Homework #3 — Phys622 — Fall 1998 Deadline: Wednesday, September 23, 1998. Turn in homework in the class or put in the box on the door of Phys 2314 by 5 p.m.

Victor Yakovenko, Assistant Professor

Office: Physics 2314 Phone: (301)–405–6151

E-mail: yakovenk@physics.umd.edu

Do not forget to write your name and the homework number! Equation numbers with the period, like (3.25), refer to the equations of Schwabl. Equation numbers without period, like (5), refer to the equations of this homework.

One-Dimensional Problems (Chapter 3)

1. [3 points each part] Schwabl’s Problem 3.5, part (a) only. Additional part:

(d) Find the force F = −∂E/∂L, where L = 2a is the width of the well, exerted by the electron in the n-th energy eigenstate on the walls of the well. Calculate the force also classically, by considering a classical particle of the energy E_{n} and finding its velocity, the frequency of collisions with a given wall, and the momentum transfer per collision. Compare with the quantum result obtained above.

2. [5 points each part] Schwabl’s Problem 3.7, part (a) only. Solve the problem in two different ways:

i. Take the limit V_{0} → ∞, a → 0, but 2aV_{0} = const = λ of the solution found in Section 3.4. ii. Integrate the Schrödinger equation (3.43) around the position of the δ-function from −ϵ to

+ϵ, where ϵ is infinitesimal: ∫_{−ϵ}^{+ϵ} dx, and find the relations between the values of the wave function ψ(x), its derivative ψ′(x), and the logarithmic derivative ψ′(x)/ψ(x) at ±ϵ. Use these relations to match the solutions of the Schrödinger equation to the left and to the right of the δ-function.

Additional part:

(c) Consider a one-dimensional potential well of a given (arbitrary) shape V (x), such that

∫ V (x) dx < 0,  (1)

and V (x) → 0 at x → ±∞. The well has the characteristic depth V_{0} and the characteristic width a that obey the following inequality:^{1)}

h̄^{2}/ma^{2} ≫ V_{0},  (2)

where m is the mass of a quantum particle moving in this potential. Such a well is called shallow. Find (approximately) the energy E_{0} and the wave function ψ_{0}(x) of a bound state of the particle in this potential. Your answers should be expressed in terms of ∫ V (x) dx (see Hints). Compare E_{0} with the depth of the potential V_{0} and the width of the wave function ψ_{0}(x) with the width of the potential a. How many bound states does this potential have?

3. [7 points] Schwabl’s Problem 3.18, part (a) only. (Similar to Physics Qualifier, Fall 1994 and Spring 1986, Problem II-1.) The antisymmetric bound state disappears when a decreases below a certain critical value a_{0}. Find a_{0}.

Find out whether the two potential wells experience an attractive or a repulsive force when the particle is in the even or in the odd bound state.

1)The presence of the parameters V_{0} and a does not imply that the shape of the potential well is rectangular, as in Section 3.4. The word “characteristic” indicates that these are the parameters that characterize the overall size of the potential V (x), which may have an arbitrary shape.

4. Let us consider transmission through an arbitrary potential V (x) located within |x| < a, so that V (x) = 0 at |x| > a. A wave function of the energy E = h̄^{2}k^{2}/2 has the following forms outside of the potential:

ψ(x) = \{ \begin{matrix}Ae^{ikx} + Be^{−ikx}, & x < −a, \\ F e^{ikx} + Ge^{−ikx}, & x > a, \end{matrix}  (3)

(compare with Eq. (3.60)). The amplitudes of the outgoing waves, B and F, are related to the amplitudes of the incoming waves, A and G, by the so-called scattering matrix S:

( \begin{matrix}B \\ F \end{matrix} ) = ( \begin{matrix}S_{11} & S_{12} \\ S_{21} & S_{22} \end{matrix} ) ( \begin{matrix}A \\ G \end{matrix} ) .  (4)

Sometimes it is more convenient to work with the so-called transfer matrix M, which gives the amplitudes to the right of the potential, F and G, in terms of those to the left, A and B:

( \begin{matrix}F \\ G \end{matrix} ) = ( \begin{matrix}M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix} ) ( \begin{matrix}A \\ B \end{matrix} ) .  (5)

(a) [3 points] Express the four elements of the M-matrix in terms of the elements of the S-matrix and vice versa. The transmission and reflection coefficients, T_{l} and R_{l}, for the wave incoming from the left are

T_{l} = \frac{|F |^{2}}{|A|^{2}} |_{G=0} = |S_{21}|^{2},  R_{l} = \frac{|B|^{2}}{|A|^{2}} |_{G=0} = |S_{11}|^{2}.  (6)

The transmission and reflection coefficients, T_{r} and R_{r}, for the wave incoming from the right are

R_{r} = \frac{|F |^{2}}{|G|^{2}} |_{A=0} = |S_{22}|^{2},  T_{r} = \frac{|B|^{2}}{|G|^{2}} |_{A=0} = |S_{12}|^{2}.  (7)

Express T_{l}, R_{l}, T_{r}, and R_{r} in terms of the elements of the M-matrix.

(b) [3 points] Suppose the potential consists of two isolated pieces: V (x) = V_{1}(x) + V_{2}(x), where V_{1}(x) is localized within x_{1} < x < x_{2}, V_{2}(x) is localized within x_{3} < x < x_{4}, and x_{2} < x_{3}. Suppose that the potential V_{1}(x) has the transfer matrix M_{1}, and the potential V_{2}(x) the transfer matrix M_{2}. Show that the M-matrix for the combination of the potentials is the product of the two M-matrices for each section separately:

M = M_{2}M_{1}.  (8)

That obviously generalizes to any number of pieces and accounts for the usefulness of the transfer matrix.

5. [5 points] Schwabl’s Problem 3.19. Find also the S- and the M-matrices of the potential. Consider both λ > 0 and λ < 0.

6. [7 points] Schwabl’s Problem 3.18, part (b). Discuss resonances in transmission.

One possible way to solve this problem is to use Eq. (8) with the transfer matrices of individual δfunctions. Here one needs to be careful. In Problem 5 you found the transfer matrix of a delta-function located at the origin of coordinates (x = 0). On the other hand, in this problem the scatterers are located at different points x = ±a. In this case one should use a generalization of Eq. (8), namely:

M = M_{−a}M_{δ}M_{a}M_{a}M_{δ}M_{−a},  (9)

where M_{δ} is the transfer matrix of a delta-function potential located at the origin of coordinates, and M_{a} is the transfer matrix of the empty space of the length a:

M_{a} = ( \begin{matrix}e^{ika} & 0 \\ 0 & e^{−ika} \end{matrix} ) .  (10)

7. Adapted from Physics Qualifier, Fall 1985, Problem II-5.

Let us consider a particle subject to a periodic potential:

V (x + a) = V (x),  (11)

where a is the period of the potential.

(a) [3 points] Show that, because of the finite translational invariance of the Hamiltonian, the energy eigenfunctions may be selected to obey the following relation:

ψ(x + a) = e^{iφ}ψ(x),  (12)

and can be written in the form:

ψ_{q}(x) = e^{iqx}u_{q}(x),  (13)

where u(x) is a periodic function of x: u(x + a) = u(x), and q is a parameter, called quasimomentum, that labels the eigenstates and their eigenenergies: E(q). Eq. (13) is called Bloch’s theorem in quantum mechanics and Floquet’s theorem in mathematics.

(b) [3 points] What momenta does the state (13) contain? In other words, if the wave function (13) is Fourier-expanded, what wave vectors would be present in the expansion?

(c) [7 points] Let us consider a particle in the potential that consists of periodically spaced deltafunctions:

V (x) = λ ∑_{n=−∞}^{+∞} δ(x − na)  (14)

(the so-called Dirac comb or Kronig-Penney model). From the Schrödinger equation for potential (14), obtain an equation that determines a relation between the quasimomentum q and energy E of the eigenfunctions (13), the so-called dispersion relation (see Hints). Study this equation graphically and show that the energy E may take values only within certain ranges, called the energy bands, and may not take value within other ranges, called the energy gaps.

Hints

2c Argue that condition (2) allows to use the method of part (a)ii.

7c One possible way to solve this problem is to write the wave function of the energy E = h̄^{2}k^{2}/2m in the form ψ(x) = Ae^{ikx} + Be^{−ikx} and argue that A and B must form an eigenvector of the transfer matrix M_{a}M_{δ} (the notation of Eq. (9)) with the eigenvalue e^{iqa}:

e^{iqa} ( \begin{matrix}A \\ B \end{matrix} ) = M_{a}M_{δ} ( \begin{matrix}A \\ B \end{matrix} ) .  (15)