Quantum Physics II (8.05) Fall 00 Assignment 11 Readings Most of the reading needed for this problem set was already given on Problem Set 9. The new readings are: Phase shifts are discussed in Cohen-Tannoudji on pages 91-93, 938-947. However, this discussion is in a context which you will only fully appreciate in 8.06. To see a description of the same physics at the level appropriate for this problem set, try Gasiorowicz (which you can get from the library or the physics reading room) pp. 178-185. The reading assignment on spin is Griffiths Ch. 4.4 and Cohen-Tannoudji Ch. IX. If you wish to read ahead, to prepare for the next problem set, read Griffiths Ch. 5.1, 5. and Cohen-Tannoudji Ch. X, not including any of the complements. Problem Set 11 1. Infinite Spherical Well (15 points) In this example, you will work out work out the bound state energies for the lowest few values of l in the infinite spherical well potential, by which I mean V = 0 for r b and V = for r > b. You will discover that in this potential, unlike in the 1/r potential, there are no degeneracies between levels with different values of l. (a) Before we get to the infinite spherical well, lets consider the potential V = 0. I mean V = 0 everywhere. You know what solutions look like in the basis of eigenstates of p x, p y, p z : they are plane waves ψ(x, y, z) exp[i(xp x + yp y + zp z )/ h]. What we want to do here, though, is lay the groundwork for imposing the boundary condition ψ = 0 at r = b. With this goal in mind, it is much more convenient to think of V = 0 as an example of a central potential and then use as basis states wave functions which are eigenstates of the Hamiltonian, L and L z, rather than of p x, p y, p z. So, these states will be given by u l (r) ψ(r, θ, φ) = Y lm (θ, φ) r where u l (r) satisfies the radial wave equation with V = 0: h d h l(l + 1) u m dr l(r) + u mr l (r) = Eu l (r), 1
with u l (0) = 0. Show that if we introduce the wave number k via the change of variables k = me/ h and then introduce the dimensionless variable z = kr, the radial wave equation becomes d l(l + 1) dz u l(z) + z u l (z) = u l (z). (1) Solutions to this equation have the form u 0 l (z) = zj l (z) where ( 1 d ) l ( ) sin z j l (z) = ( z) l. () z dz z The function j l (z) is called the lth spherical Bessel function. Write the explicit expressions for zj 0 (z) and zj 1 (z) and zj (z). Verify that zj 0 (z) and zj 1 (z) solve the radial equation. [Challenge: Prove that zj l (z) solves the radial equation, for j l (z) given in (). Do not turn this in; it is not a required part of this problem set. However, some of you may enjoy constructing a proof. There are several ways of doing this. One of them is to apply the supersymmetric methods that you learned last week.] (b) Now, consider the spherically symmetric potential with V = 0 for r < b and V = for r > b. This means that now we must impose the boundary condition u(b) = 0. i. First, consider l = 0. Find the allowed values of k, and hence the allowed energies. ii. Now, consider l = 1. Determine the allowed values of k graphically. Show that the energy of the nth energy level with l = 1 is approximately E n,l=1 ( h π /mb )(n + 1/) when n is large. iii. You can find (and by now have found) the values of z for which j 0 (z) = 0 analytically. The zeros of j l (z) for l 1 have to be found numerically. The lowest few zeroes are: j 1 (z) = 0 at z = 4.49 and z = 7.73 and z = 10.90 and... j (z) = 0 at z = 5.76 and z = 9.10 and... j 3 (z) = 0 at z = 6.99 and z = 10.4 and... j 4 (z) = 0 at z = 8.18 and... j 5 (z) = 0 at z = 9.36 and... j 6 (z) = 0 at z = 10.31 and... (For each l, there are infinitely many zeroes at ever increasing values of z. I have listed all the zeroes that occur at z < 11. You might wish to check a few of these zeroes, given that you have explicit expressions for the j l s. I will not ask you turn in such checks.) Make a level diagram for the infinite spherical well that is organized like the one I made for hydrogen in lecture. That is, arrange the energy levels in side-by-side columns, one column for each value of l. [Aside: in the same way that the level diagram for hydrogen provides initial insight into atomic structure to construct an atom with Z electrons you think of filling
levels with electrons one by one, from the bottom up the level diagram you have just constructed is a crude starting point for nuclear physics you think of filling levels with neutrons and protons one by one. The correct nuclear potential is not quite as simple as this, and there are also forces between neutrons and protons that depend on their spin which play an important role. However, this level diagram is a better starting point for nuclear physics than the level diagram for a 1/r potential would be because, unlike electrostatic forces, nuclear forces are short-ranged. You might be worried that this potential well is infinitely deep, whereas whatever potential it is that describes nuclei cannot be. You will see in the last part of the next problem that this is a good approximation to a deep, but not infinitely deep, spherical well.]. Bound States in a Finite Spherical Well (15 points) A finite spherical well is described by V (r) = V 0 for r b and V (r) = 0 for r > b. Throughout this problem, consider only states with l = 0. (a) Write the Schrödinger equation for the radial wave function u(r). (b) Find u(r) for r < b in terms of the momentum, q = m(e + V 0 )/ h. What condition on u(r) dictated your choice between the two linearly independent solutions to the Schrödinger equation? (c) Find u(r) for r > b for the case of a bound state, E < 0. [Let κ = me/ h.] What condition on u(r) dictated your choice between the two linearly independent solutions to the Schrödinger equation? (d) By demanding that u(r) and its derivative be continuous at r = b, find an eigenvalue equation for E. [It takes a simple form in terms of the variables qb and b mv 0 / h.] Explain how to solve this equation graphically. (e) What is the minimum value of V 0 for which there is a bound state? [Recall that in one dimension, a potential well always has at least one bound state. This is not the case in three dimensions.] (f) For very large values of V 0, what are the energies of the deeply bound states? [Make sure that your result agrees with what you found for l = 0 states in Problem 1.] 3. Phase Shifts (0 points) States with positive energy are not bound, and their energies are not quantized. They are important as solutions to the scattering problem. When we scatter a particle of momentum k from a central potential, we can observe the phase shifts δ l (k) for each l. (We will see how in 8.06.) This problem reviews the definition of the phase shift and then asks you to derive it in several simple cases. The radial equation for a solution with energy E and angular momentum l is d l(l + 1) m u dz l(z) + u z l (z) + h k V (z)u l(z) = u l (z) (3) where k = me/ h, z = kr and u l (0) = 0. 3
(a) First, consider the case V = 0. The solutions to the free Schrödinger equation with u l (0) = 0 are given by u 0 l (z) = zj l (z). Use () to show that ( ) 0 lπ u l (z) sin z as z. This explains why the phase shift for an arbitrary potential V is defined by ( ) lπ u l (z) sin z + δ l (k) as z. (b) Consider the potential V (r) = V 0 for r < b and V = 0 for r > b. Solve the Schrödinger equation for l = 0 and k > 0 for this potential. Remember that both u and u must be continuous at any discontinuity of V. Obtain an equation for the phase shift δ 0 (k) and plot your result for V 0 small enough that there is no bound state. Discuss the behavior of the phase shift in the limit of high energies (k mv 0 / h ) and low energies (k mv 0 / h ). In these limits, consider both the attractive potential given above, and repulsive potentials for which V (r) = +V 0 for r < b. [Answer: the equation obeyed by the phase shift is kcot(kb + δ 0 (k)) = qcot(qb), where q = m(e + V 0 )/ h. δ 0 (k) vanishes at k = 0, rises linearly (at first) with k, reaches a maximum, and then falls back to zero at large k. I leave to you deriving the answer, plotting it, and discussing its behavior.] (c) Consider the attractive potential given in (b). For l = 0 it is possible to expand the phase shift in powers of k as follows: kcotδ 0 (k) = 1 + r eff k + O(k 4 ). (4) a Relate the parameters a and r eff to the parameters of the potential b and V 0. [This is a straightforward, but rather grungy, exercise in Taylor expansion. I strongly recommend having Mathematica (or an equivalent program) do it for you...] You now have an expression for a in terms of b and V 0. a has units of length. For most values of b and V 0, a is of the same order of magnitude as b. Show, however, that (for a given, fixed, b) there are special values of V 0 for which a/b. [a and r eff are called the scattering length and effective range of the potential. Since an expansion of the form (4) holds for any well-behaved potential, low energy scattering can always be approximated by an effective square well.] (d) The potential whose scattering length a you have just analyzed is the same as the potential whose bound states you analyzed in Problem. Is there anything special about the energies of the bound states of this potential if V 0 has one of the special values for which a/b? If so, what? (e) Consider the potential V (r) = for r < b and V = 0 for r > b. Again, calculate the l = 0 phase shift. [This potential corresponds to an impenetrable sphere.] 4
4. Spin Eigenfunctions and Spin Precession (10 points) This problem should be largely a review for you. (a) Consider a spin 1/ system. What are the eigenvalues and eigenvectors of the operator S x + S y? Suppose a measurement of this operator is made, and the system is found to be in the state corresponding to the larger eigenvalue. What is the probability that a subsequent measurement of S z yields h/? (b) Consider a spin 1/ particle in a magnetic field in the z-direction. The Hamiltonian is H = γs z B = ωs z where ω = γb. Suppose that at time zero the spin is in the state ψ(0) = c + + + c. i. Find the state of the system ant time t > 0 and show that S x, S y, and S z evolve periodically in time. With what period? ii. The state itself also evolves periodically. With what period? Ie after what time is the state the same as it was at t = 0, where by the same I mean the same including any overall phase. iii. You may object: why should I care about an overall phase? It is not observable! This is correct if you only have one spin 1/ particle at your disposal. However, if you have two spin 1/ particles, you can design an experiment which shows that there are experimental consequences of the fact that the answers to (i) and (ii) differ. Outline such an experiment. [Hints: You can answer this with a simple sketch and a few lines of explanatory text. And, look in Sakurai.] 5