Chapter #5 Solutions - Introduction to Quantum Mechanics - David J. Griffiths - 2nd Edition

1. Typically, the interaction potential depends only on the vector r ≡ r1 – r2 between the two particles. In that case the Schrödinger equation separates, if we change variables from r1, r2 to r and R ≡ (m1r1 + m2r2)/(m1 + m2) (the center of mass).(a) Show that r1 = R + (μ/m1)r, r2 = R – (μ/m2)r, and ∇1 = (μ/m2)∇R + ∇r, ∇2 = (μ/m1)∇R – ∇r, where... [5.8]is the reduced mass of the system.(b) Show that the (time-independent) Schrödinger equation becomes...(c) Separate the variables, letting ψ (R, r) = ψR(R)ψr(r). Note that ψR satisfies the one-particle Schrödinger equation, with the total mass (m1 + m2) in place of m, potential zero, and energy ER, while ψr satisfies the one-particle Schrödinger equation with the reduced mass in place of m, potential V(r), and energy Er. The total energy is the sum: E = ER + Er. What this tells us is that the center of mass moves like a free particle, and the relative motion (that is, the motion of particle 2 with respect to particle 1) is the same as if we had a single particle with the reduced mass, subject to the potential V. Exactly the same decomposition occurs in classical mechanics;1 it reduces the two-body problem to an equivalent one-body problem. Get solution

2. Problem 5.2 In view of Problem 5.1, we can correct for the motion of the nucleus in hydrogen by simply replacing the electron mass with the reduced mass. (a) Find (to two significant digits) the percent error in the binding energy of hydrogen (Equation 4.77) introduced by our use of m instead of μ. (b) Find the separation in wavelength between the red Balmer lines (n = 3 → n = 2) for hydrogen and deuterium. (c) Find the binding energy of positronium (in which the proton is replaced by a positron—positrons have the same mass as electrons, but opposite charge). (d) Suppose you wanted to confirm the existence of muonic hydrogen, in which the electron is replaced by a muon (same charge, but 206.77 times heavier). Where (i.e., at what wavelength) would you look for the "Lyman-α" line (n = 2 → n = 1) (References: Problem5.1 & Equation 4.77) Typically, the interaction potential depends only on the vector r ≡ ... between the two particles. In that case the Schrodinger equation separates, if we change variables from ... (the center of mass). ... ... Get solution

3. Chlorine has two naturally occurring isotopes, ..., where μ is the reduced mass (Equation 5.8) and k is presumably the same for both isotopes. ... Get solution

4. (a) If ...are orthogonal, and both normalized, what is the constant A in Equation 5.10? (b) If ... (and it is normalized), what is A? (This case, of course, occurs only for bosons.) ... Get solution

5. (a) Write down the Hamiltonian for two noninteracting identical particles in the infinite square well. Verify that the fermion ground state given in Example 5.1 is an eigenfunction of H, with the appropriate eigenvalue. (b) Find the next two excited states (beyond the ones in Example 5.1)—wave functions and energies—for each of the three cases (distinguishable, identical bosons, identical fermions). Example 5.1 Suppose we have two noninteracting—they pass right through one another ... never mind how you would set this up in practice!—particles, both of mass in, in the infinite square well (Section 2.2). The one-particle states are ... ... Get solution

6. Imagine two noninteracting particles, each of mass in, in the infinite square well. If one is in the state ... (Equation 2.28), and the other in state ......, assuming (a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions. ... Get solution

7. Suppose you had three particles, one in state ψa(x), one in state ψb(x), and one in state ψc(x). Assuming ψa, ψb, and ψc are orthonormal, construct the three-particle states (analogous to Equations 5.15, 5.16, and 5.17) representing (a) distinguishable particles, (b) identical bosons, and (c) identical fermions. Keep in mind that (b) must be completely symmetric, under interchange of any pair of particles, and (c) must be completely antisymmetric, in the same sense. Comment: There’s a cute trick for constructing completely antisymmetric wave functions: Form the Slater determinant, whose first row is ψa(x1), ψb(x1), ψc(x1), etc., whose second row is ψa(x2), ψb(x2), ψc(x2), etc., and so on (this device works for any number of particles). Get solution

8. Suppose you could find a solution (ψ(r1, r2, …, rZ)) to the Schrödinger equation (Equation 5.25), for the Hamiltonian in Equation 5.24. Describe how you would construct from it a completely symmetric function and a completely antisymmetric function, which also satisfy the Schrödinger equation, with the same energy. Get solution

9. (a) Suppose you put both electrons in a helium atom into the n = 2 state; what would the energy of the emitted electron be?(b) Describe (quantitatively) the spectrum of the helium ion, He+. Get solution

10. Discuss (qualitatively) the energy level scheme for helium if (a) electrons were identical bosons, and (b) if electrons were distinguishable particles (but with the same mass and charge). Pretend these “electrons” still have spin 1/2, so the spin configurations are the singlet and the triplet. Get solution

11. (a) Calculate ... for the state ... (Equation 5.30). Hint: Do the ...integral first, using spherical coordinates, and setting the polar axis along r1, so that ... The θ2 integral is easy, but be careful to take the positive root. You'll have to break the r2 integral into two pieces, one ranging from 0 to r1, the other from r1 to ∞. Answer: 5/4a. (b) Use your result in (a) to estimate the electron interaction energy in the ground state of helium. Express your answer in electron volts, and add it to E0 (Equation 5.31) to get a corrected estimate of the ground state energy. Compare the experimental value. (Of course, we're still working with an approximate wave function, so don't expect perfect agreement). ... Get solution

12. (a) Figure out the electron configurations (hi the notation of Equation 5.33) for the first two rows of the Periodic Table (up to neon), and check your results against Table 5.1. (b) Figure out the corresponding total angular momenta, in the notation of Equation 5.34, for the first four elements. List all the possibilities for boron, car-bon, and nitrogen. ... ... ... Get solution

13. (a) Hund’s first rule says that, consistent with the Pauli principle, the state with the highest total spin (S) will have the lowest energy. What would this predict in the case of the excited states of helium?(b) Hund’s second rule says that, for a given spin, the state with the highest total orbital angular momentum (L), consistent with overall antisymmetrization, will have the lowest energy. Why doesn’t carbon have L = 2? Hint: Note that the “top of the ladder” (ML = L) is symmetric.(c) Hund’s third rule says that if a subshell (n, l) is no more than half filled, then the lowest energy level has J = |L – S|; if it is more than half filled, then J = L + S has the lowest energy. Use this to resolve the boron ambiguity in Problem 5.12(b).(d) Use Hund’s rules, together with the fact that a symmetric spin state must go with an antisymmetric position state (and vice versa) to resolve the carbon and nitrogen ambiguities in Problem 5.12(b). Hint: Always go to the “top of the ladder” to figure out the symmetry of a state. Get solution

14. The ground state of dysprosium (element 66, in the 6th row of the Periodic Table) is listed as 5I8. What are the total spin, total orbital, and grand total angular momentum quantum numbers? Suggest a likely electron configuration for dysprosium. Get solution

15. Find the average energy per free electron (Etot/Nq), as a fraction of the Fermi energy. Answer: (3/5)EF. Get solution

16. The density of copper is 8.96 gm/cm3, and its atomic weight is 63.5 gm/mole. (a) Calculate the Fermi energy for copper (Equation 5.43). Assume q = 1, and give your answer in electron volts. (b) What is the corresponding electron velocity? Hint: Set .... Is it safe to assume the electrons in copper are nonrelativistic? (c) At what temperature would the characteristic thermal energy (d) Calculate the degeneracy pressure (Equation 5.46) of copper, in the electron gas model. ... ... Get solution

17. The bulk modulus of a substance is the radio of a small decrease in pressure to the resulting fractional increase in volume:...Show that B = (5/3)P, in the free electron gas model, and use your result in Problem 5.16(d) to estimate the bulk modulus of copper. Comment: The observed value is 13.4 × 1010 N/m2, but don't expect perfect agreement — after all, we're neglecting ail electron-nucleus and electron-electron forces! Actually, it is rather surprising that this calculation comes as close as it does. Get solution

18. (a) Using Equations 5.59 and 5.63, show that the wave function for a particle in the periodic delta function potential can be written in the form...(Don’t bother to determine the normalization constant C.)(b) There is an exception: At the top of a band, where z is an integer multiple of π (Figure 5.6), (a) yields ψ (x) = 0. Find the correct wave function for this case. Note what happens to ψ at each delta function. Get solution

19. Find the energy at the bottom of the first allowed band, for the case β = 10, correct to three significant digits. For the sake of argument, assume α/a = 1 eV. Get solution

20. Suppose we use delta function wells, instead of spikes (i.e., switch the sign of α in Equation 5.57). Analyze this case, constructing the analog to Figure 5.6. This requires no new calculation, for the positive energy solutions (except that β is now negative; use β = —1.5 for the graph), but you do need to work out the negative energy solutions.... How many states are there in the first allowed band? ... Get solution

21. Show that most of the energies determined by Equation 5.64 are doubly degenerate. What are the exceptional cases? Hint: Try it for N = I, 2, 3, 4....., to see how it goes. What are the possible values of cos(Ka) in each case? ... Get solution

22. (a) Construct the completely antisymmetric wave function ψ (xA, xB, xC) for three identical fermions, one in the state ψ5, one in the state ψ7, and one in the state ψ17.(b) Construct the completely symmetric wave function ψ (xA, xB, xC) for three identical bosons, (i) if all three are in state ψ11, (ii) if two are in state ψ1 and one is in state ψ19, and (iii) if one is in the state ψ5, one in the state ψ7, and one in the state ψ17. Get solution

23. Suppose you had three (noninteracting) particles, in thermal equilibrium, in a one-dimensional harmonic oscillator potential, with a total energy E = (9/2)ħω.(a) If they are distinguishable particles (but all with the same mass), what are the possible occupation-number configurations, and how many distinct (three-particle) states are there for each one? What is the most probable configuration? If you picked a particle at random and measured its energy, what values might you get, and what is the probability of each one? What is the most probable energy?(b) Do the same for the case of identical fermions (ignoring spin, as we did in Section 5.4.1).(c) Do the same for the case of identical bosons (ignoring spin). Get solution

24. Check Equations 5.74, and 5.77, for the example in Section 5.4.1. Get solution

25. Obtain Equation 5.76 by induction. The combinatorial question is this: How many different ways can you put N identical balls into d baskets (never mind the subscript 71 for this problem). You could stick all N of them into the third basket, or all but one in the second basket and one in the fifth, or two in the first and three in the third and all the rest in the seventh, etc. Work it out explicitly for the cases N = 1, N = 2, N = 3, and N = 4; by that stage you should be able to deduce the general formula. ... Get solution

26. Use the method of Lagrange multipliers to find the rectangle of largest area, with sides parallel to the axes, that can be inscribed in the ellipse (x/a)2 + (y/b)2 = 1. What is that maximum area? Get solution

27. (a) Find the percent error in Stirling’s approximation for z = 10.(b) What is the smallest integer z such that the error is less than 1%? Get solution

28. Evaluate the integrals (Equations 5.108 and 5.109) for the case of identical fermions at absolute zero. Compare your results with Equations 5.43 and 5.45. (Note that for electrons, there is an extra factor of 2 in Equations 5.108 and 5.109, to account for the spin degeneracy.) Get solution

29. (a) Show that for bosons the chemical potential must always be less than the minimum allowed energy. Hint: n(∈) cannot be negative. (b) In particular, for the ideal bose gas, μ(T) (c) A crisis (called Bose condensation) occurs when (as we lower T) μ (T) hits zero. Evaluate the integral, for μ = 0, and obtain the formula for the critical temperature Tc at which this happens. Below the critical temperature, the particles crowd into the ground state, and the calculational device of replacing the discrete sum (Equation 5.78) by a continuous integral (Equation 5.108) loses its validity.29 Hint: ... where ... is Euler's gamma function and ... is the Riemann zeta function. Look up the appropriate numerical values. (d) Find the critical temperature for 4He. Its density, at this temperature, is 0.15 gm/cm3. Comment: The experimental value of the critical temperature in 4He is 2.17 K. The remarkable properties of 4He in the neighborhood of TT are discussed in the reference cited in footnote 29. ... ... Get solution

30. (a) Use Equation 5.113 to determine the energy density in the wavelength range ... and solve for .... (b) Derive the Wien displacement law for the wavelength at which the black-body energy density is a maximum: ... Hint: You'll need to solve the transcendental equation ..., using a calculator or a computer; get the numerical answer accurate to three significant digits. ... Get solution

31. Derive the Stefan-Boltzmann formula for the total energy density in blackbody radiation: ... ... Get solution

32. Imagine two noninteracting particles, each of mass m, in the one-dimensional harmonic oscillator potential (Equation 2.43). If one is in the ground state, and the other is in the first excited state, calculate ..., assuming (a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions. Ignore spin (if this bothers you, just assume they are both in the same spin state). ... Get solution

33. Suppose you have three particles, and three distinct one-particle states (ψa(x), ψb(x), and ψc(x)) are available. How many different three-particle states can be constructed, (a) if they are distinguishable particles, (b) if they are identical bosons, (c) if they are identical fermions? (The particles need not be in different states— ψa(x1) ψa(x2)ψa(x3) would be one possibility, if the particles are distinguishable.) Get solution

34. Calculate the Fermi energy for noninteracting electrons in a two-dimensional infinite square well. Let σ be the number of free electrons per unit area. Get solution

35. Certain cold stars (called white dwarfs) are stabilized against gravitational collapse by the degeneracy pressure of their electrons (Equation 5.46). Assuming constant density, the radius R of such an object can be calculated as follows: (a) Write the total electron energy (Equation 5.45) in terms of the radius, the number of nucleons (protons and newtons) N, the number of electrons per nucleon q, and the mass of the electron tn. (b) Look up, or calculate, the gravitational energy of a uniformly dense sphere. Express your answer in terms of G (the constant of universal gravitation), R, N, and M (the mass of a nucleon). Note that the gravitational energy is negative. (c) Find the radius for which the total energy, (a) plus (b), is a minimum. Answer: ... (Note that the radius decreases as the total mass increases!) Put in the actual numbers, for everything except N, using q = 1/2 (actually, q decreases a bit - as the atomic number increases, but this is close enough for our purposes). Answer: .... (d) Determine the radius, in kilometers, of a white dwarf with the mass of the sun. (e) Determine the Fermi energy, in electron volts, for the white dwarf in (d), and compare it With the rest energy of an electron.(Note that this system is getting dangerously relativistic (see Problem 5.36). ... ... ... Get solution

36. We can extend the theory of a free electron gas (Section 5.3.1) to the relativistic domain by replacing the classical kinetic energy, .... (a) Replace ...: in Equation 5.44 by the ultra-relativistic expression, lick, and calculate Eux in this regime. (b) Repeat parts (a) and (b) of Problem 5.35 for the ultra-relativistic electron gas. Notice that in this case there is no stable minimum, regardless of R; if the total energy is positive, degeneracy forces exceed gravitational forces, and the star will expand, whereas if the total is negative, gravitational forces win out, and the star will collapse. Find the critical number of nucleons, No such that gravitational collapse occurs for N > Nc. This is called the Chandrasekhar limit. Answer: .... What is the corresponding stellar mass (give your answer as a multiple of the sun's mass). Stars heavier than this will not form white dwarfs, but collapse further, becoming (if conditions are right) neutron stars. (c) At extremely high density, inverse beta decay, ...,converts virtually all of the protons and electrons into neutrons (liberating neutrinos, which carry off energy, in the process). Eventually neutron degeneracy pressure stabilizes the collapse, just as electron degeneracy does for the white dwarf (see Problem 5.35). Calculate the radius of a neutron star with the mass of the sun. Also Calculate the (neutron) Fermi energy, and compare it to the rest energy of a neutron. Is it reason-able to treat a neutron star nonrelativistically? ... ... Get solution

37. (a) Find the chemical potential and the total energy for distinguishable particles in the three dimensional harmonic oscillator potential (Problem 4.38). Hint: The sums in Equations 5.78 and 5.79 can be evaluated exactly, in this case—no need to use an integral approximation, as we did for the infinite square well. Note that by differentiating the geometric series,...you can get...and similar results for higher derivatives. Answer:...(b) Discuss the limiting case kBT ≪ ħω.(c) Discuss the classical limit, kBT ≫ ħω, in the light of the equipartition theorem.31 How many degrees of freedom does a particle in the three dimensional harmonic oscillator possess? Get solution