3 ABSORPTION, EMISSION, AND DISPERSION OF LIGHT
3.16 SUMMARY
that an effect cannot precede its cause. In the present context the “effect” and the “cause”
are the induced electric dipole moment and the applied electric field, respectively.
Equation (3.15.9) exemplifies the general result that anomalous dispersion tends to be strongest in media with strong absorption (or gain) and narrow linewidth. As discussed in Section 5.9, the refractive index associated with the lasing transition can play an important part in determining the oscillation frequency of the laser, and the effect is largest in high-gain lasers with narrow gain profiles.
kinetic energy of the electron plus the potential energy associated with its binding to the nucleus, and HI is the “interaction energy” arising from the applied electro- magnetic field.
A standard approach to the solution of the Schro¨dinger equation (3.A.1) is based on the assumption thatc can be written as a linear combination of the eigenfunctions ofH0:c¼Pnanfn, where the functionsfn(x) are solutions of the time-independent Schro¨dinger equation that follows from (3.A.1) when we take H¼H0 and write c(x,t)¼fn(x) exp(iEnt=h):
H0fn(x)¼Enfn(x): (3:A:2) Equation (2.A.1) is a special case of this equation withH0¼ (h2=2m)d2=dx2þV(x): The eigenvaluesEnare the allowed energies of the electron when there is no applied field. For simplicity we will ignore the possibility of degeneracy, so that with each allowed energy there corresponds only one eigenfunctionfn. In writingc(x,t) as a linear combination of the time-independent functionsfn(x), it is evident that the coeffi- cientsanin this linear superposition must depend on the timet:
c(x,t)¼X
n
an(t)fn(x): (3:A:3) Theanare quantum mechanical “probability amplitudes” in the sense thatjan(t)j2is the probability at timetthat the state of the atomic electron is described by the eigenfunction fn(x). Loosely speaking, we say thatjan(t)j2is the probability at timetthat the electron is in the statefn(x).
We now use the expression (3.A.3) forcin Eq. (3.A.1) and obtain ih X
n
dan
dt fn(x)¼X
n
an(t)H0fn(x)þX
n
an(t)HIfn(x)
¼X
n
Enan(t)fn(x)þX
n
an(t)HIfn(x), (3:A:4) where in the second line we have used (3.A.2). Next we use the fact that the eigenfunc- tionsfn(x) ofH0are orthogonal in the sense that
ð
d3xfm(x)fn(x)¼0 if m=n: (3:A:5) As discussed in quantum mechanics textbooks, this orthogonality is a consequence of the fact that the HamiltonianH0is a Hermitian operator (the electron energy is a real number). We assume furthermore that thefnare normalized such that
ð
d3xfn(x)fn(x)¼1, (3:A:6) which, as also discussed in quantum mechanics texts, allows us to interpret fn(x)fn(x)¼ jfn(x)j2 as a normalized probability density. We now multiply both sides of (3.A.4) byfm(x), integrate both sides over all space, and use the properties
APPENDIX: THE OSCILLATOR MODEL AND QUANTUM THEORY 133
(3.A.5) and (3.A.6). The result is ih dam
dt ¼Emam(t)þX
n
an(t) ð
d3xfm(x)HIfn(x)
;Emam(t)þX
n
Vmnan(t): (3:A:7) This is a very general equation for the change in time of probability amplitudes.
Note that in arriving at Eq. (3.A.7) we have not had to know the form of H0, but only that it has eigenvalues En and eigenfunctions fn, and HI, which is associated with the effect of the applied field on the electron, has also remained unspecified.
To proceed further with the solution of the time-dependent Schro¨dinger equation we must know something about the Vmn coefficients, usually called
“matrix elements”:
Vmn¼ ð
d3xfm(x)HIfn(x): (3:A:8) To this end we note that a displacementxof the electron from the nucleus implies an electric dipole momentd¼ex, and therefore the interaction energy2d
.
E¼2ex.
Ein an electric fieldE. Thus we assume that
HI ¼ ex
E, (3:A:9)and therefore that
Vmn¼ ð
d3xfm(x)[ex
E]fn(x): (3:A:10)We assume furthermore that E is practically constant over the region of space for whichfm(x)fn(x) in (3.A.10) is not negligibly small. That is, as in Section 3.2, ifre
is the electron’s position, we can writeE(re,t)¼E(Rþ(mn/M)x,t)E(R,t) when- ever the electric field wavelength exceeds atomic dimensions. This is the electric dipole approximation discussed in Section 3.2; in this approximationHIreduces to the potential energy (3.2.16), andEis independent of electron positionxand can be removed from the integral in (3.A.10). The positionRof the center of mass can be identified with the coor- dinate origin and we will drop it from the argument ofE. The matrix element (3.A.10) is then
Vmn(t)¼ exmn
E(t), (3:A:11)where
xmn; ð
d3xfm(x)xfn(x): (3:A:12)
According to quantum mechanics the average value (or “expectation value”) of the electron displacementxat timet, denotedkx(t)l, is
kx(t)l¼ ð
d3xxjc(x,t)j2¼ ð
d3xc(x,t)xc(x,t)
¼ ð
d3x X
m
am(t)fm(x)
!
x X
n
an(t)fn(x)
!
¼X
m
X
n
am(t)an(t) ð
d3xfm(x)xfn(x)
¼X
m
X
n
xmnam(t)an(t): (3:A:13) The variations in time of the coefficientsan(t) thus determine the variation in time ofkx(t)l.
We want to comparekx(t)lwith the displacementx(t) of the classical electron oscil- lator model [Eq. (3.2.18b)]. For this purpose we now make an approximation that greatly simplifies the solution of the time-dependent Schro¨dinger equation: We assume that at any timetthe electron has essentially zero probability of being found in any state other than the ground state or the first excited state of the atom. We denote these two states by subscripts 1 and 2, respectively. In this “two-state” approximation the set of Eqs. (3.A.7) reduces to the two equations
ih da1
dt ¼E1a1þV12a2¼E1a1ex12
Ea2 (3:A:14)and
ih da2
dt ¼E2a2þV21a1¼E2a2ex12
Ea1, (3:A:15)while the expectation value of the electron coordinate is
kx(t)l¼[a1(t)a2(t)þa2(t)a1(t)]x12: (3:A:16) In writing these equations we have assumed thatx21¼x12or, what is the same thing, that x12is a real number (Problem 3.21).
It follows from Eqs. (3.A.14) and (3.A.15) that d
dt(a1a2)¼ iv0(a1a2)þie
h x12
Ehja1j2 ja2j2i, (3:A:17)where
v0¼E2E1
h (3:A:18)
APPENDIX: THE OSCILLATOR MODEL AND QUANTUM THEORY 135
is 2p times the Bohr frequency for transitions between the ground state and the first excited state of the atom. Differentiating both sides of (3.A.17) with respect tot, and adding the complex conjugate of the result, we obtain
d2
dt2(a1a2þa2a1)¼ v20(a1a2þa2a1)þ2e
h v0x12
E(ja1j2 ja2j2) (3:A:19)and consequently, from (3.A.16), d2
dt2kxlþv20kxl¼2ev0
h x12(x12
E)(ja1j2 ja2j2): (3:A:20)This result obviously resembles the classical oscillator equation (3.2.18b), the differ- ence being in the right-hand side. To interpret this difference, we recall the circum- stances for which the classical oscillator model was invented in the period around 1900. The phenomena that Lorentz and others sought to explain involved only natural light (from the sun) or light from man-made thermal sources (lamps). As discussed in Section 1.2, the spectral intensity of any such radiation is weak. This suggests that we focus our attention on the quantum mechanical equation (3.A.20) for the case in which the excited-state probability is close to zero, i.e., forja2j21 and ja1j2 1.
Then we can approximate (3.A.20) by d2
dt2kxlþv20kxl¼2ev0
h x12(x12
E): (3:A:21)This equation still differs from (3.2.18b), but only in the constants on the right-hand side. To proceed further, let’s label the direction of the field as thezdirection:E¼E^z, where^zis the unit vector in thezdirection. Then, taking the vector dot product of both sides of (3.A.21) with^z, we have
d2
dt2kzlþv20kzl¼2ev0
h z212E, (3:A:22) wherekzl¼kxl
^zis the component ofkxlalong the direction of the electric field. Note thatz212means (z12)2and not (z2)12.In the classical electron oscillator model an electric field pointing in thezdirection induces an electron displacement in the z direction, and the Newton equation of motion for this displacement is
d2z
dt2þv20z¼ e
mE: (3:A:23)
The approximate quantum mechanical equation of motion (3.A.22) for the expectation value ofzis identical to the classical equation (3.A.23) if we replacee/min the latter by (ef/m), where
f ¼2mv0
h z212: (3:A:24)
As the notation suggests,fis the oscillator strength introduced in Sections 3.3 and 3.4 in order to bring results of the classical electron oscillator model for emission and
absorption into numerical agreement with the results of quantum theory. Equation (3.A.24) can be used to calculate the numerical value of the oscillator strength of the atomic transition of frequencyv0/2p if we know the wave functions f1(x) and f2(x) of the two states of the transition [see Eq. (3.A.12)]. Using the fact that x212¼y212¼z212 for any atomic transition, and that jx12j2¼x212þy212þz212, we can write the expression for the oscillator strength more generally as
f ¼2mv0
3h jx12j2: (3:A:25)
Thus, for example, we can write the spontaneous emission rate (3.3.7) as A21 ¼e2jx12j2v30
3pe0hc3 : (3:A:26)
Effects of level degeneracies on these expressions forfandA21are derived in Section 3.7.
The quantum mechanical validation of the classical electron oscillator model is little short of wonderful. We have shown that, under conditions of low excitation probability, an atomic electron responds to an electric field exactly as if it were bound by a spring to the nucleus, with the natural oscillation frequency corresponding to the Bohr transition frequency. And to make the predictions of the electron-on-a-spring model agree quan- titatively with quantum theory, we simply introduce the oscillator strengthf. Thus, “so far as problems involving light are concerned, the electrons behave as though they were held by springs”4—provided that excited-state probabilities are small, which is certainly the case in practically all naturally occurring phenomena. We can also call attention to theh in the denominator off, showing thatfis truly quantum mechanical—there is no classical limit for it as !h 0.
We have justified the classical oscillator model using the approximation of including only two atomic states in our calculations. It is not difficult to justify the classical model without the two-state approximation; all that is really necessary is the approximation that the atom remains with high probability in the ground state. As a practical matter, how- ever, it is usually not necessary, under conditions of low excitation probability, to include more than the ground and first excited levels of the atom. This is because atomic transitions between the ground state and the first excited state typically have a larger oscillator strength than other transitions involving the ground state, and therefore contribute most strongly to the expectation valuekxl.
When excited-state probabilities are not small, we cannot make the approximation ja1j2 1. In particular, we cannot approximate ja1j22ja2j2 in the two-state model by 1. It is precisely this difference that gives rise to the “population difference”
(N12N2) appearing, for instance, in Eqs. (3.6.9) and (3.7.1), and which arises physically because of the possibility of stimulated emission as well as absorption.
PROBLEMS
3.1. (a) Show that the spectrum of thermal radiation forT¼300K peaks at approxi- mately 10 microns.
(b) At what frequencyndoesr(n) have its maximum?
PROBLEMS 137
(c) Support or refute the statement (from S. Weinberg,The First Three Minutes, Bantam Books, New York, 1977, p. 57) that “the average distance between photons in black-body radiation is roughly equal to the typical photon wavelength.”
3.2. Assuming the classical forceF¼eEþevBacting on a chargeein electric and magnetic fieldsEandB, respectively, show that the magnetic force is small com- pared to the electric force when the charge has a velocityjvj cin a plane-wave electromagnetic field.
3.3. Assume the “spring constants” ks for the binding of electrons in atoms are approximately the same as those for the binding of atoms in molecules. Ifn¼ 51014Hz is a typical electronic oscillation frequency, estimate the range of fre- quencies typical of atomic vibrations in molecules, given typical electron – atom mass differences. Does your estimate indicate that molecular vibrations lie in the infrared region of the spectrum?
3.4. Show that if the field polarization vector in Eq. (3.4.4) is taken to be complex:
^ 1¼ 1ffiffi
2
p (^xþi^y), then the real part of the right-hand side of (3.4.4) represents a cir- cularly polarized field with the same time-averaged intensity as the given linearly polarized field, that is,E
E¼12E20. Does this field vector rotate clockwise when viewed by an observer looking into the wave (i.e., looking back toward negative z)? If so, the wave is called right circularly polarized according to the optics convention for polarization.3.5. Take the incident field to be circularly polarized (see Problem 3.4) and recalculate dW/dtto show that the result given in Eq. (3.4.18) remains unchanged.
3.6. Show that the rate of absorption of energy by an atom in a broadband field is given in the electron oscillator model by Eq. (3.5.6).
3.7. Estimate the temperature of a blacktop road on a sunny day. Assume the asphalt is a perfect blackbody.
3.8. Show that the spectrum of thermal radiation in a gas at temperatureTis unaffected by degeneracies of the energy levels of the atoms.
3.9. Compare Eq. (3.7.11) with Eq. (1.5.2) obtained in the simplified laser model described in Chapter 1. What term in Eq. (3.7.11) corresponds to the parameter fin Eq. (1.5.2)? What is the physical meaning of the differences in the form of these two equations?
3.10. Show that the number of atoms (or molecules) per cubic meter of an ideal gas at pressurePand temperatureTis given by (3.8.20).
3.11. The CO2molecule has strong absorption lines in the neighborhood ofl¼10mm.
Assuming that the cross sections of CO2molecules with N2and O2molecules are s(CO2, N2)¼1.20 nm2 and s(CO2, O2)¼0.95 nm2, estimate the collision- broadened linewidth for CO2in the atmosphere. (Note: Since the concentration of CO2is very small compared to N2and O2in air, you may assume that only N2–CO2and O2– CO2collisions contribute to the linewidth.) Compare this to the Doppler width.
3.12. Consider an atom of massmwith a resonance frequencyn0and an initial velocity vin a direction away from a stationary source of radiation of frequencyn n0. Assume that a photon of frequencyncarries an energyhnand a linear momentum hn/c, that vc, and that hn=cmv. Using the conservation of energy and linear momentum, derive the formula (3.9.4) for the Doppler-shifted absorption frequency.
3.13. Consider a radiatively broadened transition of an atom. Assuming that the degen- erate states of each energy level are equally populated, show that the stimulated emission cross section for narrowband radiation of wavelengthl¼c/n equal to the transition wavelength is simplys(n)¼l2/2p. What is the cross section for absorption? (Note: As discussed in Section 14.3, significantly different cross sections can result when the degenerate substates of each level are not equally populated, as occurs when there is “optical pumping.”)
3.14. Beer’s law in the study of dyes and optical filters states that, ifT is the trans- mission coefficient at a given wavelength and a given dye concentrationr, then ann-fold increase inrresults in a transmission coefficientTn. The equivalent statement in terms of the thicknesszof a filter is called Bouguer’s law. What func- tional dependence onrandzdo these empirical laws imply for the transmitted intensityIn(z)?
3.15. Consider the absorption coefficient a(n0) of a pure gas precisely at resonance.
Show thata(n0) is proportional to the number density of atoms when the absorp- tion line is Doppler broadened, but is independent of the number density when the pressure is sufficiently large that collision broadening is dominant.
3.16. Consider a cell of lengthLalong the direction of propagation of collimated radi- ation of frequencynnear that of an atomic line. Define the spectral brightness In(z) to be the radiant power per unit area, unit bandwidth, and steradian.
Assume that the radiation is unpolarized and that spontaneously emitted radiation is isotropic.
(a) Derive theequation of radiative transfer, dIn
dz ¼ hn
4pA21N2S(n)þ hn
c B12 N1g1 g2N2
S(n)In
;snknIn:
(b) Assuming thatN1andN2are spatially uniform, show that In(L)¼In(0)eknLþsn
kn
[1eknL],
whereIn(0) is the spectral brightness of the radiation input to the cell.
(c) If the optical depth knL1, the medium is said to be optically thin.
Assuming that there is no radiation input to the cell [i.e., In(0)¼0], show
PROBLEMS 139
that for an optically thin medium In(L) hn
4pA21N2S(n)L:
[Note that the spectrum of the emitted radiation is identical to the absorption lineshapeS(n).] This result is the basis of one method of measuring oscillator strengths.
(d) If knL1 the medium is said to beoptically thick. Show that in this case [cf. Eq. (1.2.3)]
In(L) 2hn3=c2 ehn=kBT1, whereTis the temperature of the medium.
(e) Discuss the evolution of the spectrum of the emitted radiation as knL is increased from a very small number to a very large one.
3.17. (a) Estimate the absorption coefficient for 589.0 nm radiation in sodium vapor containing 2.71018atoms/m3 at 2008C. [See J. E. Bjorkholm and A.
Ashkin,Physical Review Letters32, 129 (1973)].
(b) Assuming the same conditions as in (a), plot In(z)/In(0) vs. z for n¼n(2)0 , n¼n(2)0 +dnD, andn¼n(2)0 +2dnD.
3.18. Show that for circularly polarized light, for which^1¼ 1ffiffi
p2(^x+i^y), the remarks following Eq. (3.14.1) remain correct even though (3.14.1) itself applies only to linearly polarized light.
3.19. (a) What is the spontaneous emission rate for the helium 1S0– 2P1transition at 58.4 nm?
(b) A cell is filled with helium at a temperature of 300K, and the density is suffi- ciently low that collision broadening is negligible. Calculate the absorption coefficient for the 58.4-nm transition.
3.20. (a) The position vectorxfor an electron moving with velocity much less thancin a plane monochromatic wave1^E0cos(vtkz) is determined by the equation of motionmd2x=dt2¼e1^E0cos(vtkz). Show that the refractive index of an electron gas is given by Eq. (3.14.13).
(b) Assume that in the ionosphere the refractive index for 100-MHz radio waves is 0.90 and that the free electrons make the greatest contribution to the index.
Estimate the number density of electrons.
(c) Why is the contribution of positively charged ions to the refractive index much smaller?
(d) Choose an AM and an FM radio station in your area and compare their frequencies to the plasma frequency of the ionosphere.
3.21. Show how Eqs. (3.A.14) – (3.A.24) are altered if we do not make the assumption thatx21¼x12.