (d) Show that the energy eigenstates may be written as 1 = cos θ + sin θ and 2 = sin θ + + cos θ. (e) Consider the limit in which µε = 0. To what θ does this correspond? What are the eigenstates and eigenvalues of the Hamiltonian in this limit? 2

3 (f) In the opposite limit, in which µε is much larger than Δ, what happens to θ? What are the energy eigenstates and eigenvalues in this limit? Interpret these results. 2. Time Evolution in a Two-State Problem (12 points) This problem is a slight modification of Sakurai s 2.9. A box containing a particle is divided into a right and left compartment by a partition. If the particle is known to be on the left side with certainty, we call the state L ; if on the right, we call the state R. (Of all the different possible L states, corresponding to different spatial shapes of the wave function within the left half of the box, we consider only the one with the lowest energy. Same on the right. Hence, we have a two-state problem.) Assume that the box is symmetric, in the sense that L H L = R H R. Let us now shift our zero of energy in such a way that L H L = R H R = 0. The Hamiltonian does not vanish, however, because the particle can tunnel through the partition. This tunnelling effect is characterized by the Hamiltonian Ĥ = Δ( L R + R L ) where Δ is a real number with the dimension of energy. (a) Find the normalized energy eigenstates, and the corresponding energy eigenvalues. (b) In the Schrödinger picture, the basis kets L and R are fixed while the state vector moves with time. Suppose that at time t = 0 the state vector is given by α(0) = c L L + c R R. Find α(t), the state vector at a later time, by applying the time evolution operator. (c) Suppose that at time t = 0 the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time? (d) Suppose that I had made an error and wrote H as Ĥ = Δ L R. By explicitly solving the most general time evolution problem with this Hamiltonian (as you did in part (b) for the correct Hamiltonian) show that probability conservation is violated. 3. Time Evolution in a Three-State System (12 points) Carbon dioxide is a linear molecule (OCO) which can pick up an extra electron and become a negatively charged ion. Suppose that the electron would have energy E O if it were attached to either oxygen, or energy E C if it were attached to the carbon atom in the middle. Call these states L, C and R, for left oxygen, carbon, and right oxygen. The energy eigenstates need not, however, have either energy E O or E C because there is some probability that the electron may hop between an oxygen atom and the carbon atom. (Assume that the probability of jumping directly from oxygen to oxygen can be neglected.) 3

4 (a) Wrote down a model Hamiltonian to describe the physics of the electron outlined above. You will need to introduce a new parameter. Find the energy eigenvalues of this three-state system in terms of E C, E O, and your new parameter. (b) In this part and the next, assume E C = E O. Find the energy eigenstates. (c) Assume that at time t = 0 the electron is in state L. That is, it is localized on the left oxygen atom. This is not a stationary state. What is the probability that at some later time t, the electron will be in state L? In state C? In state R? Plot these three probabilities as functions of time. 4. Precession of a Spin-1/2 Particle (12 points) Consider a spin-1/2 particle in a uniform magnetic field applied in the positive z direction. The Hamiltonian for the spin is Ĥ = γbŝ z where γ is called the gyromagnetic ratio of the spin. (a) Write the Heisenberg picture equations of motion for the time-dependent opera tors S xh (t), S yh (t) and S zh (t). (b) Suppose that by making the appropriate measurement, we have learned that at t = 0 the spin is in an eigenstate of S n with eigenvalue h/2, where n is a unit vector lying in the xz-plane, that makes an angle β with the z-axis. Write this initial state in the S z basis. You may use results from past problem sets without rederiving them. (c) Use the results of part (a) to deduce equations describing the time evolution of the expectation values of the operators S x,y,z. Solve these equations subject to the initial conditions specified in part (b). (d) Show that your results make sense in the extreme cases β = 0 and β = π/2. Plot the time evolution of all three expectation values for β = π/4. (e) For arbitrary β, calculate the time evolution operator U (t). Calculate the (timedependent) Schr odinger picture state ket. (In the Schr odinger picture, operators are time independent.) Use the time dependent state ket you have obtained to calculate the expectation values of the operators S x,y,z, and show that your result agrees with that obtained above using the Heisenberg picture. 5. Time Evolution of the Most General Operator in a Two-State System (8 points) Consider a two-state system with arbitrary Hamiltonian. It is always possible to find a basis in which the Hamiltonian is diagonal. Let us then choose our zero of energy such that the two energy eigenvalues are +E and E, and call the corresponding eigenstates + and. Work in this basis throughout this problem. Begin by writing the 2 2 matrix representing H in this basis. 4

5 Now consider an arbitrary observable A written in the form A = a a 1 σ 1 + a 2 σ 2 + a 3 σ 3 where the a s are complex constants and the σ s are the Pauli matrices. In the Heisenberg picture, the corresponding time dependent operator is A A H (t) = U (t) U (t). Carry out the operator algebra and evaluate A H (t). If you use identities from past problem sets, state them clearly but do not rederive them. Since any operator can be written as a linear combination of the Pauli matrices and the identity, your result should take the form: Â H (t) = a 0 (t) 1 + a 1 (t) σ 1 + a 2 (t) σ 2 + a 3 (t) σ 3. If any of the coefficients turns out to be time independent, explain why. 5