1、consider a three-level system with the unperturbed hamiltonian and the perturbation since 2x2 matrix w is diagonal (and in fact idertically 0) in the basis of states (1,0,0) and (0,1,0),you might assume they are the good states, but they're not. to see this: (a) obtain the exact eigenvalues for the perturbed hamiltonian . (b) expand your results from part (a) as a power series |v| in up to second order. (c) what do you obtain by applying nondegenerate perturbation theory to find the energies of all three states (up to second order)? this would work if the assumption about the good states above were correct. moral: if any of the eigenvalues of are equal, the states that diagonalize w are not unique, and diagonalizing w does not determine the “good” states. when this happens (and it’s not uncommon), you need to use second-order degenerate perturbation theory.
2、when an atom is placed in a uniform external electric field , the energy levels are shifted—a phenomenon known as the stark effect (it is the electrical analog to the zeeman effect). in this problem we analyze the stark effect for the and states of hydrogen. let the field point in the z direction, so the potential energy of the electron is treat this as a perturbation on the bohr hamiltonian (equation 6.42). (spin is irrelevant to this problem, so ignore it, and neglect the fine structure.) (a) show that the ground state energy is not affected by this perturbation, in first order. (b) the first excited state is four-fold degenerate: using degenerate perturbation theory, determine the first-order corrections to the energy. into how many levels does split? (c) what are the “good” wave functions for part (b)? find the expectation value of the electric dipole moment (), in each of these “good” states. notice that the results are independent of the applied field—evidently hydrogen in its first excited state can carry a permanent electric dipole moment.
chapter 7 the variational principle
chapter 7 exercise
1、(a) use the function to get an upper bound on the ground state of the infinite square well. (b) generalize to a function of the form for some real number p. what is the optimal value of p, and what is the best bound on the ground state energy? compare the exact value. note: the integral formula:
2、suppose we want to find the ground state energy for the one-dimensional non-harmonic oscillator we might pick as our "trial" wave function the guassian , where b is an adjustable parameter. find the best bound on the ground state.
chapter 9 time-dependent perturbation theory
chapter 9 exercises
1、consider a perturbation to a two-level system with matrix elements where and are positive constants with the appropriate units. (a) according to first-order perturbation theory, if the system starts off in the state at , what is the probability that it will be found in the state b at ? (b) in the limit that , . compute the limit of your expression from part (a) and compare the result of problem 9.3. (c) now consider the opposite extreme: . what is the limit of your expression from part (a) ?
2、consider a particle in an infinite well, with and everywhere else. the potential in the range changes by an additional term (a) calculate the probability that a particle in the ground state (n=1) makes transition to the first excited state (n=2). (b) what is the probability that it makes a transition to the second excited state (n=3)? (c) what happens to these results as ? note: the integral formula .
chapter 11 scattering
chapter 11 exercises
1、let us consider s-wave (l=0) scattering of low energy particles. determine the total cross-section for the scattering of low energy particles by the attraction potential is a constant. hint: with partial wave method to find phase shift . in low energy case, , and
2、suppose the scattering potential using born approximation to determine the differential cross-section and the total cross-section. hint: this is a non-low approximation case.