Physics 215 Winter 2002 Introduction to Modern Physics Prof. Ioan Kosztin Homework #4 Unless otherwise stated, all the questions and problems in this assignment are from the text Modern Physics (2nd ed) , by R.A. Serway, C.J. Moses and C.A. Moyer (Saunders College Pub., Philadelphia, 1997); the question/problem and corresponding page numbers are indicated in parenthesis. 1. (P1/p224) Of the functions graphed in Figure P5.1, which are candidates for the Schrödinger wavefunction of an actual physical system? For those that are not, state why they fail to qualify. [5 pt] 2. (P2/p224) A particle is described by the wavefunction ≤≤−π=ψotherwiseLxLforLxAx04/4/)/2cos()(Physics 215 Homework #4 - 2 - (a) Determine the normalization constant A. (b) What is the probability that the particle will be found between x = 0 and x = L/8 if a measurement of its position is made? [5 pt] 3. (P5/p224) In a region of space, a particle with zero energy has a wavefunction 22/)(LxexAx−=ψ (a) Find the potential energy U as a function of x. (b) Make a sketch of U(x) versus x. [5 pt] 4. (P14/p225) A particle of mass mo is placed in a one-dimensional box of length L. The box is so small that the particle's motion is relativistic, so that E = p2/2mo is not valid. (a) Derive an expression for the energy levels of the particle. (b) If the particle is an electron in a box of length L = 1.00X10-12 m, find its lowest possible kinetic energy. By what percent is the nonrelativistic formula for the energy in error? [5 pt] 5. (P23/p226) Consider a square well having an infinite wall at x = 0 and a wall of height U at x = L (Fig. P5.23). For the case E <U; obtain solutions to the Schrödinger equation inside the well (0 ≤ x ≤ L) and in the region beyond (x>L) that satisfy the appropriate boundary conditions at x = 0 and x = ∞. Enforce the proper matching conditions at x = L to find an equation for the allowed energies of this system. Are there conditions for which no solution is possible? Explain. [5 pt] 6. (P35/p227) . From the results of Problems 33 and 34, evaluate ∆x ∆Px for the quantum oscillator in its ground state. Is the result consistent with the uncertainty principle? (Note that your computation verifies the minimum uncertainty product; furthermore, the harmonic oscillator ground state is the only quantum state for which this minimum uncertainty is realized.) [5 pt]Physics 215 Homework #4 - 3 - 7. (P40/p227) For the non stationary state of Problem 39, show that the average particle position (x) oscillates with time as )cos(0tAxxΩ+=〉〈 where ()dxxAdxxdxxx∫ψψ=∫ψ+∫ψ=2*122210||||21 and Ω = (E2 -E1)/ħ. Evaluate your results for the mean position xo and amplitude of oscillation A for an electron in a well 1nm wide. Calculate the time for the electron to shuttle back and forth in the well once. Calculate the same time classically for an electron with energy equal to the average, (E1 + E2) /2. [5 pt] 8. (P2/p253) Consider the step potential of Example 6.5 in the case where E >U. (a) Examine the Schrödinger equation to the left of the step to find the form of the solution in the range x < O. Do the same to the right of the step to obtain the solution form for x > O. Complete the solution by enforcing whatever boundary and matching conditions may be necessary. (b) Obtain an expression for the reflection coefficient R in this case, and show that it can be written in the form 221221)()(kkkkR+−= where k1 and k2 are wave numbers for the incident and transmitted waves, respectively. Also write an expression for the transmission factor T using the sum rule obeyed by these coefficients. (c) Evaluate R and T in the limiting cases of E → U and E → ∞. Are the results sensible? Explain. (This situation is analogous to the partial reflection and transmission of light striking an interface separating two different media.) [5 pt] 9. (P11/p254) The Ramsauer-Townsend Effect. Consider the scattering of particles from the potential well shown in Figure P6.11. (a) Explain why the waves reflected from the well edges x = 0 and x = L will cancel completely if 2L = λ2, where λ2 is the de Broglie wavelength of the particle in region 2. (b) Write expressions for the wavefunctions in regions 1, 2, and 3. Impose the necessary continuity restrictions on ψ and ∂ψ/∂x to show explicitly that 2L = λ2 leads to no reflected wave in region 1. [This is a crude model for the Ramsauer-Townsend effect observed in the collisions of slow electrons with noble gasPhysics 215 Homework #4 - 4 - atoms like argon, krypton, and xenon. Electrons with just the right energy are diffracted around these atoms as if there were no obstacle in their path (perfect transmission). The effect is peculiar to the noble gases because their closed shell configurations produce atoms with abrupt outer boundaries.] [5 pt] 10. (P12/p254) A model potential of interest for its simplicity is the delta well. The delta well may be thought of as a square well of width L and depth S/Lin the limit L → ∞ (Fig. P6.12). The limit is such that S, the product of the well depth with its width, remains fixed at a finite value known as the well strength. The effect of a delta well is to introduce a discontinuity in the slope of the wavefunction at the well site, although the wave itself remains continuous here. In particular, it can be shown that )0(2200ψ=ψ−ψ−+=mSdxddxd for a delta well of strength S situated at x = O. (a) Solve Schrödinger's equation on both sides of the well (x < 0 and x> 0) for the case where particles are incident from the left with energy E > O. Note that in these regions the particles are free, so that U(x) = O. (b) Enforce the continuity of ψ and the slope condition at x = O. Solve the resulting equations to obtain the transmission coefficient T as a function of particle energy E. Sketch T(E) for E ≥ O. (c) If we allow E to be negative, we find that T(E) diverges for some particular energy Eo. Find this value Eo. (As it happens, Eo is the energy of a bound state in the delta well. The calculation illustrates a general technique, in which bound states are sought among the singularities of the scattering coefficients for a potential well of arbitrary shape.) (d) What fraction of the particles incident on the well with energy E=|E0| is transmitted and what fraction is reflected ? [5 pt]Physics 215 Homework #4 - 5 - The homework is