Category: Griffith’s QM Solutions

Problem 7.2 For the harmonic oscillator , the allowed energies are where is the classical frequency. Now suppose the spring constant increases slightly: . (Perhaps we cool the spring, so it becomes less flexible.) (a) Find the exact new energies (trivial, in this case). Expand your formula asa power series in , up to second…

Problem 7.1 Suppose we put a delta-function bump in the center of the infinite square well: where is a constant. (a) Find the first-order correction to the allowed energies. Explain why the energies are not perturbed for even n. (b) Find the first three nonzero terms in the expansion (Equation 7.13) of the correction to…

Problem 5.5 (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 , with the appropriate eigenvalue. (b) Find the next two excited states (beyond the ones given in the example)—wave functions, energies, and degeneracies—for each…

Problem 5.4 (a) If and are orthogonal, and both are normalized, what is the constant in Equation 5.17? (b) If (and it is normalized), what is ? (This case, of course, occurs only for bosons.) Solution: Problem 5.4 Solution (Download)

Problem 5.3 Chlorine has two naturally occurring isotopes, and Show that the vibrational spectrum of HCl should consist of closely spaced doublets, with a splitting given by , where is the frequency of the emitted photon. Hint: Think of it as a harmonic oscillator, with where is the reduced mass (Equation 5.15) and is presumably…

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 instead of (b) Find the…

Problem 4.5 Show that satisfies the equation (Equation 4.25), for . This is the unacceptable “second solution”—what’s wrong with it? Solution: Problem 4.5 Solution (Download)

Problem 4.4 Use Equations 4.27, 4.28, and 4.32, to construct and . Check that they are normalized and orthogonal. Solution: Problem 4.4 Solution (Download)

Problem 4.2 Use separation of variables in cartesian coordinates to solve the infinite cubical well (or “particle in a box”): (a) Find the stationary states, and the corresponding energies. (b) Call the distinct energies , in order of increasing energy. Find , and . Determine their degeneracies (that is, the number of different states that…

Problem 4.1 (a) Work out all of the canonical commutation relations for components of the operators r and p: , , , , and so on.Answer: where the indices stand for , , or , and , , and . (b) Confirm the three-dimensional version of Ehrenfest’s theorem, (Each of these, of course, stands for…