scattering review

8/16/2019 Scattering review

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Ph125c Wednesday 25 April 2007 - Midterm EXAM review

Radial equation

For central forces (spherically-symmetric potentials) we may assume radial separationof the energy eigenfunctions,

r ,, R E r Y lm,.

Here Y lm, is a spherical harmonic and R E r will be referred to as a ‘radial function.’

As a result of this factorization, the 3D Schrodinger Equation may be reduced to aone-dimensional radial equation

− 2

2mr 2d

dr r 2 d

dr

ll 12

2mr 2 V r R E r E R E r .

Using the operator equivalence

d

dr r 2 d

dr R 1

r

d 2

dr 2 r R,

one can perhaps appreciate that it may be convenient to introduce

ur ≡ r R E r .

This new radial function satisfies the simpler radial equation

− 2

2mr 2d

dr r 2 d

dr

ll 12

2mr 2 V r

ur r E

ur r ,

− 2

2m1r

d 2

dr 2 r

ur r

ll 12

2mr 2 V r

ur r E

ur r ,

− 2

2m

d 2u

dr 2

ll 12

2mr 2

V r ur E ur .

Again, this now looks exactly like a 1D Schrodinger Equation with the addition of a‘centrifugal’ potential term ll 12/2mr 2. It is interesting to note that for any potentialV r whose behavior near the origin follows a power law

V r r

with ≥ −1 the centrifugal term will clearly dominate as r → 0 unless l 0. As a result,we expect expulsion of the wavefunction from the origin 0 0 for any positivevalue of l.

Examples we have studied in class: free particle, spherical square well, hardsphere, Coulomb potential.

Free particle solutions

In the absence of a real potential energy term (as opposed to the centrifugal one), itturns out that the radial equation can be morphed into

d 2 J d 2

1

dJ d

1 − l 1/22

2 J 0,

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which is Bessel’s equation. The solutions of this equation that are well-behaved atr 0 are the cylindrical Bessel functions J l1/2. Hence the radial components of thestationary wave functions can be written

R C jl

where C is a normalization constant and j l is a spherical Bessel function

jl ≡ 2 J l1/2.

Thus we end up with free-particle solutions

klmr C jlkr Y lm,,

where

E 2k 2

2m .

It should be noted that these energy eigenstates for the free particle (like plane waves)are not normalizable, as the asymptotic form of the spherical Bessel functions is

jl ≃ cos

− l

1

/2

, l

and we thus will have an integrand in the normalization integral that goes like

2d | R|2 d cos2.

As you probably know from E&M, Bessel functions are quite common in physicsand engineering so most computer mathematics packages have built-in routines togenerate them. For example in Matlab a few lines of code suffice to generate thefollowing plot of j l, for l 0 (black), l 1 (red), l 5 (blue) and l 10 (green):

0 2 4 6 8 10 12 14 16 18 20-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

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A few things to note are that only l 0 has a non-zero value at the origin(singlevaluedness of the wave-function) and that the initial maximum of j l movesto higher as l increases (centrifugal force).

Question: How are these solutions related to plane-waves,

k r N exp ik r .

Answer: in our discussion of partial wave scattering we saw the equation

eikz ∑l0

i l 42l 1 j lkr Y l0.

Note that since all terms in the sum on the RHS involve a constant wave-number k ,they are all degenerate in energy. We thus see that going from plane-waves tospherical Bessel functions is a change of basis within the degenerate subspace of

states with fixed k . We do this in order to obtain simultaneous eigenstates of p 2/2m,

L2 and L z.

The spherical square well potential

Next we move on to the 3D version of a potential well, with V r −V 0 for r a andV r 0 for r a (with V 0 0). Just as in the 1D case, we must solve the radialequation within regions of constant V r and then match solutions at the boundary:

− 2

2mr 2d

dr r 2 dR

dr

2ll 1

2mr 2 R E V 0 R for r a,

− 2

2mr 2d

dr r 2 dR

dr

2ll 1

2mr 2 R E R for r a.

Bound states of the spherical square well will have −V 0 ≤ E ≤ 0.

Inside the well region r a we can use the above results with E → E V 0,

Rr A jl2m E V 0

2 r for r a.

Outside the well r a we should be more careful. Since we are now dealing with abound state problem, we should expect the eigenstates to be normalizable. The trickis that Bessel’s equation admits additional solutions besides the j l, which aresingular at the origin but may be admissible in a region that does not include r 0.For a r ≤ , the radial solutions of choice are spherical Hankel functions of the firstkind. Recall that for bound states we have E 0 and therefore outside the well

k i −2mE

2 ≡ i,

and the corresponding solutions are

Rr B hl1

i −2mE 2

r .

As in the 1D case we match solutions at the boundary r a. Note that we have atotal of three constraints (continuity of , continuity of ′, overall normalization) in two

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unknowns ( A and B). Hence we can match the logarithmic derivative to find the energyspectrum and worry about normalization separately. This leads to

2 − 2

i

jl′ 2 − 2 a

jl 2 − 2 a

hl′1

ia

hl1

ia

as an equation for the allowed discrete energy eigenvalues, where 2 ≡ 2mV 0/2.See HW#1 solutions for more...

Scattering from a hard sphere

Here we look at scattering from a ‘hard sphere’ potential defined by

V r r a,

0 r a.

In the HW you had to compute the differential and total scattering cross sections under

the assumption that the energy of the incident particle E k

2k 2

2mis so low that ka 1, and hence that only the s-wave (l 0) term contributessignificantly in the partial wave expansion

f k 1

k ∑

l0

42l 1 expilsinl Y l0.

For l 0 the radial equation in the region r a simply reads

− 2

2md 2udr 2

E ur .

We see that this is quite simply solved by

ur A exp i 2mE 2

r B exp −i 2mE 2

r ,

where A and B are constants that determine the relative magnitude and phase of theincoming and outgoing spherical waves. Since the incoming wave reflects off of aninfinite barrier we expect | B| | A|, so we may write

ur exp −i 2mE 2

r − exp2i0exp i 2mE 2

r

exp−i0exp −i 2mE 2

r − expi0exp i 2mE 2

r

sin 2mE 2

r 0 ,

which defines the scattering phase-shift 0. The boundary condition then determines

0 −a 2mE 2

→ −ka,

where E 2k 2/2m. In terms of this, we have

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f k 1k

4 exp−ika sin−kaY 00

− exp−ika sinka

k ,

, | f k |2

sin2ka

k 2 ,

tot 4 sin 2ka

k 2 .

In the limit ka 1,

tot → 4a2,

which is simply the geometric cross-sectional area of the hard-sphere potential region.

The above calculation should remind you a lot of problems we did in Ph125b under theheading “sectionally constant potentials in 1D.” The main difference is that, for the

radial equation, there can never be more than one free limit. Last term we alsomentioned, but did not explicitly show, that stationary scattering states can besuperposed to make wave packets that display the kind of scattering ‘dynamics’ onewould intuitively hope for. The dynamics happens automatically as a consequence of the structure of the stationary scattering states.

Wave-packet dynamics from stationary scattering states

Consider a simple (negative) potential step,

V x 0, x ≤ 0,

−V 0, x 0.We saw last term that the stationary scattering states, for a particle incident from theleft, are

x expikx − r exp−ikx, x ≤ 0,

t expik ′ x, x 0,

k 2mE /,

k ′ 2m E V 0 /,

where in order to keep things interesting we consider E 0. It is a simple matter toderive the refelction and transmission coefficients, as we have seen before. Theboundary matching conditions are

1 − r t , ik ikr ik ′t ,

from which we get

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k 1 r k ′1 − r ,

r k ′ − k

k ′ k ,

t 1 − r 2k k ′ k

.

Suppose we want to consider an initial gaussian wave-packet (at t 0) centeredfar to the left of the origin (compared to its width) but travelling towards the right:

x, t 0 22−1/4 exp − x − 2

42 expik 0 x.

This has 〈 x , Δ x , and 〈 p k 0. We need to decompose this in stationaryscattering states:

x, 0 → 0

dE ̄ L E E , L x,

where ̄ L E are the coefficients of stationary scattering states with energy E , incidentfrom the left. We can obtain the coefficients for our desired state by taking inner

products:

̄ L E −

dx E , L

∗ x x, 0

22−1/4

−

0

dx exp − x − 2

42 expik 0 − k x

− r 22−1/4 −

0

dx exp − x − 2

42 expik 0 k x

t 22−1/4

0

dx exp

− x − 2

42 expik 0 − k ′ x.

In order to evaluate these coefficients we resort to numerical computation. Once wehave them, we can evolve our state forward in time via

x, t 0

dE ̄ L E E , L xexp−iEt /,

and see what we get (MATLAB images in supplemental material for 4/11).

Question: suppose we limit ourselves to an s-wave scattering scenario. What isthe closest analog of the above 1D problem?

Scattering cross-sections from wave-functions

In 3D (as opposed to 1D) we want to calculate not only transmission and reflectioncoefficients, but an entire differential cross-section. With an incident flux F i, thenumber of particles (per unit time) dn that will be scattered into solid angle d aboutthe direction , may be expressed

dn F i ,d ,

where , is known as the differential scattering cross-section. The integral of ,d ,

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tot ≡ d ,is known as the total scattering cross-section.

In the quantum theory of scattering, we compute , (and thus tot ) fromsolutions of the Schrödinger Equation with potential V r . The idea is to have an

incident wave packet, initially (for large negative times) localized in the region of largenegative z and travelling towards the coordinate origin. As t approaches 0 this wavepacket encounters the scattering potential centered at r 0, and becomes distorted byit. At late times (t → ) the particle will have left the potential region and once againpropagates freely, but with modified wave function. In general the post-scatteringwave function will contain both a ‘transmitted’ wave packet that continues to propagatetowards z , and a ‘scattered’ component that radiates outwards from the potentialregion – think about water waves that encounter a scattering object, and theappearance of the resulting ripples [Merzbacher Fig. 13.2]:

Note that while one tends to think about the scattering ‘process’ in time-dependentterms, we can actually formulate everything in stationary terms! For a given scatteringpotential V r an arbitrary time-dependent solution of the Schrödinger Equation maybe written

r , t dk ck k r exp−iE k t /,where k is a continuous index for the eigenfunctions k r that satisfy

− 2

2m ∇ 2 V r k r E k k r .

Assuming we can find such stationary solutions for a given V r , we could constructan overall solution by choosing the coefficients ck such that r , t at large negative t looks like the desired incident wave packet, and then simply allowing the oscillatingphase factors to do their thing at all subsequent times! Note that we are guarenteed tobe able to do this as the k r form a complete basis.

It can be shown (Merzbacher §13.3) that for any V r with finite spatial extent,stationary solutions k r can be found that have the asymptotic r → form

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k r r → expikz f k , e

ikr

r .

Here only the function f k , depends on V r , and the common term expikz ischosen to facilitate the representation of a wave packet impinging along the z axis for t → −. These k r thus have the (un-normalized) form of a ‘transmitted’ plane wave

expikz propagating towards z → plus an outgoing spherical wave with angular distribution f k , determined by the form of the potential. It can finally be shown that

, | f k ,|2,

so our task reduces to computing the angular distributions f k , for a given potentialV r .

Scattering by a central potential; partial waves

For the free particle we have seen that the radial functions correspond to sphericalBessel functions, which satisfy

uk ,lr r → C sin kr − l

2 (free particle).

In quantifying the degree to which a given potential ‘distorts’ the particle wave packet,it is convenient to take this as a reference point and define scattering phase shifts laccording to

uk ,lr r → C sin kr − l

2 l (scattering by V r ).

In terms of these, we may write

k ,l,mr r → −Y l

m, exp−ikr expil/2 − expikr exp−il/2 2il

2ikr ,

which looks like the sum of an incoming free spherical wave and phase-shiftedoutgoing spherical wave. It can be shown from this that for an incident plane wave,

eikz ∑l0

i l 42l 1 j lkr Y l0,

we have

f k , 1k ∑

l0

42l 1 expilsinlY l0.

Here we have written Y l0 to emphasize that spherical harmonics with m 0 have no

dependence on .

In considering this summation over l, it is important to recall the behavior of freespherical waves near the origin. The general property of Bessel functions

jl→ 0

l

2l 1!!

results in negligible probability density ||2 for ≡ kr

r 1k

ll 1 .

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As a result, for very small k (low relative kinetic energy between the interactingparticles) only terms with small l will contribute to the above summation for f k . In theextreme, some low-temperature processes in solid-state and atomic physics involveonly s-wave (l 0) contributions to the scattering cross-section!

Coulomb potential

Moving on to the specific case of the Coulomb potential V r r −1, we can take intoaccount what we know about the expected asymptotic behaviors of u by setting

u l1 exp−w,

and derive the corresponding radial equation for w :

d 2wd 2

2 l 1 − 1 dw

d

V E

− 2l 1

w 0,

d 2wd 2

2 l 1 −

dwd

0 − 2l 1

w 0,

w′′ 2l 1 − w′ 0 − 2l 1w 0.

Plugging in a power-series expansion

w ∑k 0

ak k

leads to the recursion relation

ak 1ak

2k l 1 − 0k 1k 2l 2

.

Just as we did earlier this term with the harmonic oscillator, it is possible to show thatthis series must truncate in order to avoid divergent behavior as →

. It follows that

w is a polynomial of finite order. If we let N denote the order of some particular w, we have

a N ≠ 0, a N 1 0.

Hencea N 1a N

2 N l 1 − 0 0,

0 2 N l 1.

Recall that we defined 0 to be a constant

0 ≡ V

E ,

where V r −1 and therefore V | E | −1. Thus,

0 | E |−1/2

and we may conclude that

E 1 N l 12

.

For this reason,

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n ≡ N l 1

is conventionally chosen as the third quantum number (together with l and m of thespherical harmonic), often called the principal quantum number, to specify a givenenergy eigenfunction. It is important to note that the energy eigenvalues E n−2 willgenerally be degenerate, having both 2l 1-fold essential degeneracies as well as

n-fold accidental degeneracies. The former factor comes from the fact that the energyof a state evidently does not depend on the z-projection of oribital angular momentumm. Hence the essential degeneracy is a consequence of rotational symmetry and iscommon to all central-force problems. On the other hand, Coulomb potentialsV r r −1 have the additional property that certain eigenfunctions that differ in boththeir total angular momentum l and in the polynomial order N of their radial functionsw never-the-less evaluate to exactly the same energy if their n quantum numbersare identical. For a given n, the overall degree of degeneracy is

∑l0

n−1

2l 1 n2.

Hydrogenic atoms

Any atom with a single valence electron (alkali atoms such as Li, K, Na, Rb, etc.; alsothings like ‘hydrogenlike uranium,’ which has a nucleus with large positive charge andhas been modified to have only one electron) will have a similar energy-level structureto that of hydrogen. Let us write the net nuclear charge as Ze and the nuclear mass asmn, where −e is the elementary charge of an electron. For hydrogen Z 1.

The hydrogenic energy spectrum is given by

E n − Z 2me4

22

n2

≡ − E I /n2.

where E I ≈ 13.6 eV for actual hydrogen ( Z 1). Note that the ground (lowest) energylevel corresponds to n 1. Evidently there are infinitely many bound levels, whosespacing becomes more and more dense as E n → 0 (as n → ). As mentioned above,each “shell” of given n contains n sub-shells of differing angular momentuml 0… n − 1 [Cohen-Tannoudji et al, Chapter VII Fig. 4]:

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You will often see the lower values of l associated with the spectroscopic notations

l 0 ↔ s

l 1 ↔ p

l 2 ↔ d

l 3 ↔ f

l 4 ↔ g

l … ↔ (alphabetical).

Just as was the case with the harmonic oscillator, the Coulomb potential has a naturallength-scale. Dimensionally, one finds that the ‘Bohr radius’

a 2

me2

should set the natural unit of length for hydrogenic atoms. Indeed it turns out that theground-state wave-function with n 1, l 0, m 0 can be written

1,0,0r ,, Z 3

a3

1/2

exp − Zr a ,

and

〈r 4 Z 3

a 0

exp − 2 Z a r r

3dr 32

a Z

.

The maximum of

r 2 d |r ,,|2 r 2 exp − Z a r 2

(the probability density to find the electron at radius r from the nucleus – do you seewhy?) actually occurs at a/ Z . It is sometimes useful to note that in terms of the Bohr

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radius,

E n − Z 2e2

2an2 , Z na .

For hydrogen a ≈ 0.529 10−8 cm. Note that a is inversely proportional to the reducedmass, so for a heavier orbiting particle (say a muon instead of an electron) the Bohr

radius will be even smaller!

Fine and Hyperfine Structure

In order to obtain a more accurate picture of hydrogenic energy spectra we must addthree things to the simplest model based only on the Coulomb potential:

1. relativistic effects

2. electron spin

3. nuclear spin

The first two items combine to contribute what are known as the fine-structure terms in

the atomic Hamiltonian, while addition of the third produces hyperfine terms. A correct relativistic description of the hydrogen atom (via the Dirac equation) can

be used to derive leading-order (in ) corrections to the basic Coulomb Hamiltonian:

H mec2 p2

2me V r −

p4

8me3c2

12me2c2

1r

dV r

dr L S

2

8me2c2 ∇ 2V r

≡ mec2 H 0 W mv W SO W D .

It should be noted that here m e refers to the bare electron mass rather than a reducedmass (Cohen-Tannoudji et al. mention that there is some significant complication withtwo-body systems in relativistic theory, on p. 1214). The rest-mass energy term mec2

clearly shifts all energy levels equally, and therefore has no effect on the hydrogen

spectrum per se. The next term in square brackets reproduces the CoulombHamiltonian. The remaining terms can be assigned specific physical interpretations.

With the spectroscopic notation nL j, we for example discussed the fine-structureenergy corrections

Δ E 2s1/2 − 5128

mc 24,

Δ E 2 p1/2 − 5128

mc 24,

Δ E 2 p3/2 − 1128

mc 24.

[Cohen-Tannoudji et al Ch. XII Fig. 2]

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The hydrogen nuclear magnetic moment operator may be written

I

g p N

I,

where I is the usual spin-1/2 vector operator, g p ≈ 5.585 is the proton “g-factor”(having to do with its internal quark structure) and N is the “nuclear magneton”

N q p

2m p

(where q p −qe). It should be noted that N is smaller than the electron’s “Bohr magneton”

B qe

2meby a factor of m e/m p 1/2000. The electron’s magnetic moment operator is

S

ge B

S,

with g e 2.

The hyperfine Hamiltonian is given by

W hf − 04

q

mer 3 L

I

1r 3

3 S

n̂ I

n̂ − S

I 8

3

S

I r ,

where n̂ is the unit vector r/|r|. It can be shown that the first two terms here are of order 2000 times smaller than W SO, and the third term is likewise about 2000 timessmaller than W D (which also contains a delta-function). The first term of W hf reflectsthe interaction of the nuclear magnetic moment with the magnetic field 0/2qL/mer 3 created by the orbiting electron. The second term represents themagnetic dipole-dipole interaction between the nuclear and electronic spins. The thirdterm, known as Fermi’s ‘contact term,’ has to do with the internal magnetic structure of the proton.

The important thing to note about the hyperfine Hamiltonian is that it leads us toconsider coupling of angular momenta between the nuclear spin I and the total

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electron angular momentum J , yielding

F I J.

The rationale for this coupling scheme (as opposed, e.g., to coupling S and I first andthen adding L) is that the spin-orbit coupling is so much larger than the hyperfineinteraction. As a result, in the absence of an applied magnetic field, W hf leads to small

additional splittings within the nL j fine-structure levels [Cohen-Tannoudji et al, Ch. XIIFigs. 3 and 4]:

( A/2 ≃ 1420405751.768 0.001 Hz, corresponding to the famous 21 cm line in

hydrogen).

Zeeman effect of the 1s ground state hyperfine structure

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[Cohen-Tannoudji, Diu, and Laloe Ch. XII section E]

In this lecture we’ll see our first good physical example of a situation where “goodquantum numbers go bad” as a function of some external parameter. The basic ideato keep in mind is that we’ll have an overall Hamiltonian

H H 0 W ,

where H 0 and W have comparable eigenvalues, we suppose H 0, W ≠ 0, and is areal-valued scalar parameter that can be varied by changing experimental conditions.When 1, we can assume that the eigenstates of H are essentially those of H 0,with W merely contributing small energy corrections. Therefore the eigenvalues of H 0are a good quantum number. On then other hand when 1, we expect to be able totreat H 0 as a perturbation on W . Then the eigenvalues of W should be a goodquantum number in the regime of large .

For the Zeeman effect we will consider H 0 to be the hydrogen Hamiltonian withfine and hyperfine couplings, for which n, j, F are the good quantum numbers (recallthat the 2s1/2 − 2 p1/2 splitting is strictly due to the Lamb shift, which does not come fromany Hamiltonian we have considered!). Now we are going to add a further perturbationassociated with an externally-applied magnetic field. The overall form of the ZeemanHamiltonian should be pretty familiar by now,

W z − B 0 l s I .

[Before moving on let us note that in principle there ought to be an additional term,quadratic in B 0, in the Zeeman Hamiltonian – see Cohen-Tannoudji et al. p.1233 for

more details.] Here B 0 is the applied magnetic field (created by some sort of laboratorymagnet), and the ’s are magnetic-moment operators. We already know that

s

qge2me

S, I

− qg p2m p

I,

where g e 2 and g p ≈ 5.585. Just as with any current ‘loop,’ there is also a magneticmoment associated with the orbiting electron charge:

l

q

2meL.

(Comparison of this expression with the ones for s, I may help motivate the definition

of g factors). If we simply take the z coordinate axis to coincide with the orientation of

the applied magnetic field B 0, we may simplify

W z 0L z 2S z N I z

B0,

where B 0 B0 ẑ and thus

0 − q2me B0,

N q

2m pg p B0,

are the Larmor frequencies. Let’s begin by considering the weak-field case where B 0 isassumed to be small.

Note that N 0 since m p me. In order to simplify drastically the followingdiscussion, we will henceforth neglect the N I z term. In the 1s shell, we can likewise

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drop the 0L z term since l 0 for these states. Hence we are left considering theeffect of the perturbation

W Z → 02S z

on the set of states

|n 1 , j 1/2, I 1/2; F 1 , m F 1,

| n 1 , j 1/2, I 1/2; F 1 , m F 0,

| n 1 , j 1/2, I 1/2; F 1 , m F −1,

| n 1 , j 1/2, I 1/2; F 0 , m F 0.

As only the last (F 0) state differs in energy from the rest, we should begin by tryingto diagonalize S z in the F 1 manifold.

Using our favorite method (e.g. a table of Clebsch-Gordan coefficients) we canwrite the F 1, mF basis states in terms of the eigenstates of S z,I z for s 1/2 and I 1/2 :

| F 1 , m F 1 1

2 s ⊗

1

2 I ,

| F 1 , m F 0 12

12 s

⊗ − 12 I

− 12 s

⊗ 12 I

,

| F 1 , m F −1 − 12 s⊗ − 1

2 I .

It is then easy to verify that

S z| F 1 , m F 1 2 | F 1 , m F 1,

S z| F 1 , m F 0 12

2

12 s

⊗ − 12 I

− 2

− 12 s

⊗ 12 I

2 | F 0 , m F 0,

S z| F 1 , m F −1 − 2 | F 1 , m F −1.

Within the F 1 manifold then, we see that

S z

2

1 0 0

0 0 0

0 0 −1

and hence is already diagonal.

In the F

0 manifold,S z| F 0 , m F 0 S z 1

2

12 s

⊗ − 12 I

− − 12 s

⊗ 12 I

1

2

2

12 s

⊗ − 12 I

2 − 1

2 s⊗ 1

2 I

2 | F 1 , m F 0.

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|n 1 , l 0 , j 1/2, I 1/2; m j 1/2, m I 1/2 ,

|n 1 , l 0 , j 1/2, I 1/2; m j 1/2, m I −1/2 ,

|n 1 , l 0 , j 1/2, I 1/2; m j −1/2, m I 1/2 ,

|n 1 , l 0 , j 1/2, I 1/2; m j −1/2, m I −1/2 .

The effect of the Zeeman term

W Z 02S z

is very simple to treat exactly, since with l 0 the simultaneous eigenstates of J 2,J zare the same as those of S 2,S z (since J L S S). Hence the energies of the basisstates (which start out being degenerate since we have not yet applied hyperfinecouplings) shift according to

|n 1 , l 0 , j 1/2, I 1/2; m j 1/2, m I 1/2 ↔ 0,

|n 1 , l 0 , j 1/2, I 1/2; m j 1/2, m I −1/2 ↔ 0,

|n 1 , l 0 , j 1/2, I 1/2; m j −1/2, m I 1/2 ↔ −0,

|n 1 , l 0 , j 1/2, I 1/2; m j −1/2, m I −1/2 ↔ −0.

Next we must add the hyperfine term as a perturbation to this basis. As we currentlyhave two 2-fold degenerate subspaces (corresponding to m j 1/2), we should againstart by diagonalizing...

As discussed above, in the 1s shell the hyperfine Hamiltonian may be written

W hf AS I

AS x I x S y I y S z I z

A S z I z 14 S S − I I − − 14

S − S − I − I −

A S z I z 12 S I − S − I .

Considering only the restriction W ̄ hf of W hf to the degenerate subspaces withwell-defined m j (for the purposes of degenerate perturbation theory), the S and S −terms may be ignored since they only connect states of different m s m j. Hence,

W ̄ hf → AS z I z,

and we again find that there is no need to change bases. As a result, we have theenergy spectrum

|n 1 , l 0 , j 1/2, I 1/2; m j 1/2, m I 1/2 ↔ 0 A2

4 ,

|n 1 , l 0 , j 1/2, I 1/2; m j 1/2, m I −1/2 ↔ 0 − A2

4 ,

|n 1 , l 0 , j 1/2, I 1/2; m j −1/2, m I 1/2 ↔ −0 − A2

4 ,

|n 1 , l 0 , j 1/2, I 1/2; m j −1/2, m I −1/2 ↔ −0 A2

4 .

Under the assumption A2 0 (valid in the regime of large B 0), we thus have thefollowing diagram [C-T et al, Ch. XII Fig. 7]:

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Here the expressions are really only valid in the region with solid lines, but the dashedlines provide extrapolations of the first-order perturbation results back to B 0 0. Weknow from results above, of course, that the real energy diagram for small B 0 shouldhave a singlet and a triplet!

As it turns out, it is not so bad to consider the overall term

W hfZ W hf W Z

as a whole, and to use it directly as a single perturbation term to the fine-structure (doyou see why this is different from what we have done so far?). This removes allapproximations about the relative size of B 0 and the hyperfine term, while still treatingthe combination of the two as a perturbation on H 0 W f . The resulting energy diagramlooks like this [C-T et al, Ch. XII Fig. 9]:

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Time-dependent perturbation theory

Say we have a time-dependent Hamiltonian

H t H 0 W t .

There will not generally be stationary states for the corresponding Schrödinger Equation, because of the explicit time-dependence. To solve something like aninitial-value problem then, we must resort to actual integration of the S.E. since theusual trick of decomposition into energy eigenstates and multiplication by oscillatingphase factors does not apply. In principle, H t still admits a time-developmentoperator

i d dt

| H t | ,

| t T t , t 0| t 0 ,

but obtaining an explicit form for T t , t 0 can be arbitrarily difficult because of theexplicit time-dependence of W t . So in general we need approximation methods.

Under the assumptions 1 and H 0 W t , we may be able to use perturbationtheory. The spirit of time-dependent perturbation theory is to consider W t as a weak“driving” term that induces transitions among the eigenstates of H 0 – recall that sinceW t is time-dependent there (usually) isn’t really a sense in which we could thinkabout leading-order corrections to the eigenspectrum. (One exception to this is whenW t is strictly periodic in time, in which case it is sometimes possible to definequasi-stationary states called Floquet states. These are something like time-domain

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versions of the Bloch states for a spatially-periodic potential...) Hence our approachwill be to derive approximate expressions for transition probabilities such as

Pk ← st |〈k | t |2, | t 0 | s ,

|〈k | T t , t 0 | s |2,

where | k and | s are eigenstates of H 0, the system is initially prepared in state | s attime t 0, and we are interested in the probability that the perturbation has “kicked” thesystem into state | k at (by?) some later time t .

Sinusoidal or constant perturbations

In the 0 case, we have

Pk ← st ≈ 2|〈k | W 0 | s |

2

2

sinkst /2ks/2

2

≡ 2|〈k | W 0 | s |

2

2 Ft ,ks,

which will serve as a definition for Ft ,ks. In the case of finite , we can obtain a verysimilar form for P k ← st near resonance:

Pk ← st ≈ 2|〈k | W 0 | s |

2

42expiks − t − 1

ks −

2

2|〈k | W 0 | s |

2

42 Ft , − ks.

Note the following Taylor expansion of Ft , Δ, where Δ ≡ ks − , for small Δ t :

Ft , Δ 1 iΔt − 1

2 Δ 2t 2 − 1

Δ

2

iΔt − 1

2 Δ 2t 2

Δ

2

≈ t 2.

Also, note that Ft , Δ → 0 for Δ t 2.

Let us now consider what the transition probability will be at a fixed time t , but for arange of values of (around resonance) [C-T et al, Ch. XIII Fig. 3]:

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For constant perturbations we have a similar picture [C-T et al, Ch. XIII Fig. 4].Note that the width of the large central peak of this modified sync-function (in either case) becomes more and more narrow with increasing t , while the height of itincreases as t 2 as we saw from our Taylor expansion above.

When thinking about these expressions it is important to remember that our first-order perturbative treatment is only valid so long as P k ← st 1. In addition theresonant approximation used for the sinusoidal case requires that we be able toneglect the ks

−1 term in favor of the − ks−1, which if you think about it interms of sync-functions localized at ks leads to the further condition

t 1ks 1 .

Hence, it seems that our picture has a limited time-window of validity, for times not tooshort (resonance approximation) and not too long (small transition probabilities).

Coupling to a continuum; Fermi’s Golden Rule

So far in our discussion we have implicitly assumed that | k and | s lie in a discrete(bound) part of the eigenspectrum of H 0, as opposed to a continuum. Transitions frombound to free states are equally important, however, so in this section we outlineappropriate modifications to the perturbative treatment and arrive at the expression for transition rate known as Fermi’s Golden Rule.

Assume now that the final state | k lies in a continuous part of the spectrum of H 0.Rather than talking about transitions to a particular final state in the continuum, weshould think about transitions “into” a domain D of some finite size, centered on k :

Pk , t k ′∈ D

dk ′ |〈k ′ | T t , t 0 | s |2.

For example k could represent 3D momentum p for an unbound state of a particle, inwhich case D could be a small ball in momentum space and

dk ′ ↔ d 3 p.

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In practical terms the size of D should be chosen to reflect the resolutions of instruments to be used in actual measurements of the transition rates.

Recall that in the expressions derived above, the energy difference between s andk plays a crucial role (Fourier transform form of 〈W t ). Hence we will generally wantto parametrize the range of final states D in terms of an energy parameter E , in

addition to whatever other parameters are necessary:dk ′ , E d dE .

Here , E is known as the density of final states, and arises from the change of coordinates. For example, if dk ′ d 3 p,

d 3 p p2dp d

E p2

2m ,

dE 1m p dp,

p2dp m 2mE dE ,

d 3 p m 2mE d dE ,

, E m 2mE .

Hence we may now write

Pk , t D, E d dE |〈, E | T t , t 0 | s |

2.

For our favorite case of a constant or sinusoidal perturbation, and to first order,

Pk , t ≈ 142

D, E d dE |〈, E |W 0 | s |

2F t , E − E s

− .

Now comes a crucial assumption that we can consider times t long enough for

F t , E − E s

− sinΔt /2Δ/2

2

,

Δ ≡ E − E s

− ,

to approach its limiting form of

Ft , Δt →→ 2t Δ.

(Note that we must have in some sense a sufficiently small 〈, E |W 0 | s in order for this to be possible within first-order perturbation theory.) In this limit, we can integrate

Pk , t ≈ 142

D, E d dE |〈, E |W 0 | s |

2F t , E − E s

−

t 2

D, E d dE |〈, E |W 0 | s |

2 E − E s

−

t 2

∈ D

, E s d |〈, E E s |W 0 | s |2.

In arriving at the third line we have used the -function to pick out

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E : E − E s

− 0,

E E s .

Hence Pk , t 0 unless the domain D around k includes states with energy E s .Somewhat surprisingly (I think), we also find that Pk , t is proportional to time t

(indicating that this is not the leading order of some oscillation between s and k )meaning that we can define a constant transition rate

ws → k ≡ t −1 Pk , t ,

and a transition probability density per unit time and per unit interval in ,

ws → k ≡ ws → k

2

|〈, E E s |W 0 | s |2, E s (resonant),

2

|〈, E E s |W 0 | s |

2, E s (constant).

This, finally, is Fermi’s Golden Rule.Note that any time we have a constant rate of decrease for a probability, the

long-time dynamics should be an exponential decay. For example, if we think of thedecay of a discrete initial state coupled to a continuum,

d dt

P s −wPs,

Pst Ps0exp−wt .

Hence the linear behavior predicted by our first-order treatment could correspond tothe early linear behavior of an exponential [C-T et al, Complement DXIII Fig. 2]:

Although our treatment here has been only first-order, related calculations valid tohigher order can be carried out for numerous settings in which a discrete state iscoupled to a continuum, and this picture of an exponential decay holds true! Incontrast we know from first term that any coupling between a pair of discrete statesleads to Rabi oscillation, that is, to a “reversible” oscillation of population between thetwo coupled states. In exponential decay however, it appears that population leavesthe initial discrete state and simply “dissolves” irreversibly into the continuum.

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