Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid Announcement This handout includes 9 problems. The first 5 are the problem set due. The last 4 cover material from the final few lectures of the course. This material will be addressed on the final (I guarantee it) and these problems should aid you in your study of this last part of the course. However, these problems are not to be turned in; i.e. they will not be graded. Solutions to all 9 problems will be made available. Readings The suggested readings for the last part of 8.05 are: Griffiths Ch. 4.4, 5.1, 5.3. Also 6.3 not including 6.3.1; 6.4. We will return to the material from Ch. 6 in 8.06 after we have studied perturbation theory. Only the parts I list are relevant now. Cohen-Tannoudji, Ch. IX (including Complement A), X (including Complements A and B) and XIV (not including any Complements; note also that we will not follow the study of systems of identical particles as far as is done in this Chapter.) Problem Set 12 1. CSCOs for Spin-Orbit Coupling (10 points) Including the spin-orbit coupling, the electronic Hamiltonian for hydrogen with no external fields is 2µ 2 Ĥ = ˆ B L ˆ S ˆ H 0 + r 3 h 2, where Hˆ 0 = pˆ2 /2m e 2 /r as usual and Ŝ is the electron spin. For this problem, simply take this as a definition of H. ˆ (a) Evaluate the commutators [ H, ˆ L ˆ 2 ], [ H, ˆ L ˆ z ], [ H, ˆ Ŝ 2 ], [ H, ˆ S ˆ z ], [ H, ˆ Ĵ 2 ], [ H, ˆ J ˆ z ] where Ĵ = L ˆ + S ˆ is the total angular momentum. What is the largest set of these operators including H ˆ that is mutually commuting? (b) Consider states whose radial wave functions are eigenstates of the radial part of Ĥ 0 with principal quantum number n = 1 or n = 2. What values of l, s, j and m j are possible? If we include the effects of the spin-orbit coupling, these states are not all degenerate. How many distinct energy levels are there? What is the degeneracy of each? Draw a diagram showing the levels in order of energy. [Eg determine which levels have higher energy and which have lower energy, but do not calculate the magnitude of the energy differences between levels.] 1
(c) Now turn on an external magnetic field B = Be z, so that the term µ B H B = B ( L ˆ + gs) ˆ is added to the Hamiltonian, where g 2. Repeat part (a). 2. Clebsch-Gordan Coefficients for l = 1 and s = 1/2. (12 points) Consider a spin 1/2 particle in a state with orbital angular momentum l = 1. The aim of this problem is to calculate the Clebsch-Gordan coefficients that allow us to construct states of definite total angular momentum J from simultaneous eigenstates of spin and orbital angular momentum. Label the eigenstates in the uncoupled basis (ie eigenstates of L 2, S 2, L z and S z ) by lsm l m s. Label the states in the coupled basis (ie eigenstates of J 2 and J z ) by j, m. (a) Find the state with maximum j and m (= j max ) in terms of the lsm l m s states. (b) Use J = L + S to generate all the j max, m states. (c) Use orthonormality to find the state j max 1, j max 1. (d) Use J to generate all the states j max 1, m. (e) Repeat steps (c) and (d) for smaller j s as many times as necessary. (f) What is the expectation value of L z in the state with j = 1/2, m = 1/2? What is the expectation value of S z in this state? (g) Suppose that this particle moves in an external magnetic field in the z-direction, B = Be z. Assume the particle is an electron, and take g = 2. The Hamiltonian describing the interaction of the electron with the field is µ B L + 2 ˆ H B = B ( ˆ S). What is H B in each of the eigenstates j, m? (h) For the eigenstate j = 1/2, m = 1/2, what are the possible values of the magnetic energy and what are their probabilities? [For those of you who want more practice, work out the Clebsch-Gordan coefficients for the addition of orbital angular momentum l and spin 1. I decided not to put it on the problem set, as it involves a fair amount of algebra. It is, however, very good practice.] 3. A Spin-Dependent Potential (8 points) A particle with spin 1 moves in a central potential of the form S L V (r) = V 1 (r) + h 2 V 2 (r) + (S L) 2 V 3 (r). h 4 Suppose that the particle is in an orbital with a particular value of l. This means that it could have one of three possible values of j. For each of these three possibilities, what is the potential V (r) felt by the particle? [Your answers should depend on V 1 (r), V 2 (r), V 3 (r), and the value of l.] 2
4. Spin-Orbit and Zeeman Effects Together (15 points) This problem demonstrates the Zeeman effect for an atom with a single valence electron in a state with l = 1. [If you wish, replace the previous sentence with Consider an electron with l = 1 in a central potential. ] In the presence of a uniform magnetic field B = Be z along the z-axis, the magnetic terms in the Hamiltonian may be written as H = µ B B(L z + 2S z ) + 2W 2 L S. [Here, W is a constant whose magnitude is approximately W Z 4 h 2 eff e 2 /m 2 e c 2 a 3 0 where a 0 is the Bohr radius and Z eff is the effective charge of the nucleus as seen by the electron.] Denote the three eigenstates of L z by φ 1, φ 0 and φ 1. Denote the two eigenstates of S z by + and. Consider the basis of product states: ψ 1 = φ 1 +, ψ 2 = φ 1, ψ 3 = φ 0 +, ψ 4 = φ 0, ψ 5 = φ 1 +, ψ 6 = φ 1. (a) Show that L S = 1 2 (L +S + L S + ) + L z S z. (b) Find H ψ 1... H ψ 6. [For example, H ψ 3 = 2W ψ 2 + µ B B ψ 3.] (c) Write the Hamiltonian as a six by six matrix using the basis of product states. (d) What are the eigenvalues of the Hamiltonian? What are the eigenvalues for B W/µ B? [This is called the weak field Zeeman effect. All six energies should look like (constant proportional to W ) + (constant)(b).] What are the eigenvalues for B W/µ B? [Strong field Zeeman effect.] (e) Plot the six eigenvalues as a function of B, with B ranging from zero to B W/µ B so that you can show the behavior of the energies in the weak field, intermediate field and strong field regimes. (f) The eigenstate for a given eigenvalue changes as a function of B. I will not ask you to work out the eigenstates in detail in the intermediate field region. Instead, just state what eigenstate corresponds to each of the six eigenvalues in the weak field case, and in the strong field case. [You can do this by careful labelling of the diagram you have drawn in part (e).] 5. Total Spin States (15 points) (a) Suppose you are given two (non-identical) particles, each with angular momentum j = 1. How many distinct angular momentum eigenstates does this system possess? What are the possible values of the total angular momentum J of this system? (b) Repeat part (a) for two particles, each with angular momentum j. (c) Suppose you are given three (non-identical) particles, each with angular momentum j = 1. How many distinct angular momentum eigenstates does this system possess? What are the possible values of the total angular momentum J of this system? 3
(d) Repeat part (c) for three particles, each with angular momentum j = 3/2. (e) For part (a), what values of total angular momentum are allowed if the particles are identical fermions with the same radial wave function? Identical bosons with the same radial wave function? [Hint: A general way of solving such problems involves counting and analyzing the possible m-values. Here s how it works for three particles with j = 1/2. In a basis where j 1z, j 2z and j 3z are diagonal with eigenvalues hm 1, hm 2 and hm 3, states can be labelled by the eigenvalues m 1, m 2 and m 3 (which can each be either 1/2 or 1/2). Make a table of all 8 states. Notice that M m 1 + m 2 + m 3 is the eigenvalue of J z j 1z + j 2z + j 3z. Each state of definite M must be part of a multiplet of definite total angular momentum J (with M values ranging from J to J ). Look at the list of M values in your table, and conclude that the M values corresponding to J = 3/2, J = 1/2 and J = 1/2 are represented. So, we conclude that the total angular momentum states of three spin 1/2 particles are J = 3/2, J = 1/2 and J = 1/2. Another way to do this problem is to use the composition law for two angular momenta twice. Schematically, End of hint.] 1 1 1 1 1 1 1 1 3 = [0 + 1] = [0 2 ] + [1 2 ] = + +. 2 2 2 2 2 2 2 [Don t forget: Problems 6-9 are NOT due. They are not to be handed in. They are meant to help you learn the material covered in the last few lectures of the course for the final. Solutions will be provided.] 6. Bosons and Fermions in a Square Well Consider a set of 4 identical particles confined in a one-dimensional infinitely high square well of length L. (a) What are the single particle energy levels? What are the corresponding single particle wave functions? Name the wave functions φ 1 (x), φ 2 (x)... (b) Suppose the particles are spinless bosons. What is the energy and (properly normalized) wave function of the ground state? Of the first excited state? Of the second excited state? [Example: The ground state is Ψ(x 1, x 2, x 3, x 4 ) = φ 1 (x 1 )φ 1 (x 2 )φ 1 (x 3 )φ 1 (x 4 ).] (c) If the particles are spin 1/2 fermions, what is the energy and wave function of the ground state? The first excited state? The second excited state? Hint: use some abbreviated notation such as a Slater determinant. For example, 4
the ground state is with energy φ1(x 1 ) φ 1(x 2(x 1 ) φ 1 ) φ 2(x 1 ) 1(x 2 ) φ 2(x 2 ) φ 1 φ 1(x 2 ) φ 2(x 2 ) Ψ(x 1, x 2, x 3, x 4 ) = 4! φ 1(x 3 ) φ 1(x 2(x 3 ) φ 3 ) φ 2(x 3 ) φ1(x 4 ) φ 1(x 2(x 4 ) φ 4 ) φ 2(x 4 ) 2 π 2 E = (1 + 1 + 4 + 4). 2mL 2 (1) 7. Griffiths, Problem 5.5 8. Two Electrons Consider two electrons. Their spatial wave function may be either symmetric or antisymmetric under the interchange of the electrons coordinates. Since the electrons are fermions, the overall wave function must be antisymmetric under the simultaneous interchange of both space coordinates and spin. (a) Suppose the spatial wave function is antisymmetric. What are the allowed spin wave functions of the electrons? What are the eigenvalues of the square and the z-component of the total spin operator S = s 1 + s 2 in these states? (b) Repeat part (a) in the case where the space wave function is symmetric. (c) Suppose that (up to this point in the problem) all the states enumerated in parts (a) and (b) have the same energy. Now add the following term to the Hamiltonian: H = Cs 1 s 2. [That is, the electrons interact by a spin-spin force due to the interaction of the magnetic moment of each with the magnetic field generated by the other.] What are the eigenstates of the system including the interaction H? What are the energies of the states in parts (a) and (b)? 9. Two Electrons in a Spin Singlet State Consider two electrons in a spin-singlet state. (a) If a measurement of the spin of one of the electrons shows that it is in an S z eigenstate with S z -eigenvalue /2, what is the probability that a measurement of the z-component of the spin of the other electron yields /2? (b) If a measurement of the spin of one of the electrons shows that it is in an S y eigenstate with S y -eigenvalue /2, what is the probability that a measurement of the x-component of the spin of the other electron yields /2? 5