Multiple-Choice Problems
017 qmult 00100 1 1 4 easy memory: Fermi, person identification Extra keywords: Fermi, person identification
1. Who was Enrico Fermi?
a) An Italian who discovered America in 1492. b) An Italian who did not discover America in 1492.
c) An Italian-American biologist. d) An Italian-American physicist. e) Author of Atoms in the Family.
017 qmult 00200 1 4 5 easy deducto-memory: golden rule Extra keywords: Sc-288
2. “Let’s play Jeopardy! For $100, the answer is: This quantum mechanical time-dependent perturbation result was discovered by Pauli, but named by Fermi.”
a) What is the categorical imperative, Alex? b) What is the sixth commandment, Alex?
c) What is the no-fault insurance, Alex? d) What is the iron law, Alex?
e) What is the golden rule, Alex?
017 qmult 00300 1 4 5 easy deducto-memory: golden rule validity
3. “Let’s play Jeopardy! For $100, the answer is: This aureate time-dependent perturbation result requires, among other things, that
δElevel separation<< 2π h
− t − tchar
<
∼ ∆Ebandwidth,
where δElevel separation is of order of the separation between energy levels in a continuum band
of energy levels, t − tchar is the time since the perturbation became significant (i.e., tchar), and
∆Ebandwidth is the characteristic energy width of the continuum band.”
a) What is 2nd order perturbation, Alex? b) What is 3rd order perturbation, Alex?
c) What is the optical theorem, Alex? d) What is Pauli’s exclusion principle, Alex?
e) What is Fermi’s golden rule, Alex?
017 qmult 00100 1 1 4 easy memory: exponential decay of state
4. Fermi’s golden rule if it applies to transitions to all states from an original state and for all time after a perturbation is applied (which may be from the time the original state forms) causes the original state to have:
a) no transitions.
114 Chapt. 17 Time-Dependent Perturbation Theory b) a linear decline in survival probability.
c) a power law decline in survival probability. d) an exponential decline in survival probability.
e) an instantaneous decline in survival probability.
017 qmult 00800 1 1 5 easy memory: harmonic perturbation, sinusoidal Extra keywords: harmonic perturbation, sinusoidal time dependence 5. Harmonic perturbations have:
a) a linear time dependence. b) a quadratic time dependence.
c) an inverse time dependence. d) an exponential time dependence.
e) a sinusoidal time dependence.
017 qmult 01200 1 1 1 easy memory: principal value integral
6. One can sometimes in integrate over a first order singularity and get a physically reasonable result. This kind of integral is called:
a) a principal value integral or Cauchy principal value integral. b) an interest integral.
c) a capitol integral. d) a bull-market integral.
e) a bear-market integral.
017 qmult 02000 1 1 5 easy memory: electric dipole selection rules Extra keywords: electric dipole selection rules
7. The selection rules for electric dipole transitions are: a) ∆l = 0 and ∆m = 0. b) ∆l = ±2 and ∆m = ±1. c) ∆l = −1 and ∆m = 1. d) ∆l = ±1 and ∆m = 0. e) ∆l = ±1 and ∆m = 0, ±1.
Full-Answer Problems
017 qfull 00010 1 5 0 easy thinking: time-dependent Sch.eqn.
1. Is the time-dependent Schr¨odinger equation needed for time-dependent perturbation theory? 017 qfull 00020 2 5 0 moderate thinking: energy eigenstates
2. Are stationary states (i.e., energy eigenstates) needed in time-dependent perturbation theory? Please explain.
017 qfull 00030 2 5 0 moderate thinking: energy eigenstates
3. What is done with the radiation field in quantum electrodynamics. 017 qfull 00100 2 3 0 easy math: Fermi’s golden rule integral
Extra keywords: Fermi’s golden rule integral, Simpson’s rule
4. In the simplest version of the derivation of Fermi’s golden rule one uses the integral Z ∞
−∞
sin2x x2 dx = π
Chapt. 17 Time-Dependent Perturbation Theory 115 which can be evaluated using complex variable contour integration (Ar-364). One of the features of this integral that is used in the justification of the golden rule is that most of the total comes from the central bump of the integrand: i.e., the region [−π, π]. It would be good to know what fraction of the total comes from the central bump. Alas,
Icen =
Z π
−π
sin2x x2 dx .
is not analytically solvable.
a) Find an excellent approximate value for Icen. HINTS: It’s probably no good trying to
find a good approximation for the central bump directly since it is most of the total. An approximate value could easily turn out to be off by of order |Icen− π|. Try finding a value
for the non-central bump region.
b) Now—if you dare—evaluate Icen numerically and compare to your analytic result from
part (a). HINT: I use double precision Simpson’s rule myself. 017 qfull 00200 3 3 0 tough math: time dependent perturbation, square well
Extra keywords: (MEL-141:5.3), time dependent perturbation, infinite square well
5. At time t = 0, an electron of charge ˜e is in the n eigenstate of an infinite square well with potential
V (x) =n0, x ∈ [0, a]; ∞ x > a.
At that time, a constant electric field ˜E pointed in the positive x direction is suddenly applied. (Note the tildes on charge and electric field are to distinguish these quantities from the natural log base and energy.) NOTE: The 1-d infinite square-well eigenfunctions and eigen-energies are, respectively ψn(x) = r 2 asin nπ a x and En = h −2 k2 2m = h −2 2m π a 2 n2,
where n = 1, 2, 3, . . . The sinusoidal eigenfunctions can be expressed as exponentials: let z = πx/a, and then
sin(nz) = e
inz− e−inz
2i .
a) Use 1st order time-dependent perturbation theory to calculate the transition probabilities to all OTHER states m as a function of time. You should evaluate the matrix elements as explicitly: this is where all the work is naturally.
b) How do the transition probabilities vary with the energy separation between states n and m?
c) Now what is the 1st order probability of staying in the same state n? 017 qfull 00300 3 5 0 tough thinking: usual and general Fermi’s golden rule
6. Say we have time-dependent perturbation
H(t) = 0, t < 0; H, t ≥ 0,
and initial state |φji, where |φji is the eigenstate belonging to the complete set {|φii}. The
116 Chapt. 17 Time-Dependent Perturbation Theory
a) Work out as far as one reasonably can the 1st order perturbation expression for the coefficient ai(t) in the expansion of |Ψ(t)i in terms of the set {|φii}. Include the case
of i = j. HINT: The worked out expression should contain a sine function. Define ωij = (Ei− Ej)/ h−.
b) Given i 6= j, find the transition probability (to 1st order of course) from state j to state i. c) What is this probability at early times when ωijt/2 << 1 for all possible ωij? Describe the
behavior of the probability as a function of time for all times. (You could sketch a plot of probability as a function of time.) What is the behavior for ωij= 0 (i.e., for transitions to
degenerate states)?
d) You want to calculate the summed probability of transition to some set of states (which may not be all possible states) that are dense enough in energy space to form a quasi- continuum or even a real continuum of states. The set does not include the initial state j. The summed probability for energy interval Ea to Eb can be approximated by an integral:
P (t) =X i6=j Pi(t) ≈ Z Eb Ea P (E, t)ρ(E) dE ,
where ρ(E) is the density of states per unit energy and where the time-independent part of the matrix element Hij is replaced by H(E) which is a continuous function of energy.
What is the total transition probability to all states in the set assuming integrand is only significant in small region near Ej. The region is small eneough that |H(E)|2 and
ρ(E) can be taken as constants and that the limits of integration can be set to ±∞. (Note you will probably need to look up a standard definite integral.)
In fact, 90 % of the integral (assuming |H(E)|2 and ρ(E) constant) comes from the
energy range [Ej− 2π h−/t, Ej+ 2π h−/t]. (Can you show this by a numerical integration?
No extra credit for doing this: insight is the only reward.) We can see that at some time the 90 %-range will be so narrow that the approximation |H(E)|2and ρ(E) constant
will probably become valid. They should clearly be evaluated at Ej. Practically, this
often means that the approximation becomes valid when almost all of the transitions are to nearly degenerate states. Of course, the 90 %-range can become so narrow that the approximation of a continuum of states breaks down and then the integration becomes invalid again. In fact, for the integration to be valid we require |H(E)|2 and ρ(E) to be
constant over ∆E such that ∆E>
∼ 2π h−/t and that the energy separation δE between the final satisfy δE << 2π h−/t. Thus we require
δE << 2π h−/t<
∼ ∆E .
What is the rate of transition for (i.e., time derivative of) the total transition probability? The transition rate result is one of the usual forms of Fermi’s golden rule. Although it is restricted in many ways, it is still a very useful result: hence golden. e) Let’s see if we can derive a generalized golden rule without the restriction that the
perturbation is constant after a sudden turn-on. To do this assume that the perturbation Hamiltonian has the form
H(t) = Hf (t) ,
where H is now constant with time and f (t) is a real turn-on function with the property that f (t) is significant only for t ≥ tch, where tchis a characteristic time for turn-on. Let
time zero be formally set to −∞ for generality.
First, derive Pi(t) with explicit integrals. Second, assume again that there is a continuum
Chapt. 17 Time-Dependent Perturbation Theory 117 can be replaced by H(E). Third, argue that the time integrals must be sharply peaked functions of E about the initial E = Ej for t ≥ tch. Fourth, re-arrange the integrals and integrate over
energy making use of the third point. You can then make use of the result
δ(x) = Z ∞
−∞
e±ikx
2π dk ,
where δ(x) is the Dirac delta function (Ar-679). What is the total transition probability for t ≥ tch? What is the total transition rate t ≥ tch? When does this generalized golden rule