Modern Approach to Quantum Mechanics (A) 2E

Problems I 533 Problems 14.1. By taking the dot product of Ampere's law, ( 14.21 ), withE and using the vector identity v . (E X B) = B . (V X E) - E . (V X B) show that energy conservation follows from Maxwell's equations in the form -aaur+ .v.., · Sp =-J.· E where ] ') ') c' u = -(E- + B-) and Sp = - (Ex 8) 8n 4n Discuss the physical significance of each term in this equation. Suggestion: Integrate the equation over an arbitrary volume and use Gauss's theorem to make the physical significance more transpare nt. 14.2. Show that f 3d r (e-JikV-·r)*(e-.iJkV-'·r) = ok,k' given period ic boundary conditions. See (14.34). 14.3. The nonrelativistic wa\\·e equation is the Schrodinger equation in-aat{t! = 1- (li )2 1/1 + Vt{l 2m -:V I =For a relativistic free particle. for which £ 2 +p2c2 m2c4, a natural wave equa- tion is or which is called the Klein-Gordon equation. Use this equation to show that there is a local conservation law of the form =()p + V · j = 0 with j h. (1/f*Vvr- 1jrV1jr'\") ot 2mt Page 550 (metric system)

534 I 14. Photons and Atoms Detemune the f01m of p (r , t ). From this form for p , give an argument for why the Klein-Gordon equation is not a good candidate for a one-particle relativistic wave equation in place of the SchrOdinger equation, for which p = 1/J*1/1. 14.4. The resolution of the problem outlined in Problem J4.3 is to treat the solution to the Klein-Gordon equation as a quantum fi eld. (a) Verify that if we write ~- ~ ( ~ ei(k·r- wt) ~te-i(k·r-wt) ) ip (r , t )- l : c - ak ,JV +ak ,JV k 2w V V then fP is a solution to the Klein-Gordon equation (mc)~ an 2 i~p-- ~ y 0 2! p - -1 -2cp- - c2 iJt 2 fi provided w = / k2 + (mcjfi)2 c. (b) One can show that the Hamiltonian for this system is given by Show that if [ak, ak,] = 0, f£2~, a~,]= 0, and [£lk, a~,]= ok,k'• then the Hamil- tonian becomes Argue that the field fP creates and annihilates (spin-0) particles of mass m and energy E = j p2c2 + m2c4 and that these particles are indeed bosons; that is. it is possible to put more than a single particle in a state with momentum p . 14.5. In order to see why the particles created by a scalar field must be bosons. consider an alternative procedure for quantizing this field. Try writing +~ ~ ~ ( ~ ei(k·r-(t>r) ~te-i(k ·r-wt)) ip(r , t) = ~ c bk l1i b k - - l=t i- k vV vV where the annihilation operators bk and the creation operators b~ obey the anticom- mutation relations where the anticommutator is defined by Page 551 (metric system)

Problems I 535 (a) Show that these anticommutation relations require that there cannot be more than one particle in a state: and thus the particles are fermions. (b) How would the result of your calculation for the Hamiltonian of the preced- ing problem for the scalar field be modified by using these anticommutation relations? It can be shown formally that there is no viable scalar field theory (the field essentially vanishes) when the field is quantized using anticom- mutation relations. On the other hand, for a spin-~ Dirac field, the situation is reversed: commutation relations for the annihilation <ljld creation opera- tors lead to problems-negative energies for free particles and, consequently, a lack of stability-while anticommutation relations produce a Hamiltonian with positive energies. 18 14.6. Calculate (nk,J and b.nk,s for the coherent state 14.7. Show that the probability Pnk.s of finding nk,s photons in the coherent state is given by the Poisson distribution 14.8. For the coherent state 2 00 a\"k.s Ia ) = e - lal /2 \"L....... -,r:--r Ink,s) v\"k..•= O \"k,s' the expectation value of the electric field is j2n:nw . +(aiEA ia) = -2 -V- lale(k , s) sm(k · r- wt o) 18 SeeR. F. Streater and A. S. Wightman. PC1; Spin & Statistics, andAll That, W. A. Benjamin, New York, 1964. Page 552 (metric system)

536 I 14. Photons and Atoms where a= la lei8. Show that 2JT liw + 41a l 2 .2 wt + 8)] - - v{a lEA 2Ia) = [1 sm (k · r - provided the contributions of photon slates with k' =J k, s' =J s are neglected, as was the case in the beginning of Example 14.1. Use the result of Example 14.2 to show that the fluctuations in electric field strength for the coherent state are the same as for the vacuum state IOk,s)· 14.9. Calculate {a IBia) for the coherent state Ia). Ve1ify that the expectation values forE (see Problem 14.8) and B obey Faraday's law. The next three problems require time-dependent pe.tturbation theory strictly within nomelativistic quantum mechanics. The electromagnetic field in these prob- lems is to be treated as a classical field. The Hamiltonian in Problem 14.10 and Problem 14.11 is altered by the presence of an electric field E by the addition of a tem1 where fle is the electric dipole moment operator. 14.10. A particle with charge-to-mass ratio ej m in a one-dimensional harmonic oscillator with spring constant k is in the ground state. An oscillating uniform electric field y;;E(t) =Eo cos wt wo = /k is applied parallel to the motion ofthe oscillator for t seconds. What is the probability that the particle is excited to the sta~e In )'? Evaluate the probability of making a transition when w = w0 , the resonance condition. 14.11. A hydrogen atom is placed in a time-dependent homogeneous electric field where E0 and r are constants. At t = 0 the atom is in its ground state. Calculate the probability that it will be in a 2p state as t--+ oo. 14.12. A spin-1 particle is immersed in a constant magnetic field B0 in the z direction and an oscillating magnetic field B1 cos wt in the x direction. The spin Hamiltonian can be written in the fom1 fi = Sw0 2 + w1cos cutS.~,. See (4.34). Assume w1 « w0 and treat the time-dependent pmt as a perturbation. Calculate the probability that the =particle is in the spin-down state in timet if it is in the spin-up state at t 0. Evaluate Page 553 (metric system)

Problems I 537 your result in the resonance region when w is near w0. Compare your pe11urbative result with Rabi's formula (4.44). Suggestion: You may wish to review the material in Section 4.4. where an approximation made in deriving Rabi's formula is described. 14.13. (a) Show for the one-dimensional harmonic oscillator that the position and mo- mentum operators in the Heisenberg picture are given by XA H ( 1) = XA H (O) COS Wt + PxH (O) . Wt -- SID mw PxH (I) = Pxu (0) cos WI - mwxH(O) sin (t)/ respectively, where xH(O) = .r is the usual position operator in the SchrOdinger picture and fixH (0) = f7x is the usual momentum operator in the Schrodinger picture. (b) Show that the equal time commutator of the position and momentum operators in the Heisenberg picture is given by What happens if times for the two operators are different? 14.14. As discussed in Section 10 .5, the allowed energies of the isotropic three- dimensional harmonic oscillator are given by (n ~)E11 = + hw n. = 0, 1, 2, ... where n = 2n, + L. Thus then = 0 ground state is an l = 0, or s, state, while then = 1 first excited state is an l = 1, or p, state. For a particle with charge q = e confined in this potential, calculate the transition rate R for the transition between the first excited state and the ground state with the emission of a photon. 14.15. An unstable spi n- ~ particle of mass m 1 initially in the spin state l+z) under- goes a magnetic dipole transition to a spin- ~ particle of mass m 2, emitting a photon. Use the spin Hamiltonian (14.177) to show that the angular distribution of the pho- tons is isotropic, provided we sum over the probabilities of making transitions to both the l+z) and 1-z) states. 14.16. Show that the number of states, apart from spin, for an electron with energy between E and E +dE is given by v. -(2-Jf)-3-/'i.3-me p dQ dE Page 554 (metric system)

538 I 14. Photons and Atoms 14 .17. Show that the differential cross section for the photoelectric effect in which an electron in the ground state of hydrogen is ejected when the atom absorbs a photon is given by where q = k; - kf· Assume the ejected electron is sufficiently energetic that the wave function for the electron can be taken to be the plane wave where p1 = fik 1 is the momentum of the electron and fill; is the momentum of the incident photon. Suggestion.: The cross section is the transition rate divided by the incident photon flux, which is equal to cj V in the box nonnalization used in this chapter. 14.18. (a) Show that the magnetic dipole Hamiltonian (14.175) yields a zero matrix element for the 3d to Is transition in hydrogen. Suggestion.: Express the angular momentum operator L in terms of L+, L_, and fz. (b) Show that the electric quadmpole Hamiltonian can yield a nonzero matri;Y; element for this 3d to Is transition. Suggestion: Use the \"trick\" (14.152) to express the first term on the right-hand side of (14.172) solely in terms of position operators. 14.19. Use dimensional analysis to argue that the Casimir force per unit area between two uncharged conducting plates varies as d-4, where d is the separation of the plates. How does the gravitational attraction of the plates vary with the separation of the plates? Page 555 (metric system)

APPENDIX A Electromagnetic Units Let's stmt with Maxwell's equations in SI units: (A. I) p (A. 2) (A.3) V·E = - eo (A.4) aB VxE = - -at v x B = J.Lo j + fJ-oto CJ E - &t In these units, the force on a charged particle q is given by F = qE +q ~ X B (A.5) We can use Gauss's law (A.I ) most easily in its integral form rJ. E . dS = qeuclosed (A.6) eo to determine the magnitude of the electric field from a point charge q. Us ing a spherical Gaussian surface of radius r centered on the charge, we find (A.7) or the familiar (A.8) Page 556 (metric system5)39

540 I A. Electromagnetic Units Thus the magnitude of the force F between two charges q1 and q2 separated by a dt rnnce r is (A.9) \\\\ rucb we can express in the form ( A . l O) with the constant k1 = 1j4rrs0. In SI units, the unit ofcharge, the coulomb, is defined to be equal to 1 ampere-second, while the unit of current, the ampere, is actually determined by magnetic measurements. This is a natural set of \"experimentalist's.. units because it is possible to make very accurate current mea..<>urements and hence fix the unit of current, and thus the unit of charge, quite precisely. Once we know the unit of charge, the force between two charges is determined. The value of the constant k1is found experimentally to be roughly 9 x 109 newton-meter2/coulomb2. See (A.19). Another system of units that is commonly used for Maxwell's equations is Gaussian units. These units can be described as \"theorist's\" units, since they are somewhat impractical for use in the laboratory but much more useful than SI units for revealing the true structure and beauty of electricity and magnetism. In addition. they are more commonly used than SI units for describing microscopic phenomena In these units we begin by first determining the unit of charge, not the unit of current. In Gaussian units we define the constant k1 to be unity. Then the force F between two charges is simply given by F = qlq2 (A.l l) r2 where the unit of charge is determined by the requirement that two units of charge separated by a distance of 1 centimeter exert a force on each other of 1 dyne. This unit of charge is called a statcoulomb. The corresponding electric field produced by charge q is just q (A . l 2 ) E=- r2 and consequenlly fcompare (A.8) and (A.l )] V · E =4rrp (A.13) in these units. Let's see what happens to the rest of Maxwell's equations. In SI units, Ampere·s law, (A.4), can be expressed in integral form as f J :t JB · dr = J.Lo j · dS + J.Lo£0E · dS ( A . l 4J Page 557 (metric system)

A. Electromagnetic Units I 541 where Jj · dS = '~ncloscd (A.15) is the current enclosed by the closed line contour on the left-hand side of (A.l4). For a collection of charges that form a current, the magnetic portion of the force (A.5) due to a differential length dl of current is given by dF = l d l x B. Since the magnitude of the magnetic field produced by a long current-carrying wire is given by B = J.Lol (A.l6) 2rrr ~ in SI units, the magnitude of the force per unit length between two parallel wires separated by a distance r carrying currents / 1 and /2 is just (A.l7) We can also express this force per unit length in the form (A.18) 4rr Io-In SI units it is the constant k2 that is defined to be J.Lo/ = 7 newton/ampere2. This then determines the unit of current by the requirement that two long wires each carrying a current of 1 ampere separated by a distance of I meter exert a force per o-unit length on each other of 2 X 1 7 newton/meter. Note that (4rr)k1 1 12 (A.l9) 4rrk2 = Eo J.Lo = Eof.Lo = c where cis the speed of light. Thus measuring the speed of light determines experi- mentally the value of k1 in SI units. ln Gaussian units, on the other hand, where k 1 is defined, it determines the value of k2• We still have a little freedom in our choice of units left to play with. In particular, even though the force between the two current-carrying wires is determined, the magnetic field produced by the wires can still be adjusted. We introduce a constant A. so that the magnetic portion of the Lorentz force is given by F = A.- Iqv x B, or dF = A.-I I d l x B. Since it is the forces that we measure directly, not the magnetic fields, we can avoid changing any physics by adjusting the value of the magnetic field produced by the wire so that B =A J.Lol (A.20) 2rrr Page 558 (metric system)

542 I A Electromagnetic Units More generally, we need to modify the right-hand side of Ampere's law, which we now express as ( . IJE)VxB= A. J 4nk2 +- -at (A.2 1) c2 Jn Gaussian units we use this freedom to choose).= c. This choice shows the inher- ently relativistic nature of magnetic effects lcompare the magnetic force q (vj c) x B with the electric force qEj and brings the relativistic nature of electricity and mag- netism to the fore with the explicit appearance ofthe speed of light. It also means that in Gaussian units the electric field E and magnetic field B have the samcdimensions.1 Thus in Gaussian units (A.21) becomes (A.22) Of course, E and B having the same dimensions means that Faraday's law, (A.3), must also be adjusted in Gaussian units. We introduce a third constant k3: V X J<: = aB (A.23) -k-, -at Since the gradient on the left-hand side of (A.23) has the dimensions of 1/length, k3 must have the same dimensions as l j c. In fact, in order for electromagnetic fields to propagate at the speed of light, the constant k3 must equal l j c. Thus the full Maxwell's equations in Gaussian units are given by V · E=4rrp (A.24) (A.25 ) V·B =O ( A.26 ) J aB (A.27) V x E = - - -ar c 4rr. 1 DE V X B =-J+ -- C • C i)[ with the force law F= qE +q(vj c) x B (A.2 8) Trading t:0 and tJ-o for the explicit appearance of c seems to be a step in the right direction. In fact, these units make more advanced full y covariant presentations of 1 Since the torque on a magnetic moment J.L is given by J.L x B, scaling up the magnetic fielo by a factor of c means that the magnetic moment of a current loop picks up a factor of 1/c 111 comparison with its value in SI units, as indicated in (1.1). Page 559 (metric system)

A. Electromagnetic Units I 543 ~laxwe ll 's equations in terms of relativistic four-vectors and tensors both straight- fan\\ ard and elegant.2 Fo r completeness, we should mention one other system of units that is also attractive to theorists. If you are interested in making the full Maxwell's equations as simple and as elegant as possible, you can eliminate the factors of 4;r that appear in the charge and current source terms of Gauss's law and Ampere's law at the expense of a slightly more complicated expression for the fields produced by these charges and currents. In this other system of units, known as Heaviside-Lorentz units, we =take k 1 II 4;r. The coiTesponding electric field produced by a point charge q is then E = q14;r r2. The explicit appearance of these factors of 4;r in the forms for the fields . however, is compensated for by their disappearance in Maxwell's equations. From (A.l9) we see that~ = l j 4;rc2, and therefore in Heavisid~-Lorentz units Maxwell 's equations are given by V·E=p (A.29) V·B=O (A.30) n X B = -j + -cI oa-Er (A.3 I) v (A. 3 2 ) C 188 V x E=-c-a-r with the force law still given by (A.28). Heaviside-Lorentz units are sometimes re ferred to as rationalized Gaussian units. 2 A good trick (see Sections I0.2 and 11.5) for evaluating expressions such as energies of the hydrogen atom is to replace e2 with (e2jlic) fic since the quanti ty e2jlic =a (e2j4ru:0tic in Sl untts) is a dimensionless quantity whose value is roughly 1/137. In this way you may never need • recall that the charge on an electron is 4.8 x 10- 10 ~tatcoulombs in Gaussian units rather than w-tht> more familiar 1.6 x 19 coulombs in SI units. Page 560 (metric system)

Page 561 (metric system)

APPENDIX B The Addition of Angular Momenta In this appendix we would like to investigate a simple way of addmg the angular momenta }1 and h together. Our goal is to detennine the linear combinations of the basis states for two angular momenta that form eigens~are-. ([ total angular momentum. Rather than give a general proof that the values for the to~al angular momentum j run from 1} 1 -h i to h + h in integral steps. we \\\\111 take the 'pecific 4example of adding spin and spin 1 to illustrate a procedure that can be readily extended to the addition of any two angular momenta. We label in the usual way the two basis states for a spin-~ particle b~ I ~ . ~ ) and 11, - 4)and the three basic states for a spin-1 particle b) I~ I . I. :> . and I~ -1). By taking the direct-product of these basis states, we can form six t\\\\0-particle bao;is states: If, ~ )t ® 11, lh I~, 4) I® II, Oh lf-!t® l l. -lh 14, -i) l ® 11. Oh 11. - 4>t®I I, 1h 1 ~--! t® IL-1h (B. l) We have arbitrarily chosen to call particle 1 the spin-i particle and particle 2 the spin-1 particle. As usual, we can drop the direct-product s~ mbol ® without gener- ating any confusion. i , 4) -f)The two basis states I 111, Ih and It , til, -1)2 are sometimes refened to as \"stretched\" configurations. What is special about these configurations is that the z component of the total angular momentum takes on its maximum and minimum values for these two states, respectively. For example, applying (B .2) to these kets, we find 545 Page 562 (metric system)

546 B. The Addition of Angular Moment a ( A + All ' = A ll lh + All ti l, l}z hz fzz) l2, 2) ,11, l}z J1zl2, 2),1 I, fzzl2 • 2) = 1nl1. ~ } Il l, l}z + nl4, 4hll, lh = ~ti1 4 , 4)Ji l, 1)2 (B .3) Similarly, Clearly, none of the other four states in our set of basis states (B.1) has these eigenvalues for Jz . Since the total angular momentum operator (B .S) commutes with the operator Jz, these two operators must have eigenstates in com- 11,- 1)4}mon. Consequently, the states If, d1, 1h and Il l, - 1}2 must each be an Jeigenstate of the total angular momentum 2 . We label these states in general by IJ, m). Since m = ~ and m = - ~ for these two states, respectively, you may be tempted to guess that these are both j = ~ states. To verify that this is the case, we 2J Jexpress 1 · 2 in terms of angular momentum raising and lowering operators, just as we did in (5.10): (B .6) Then we can apply the operator (B.5) to these states. For example, = [ 21 ( 2I + 1) ti2 + 1(1 + l)h.·2 + JAl+ fAz- + ]Al- izA + + J2.lAtzl2Az lz1. z1}ti l, l}z [1u= + 1) + 1(1 + 1) + 24(1)Jh214,· 4) 111, l}z (B.7) where we have taken advantage of the fact that the raising operator for each of the pruticles yields zero when it acts on a state that has the z component of the angular momenmm for that particle equal to its maximum value: (B .8) Thus (B .9) Page 563 (metric system)

B. The Addition of Angular Momenta I 547 Similarly, you can verify that (B.lO) or (B.ll) So far we have found two of the four j = ~ states. We can determine the other two by applying the lowering and raising operators for the total angular momentum to the I~, ~) and I~, - ~) states, respectively. Using (3.60), we find (B.1 2) Since (B.I3) we also know that +~ 1 1 ~I 1 J- 12, 2)J11, h-12, 2)111, lh = ~1 1 lh lh lt-12· 2) til, = )4 G+I) -1 G- 1) filt. - i>tll, lh + j l(l + 1) - 1(1 - I) nl4, 1>tll, Oh (B . l 4) (B.l5) Equating (B .12) and (B.14), we find that 4>1~. = JII~, -1>111, lh + fii~, ~)t il, Oh Either by applying the lowering operator again to (B.I5) or applying the raising operator to (B.ll), we can show that (B.l6) =Thus we have determined all four of the j ~ states. Since our basis (B. I) is six dimensional, there are two states left over. These 4states turn out to be total angular momentum j = states. We can generate them by 4, 4taking advantage of the fact that ( I~, ~) = 0, that is, the amplitude to find a state 1 4, 1with j = ~, m = in the state j = =m is zero. This is enough information to deduce that (B.J7) Page 564 (metric system)

548 I B. The Addition of Angular Momenta i, -up to an overall phase. Note that the two basis states I ~ ) 111, lh and 11, ~ ) 111. 0 : are candidates to be involved in this superposition, since they are the only two of lhe ,b,. basis states with the eigenvalue for the z component of the total angular 1·momentum equal to Lastly, we can determine the total angular momentum state {. -~),either by applying the lowering operator .7_ = ] 1_ + ]2_ to the state (8.1 7) o; b~ ~hoosing the linear combination of the basis states with total m = - ~ that b orthogonal to the state (8.16). In this way we find (B.1 8) Of course, we haven't really proved that the two states (8.17) and (B.l8) are j = ~ -.tares, although this is consistent with the fact that there are two states remainin2 -after we constructed the j J~ 2 = ~ states. Real proof comes ~ to one of from applying them. In conclusion, note that our initial two-particle basis consisted of six states and that we have now determined the linear combinations of these s tates that are eigen- 1.;,tates of total angular momentum with j =~ (four states) and j = (two states).The procedure that we have used to determine these stales can be utilized to add together any two angular momenta. For example, ifthc system consists oftwo spin-1 particles. the two-particle basis is nine dimensional. The total angular momentum takes on the values 2, 1, and 0. We can start with the stretched configuration 12. 2) = 11. l)111, 1 : and apply the lowering operator (B.I3) to determine the other four j = 2 states. We then use orthogonality relation (1, 112, I)= 0 to determine the II, I) state and ap- ply the lowering operator to determine the -other two j = I states. Finally. we can take advantage of the orthogonality relations (0, 012, 0) = 0 and (0, 011. 0) = 0 to determine the siugle j = 0 state. We need two orthogonality relations in this case because there are three two-particle states, including II, 1}111, -1h, 11, -1) 111, l12. and 11. 0)111, Oh, that can comprise the 12, 0), II, 0}, and 10, 0} states. The amplitudes (j, ml(l.it, m 1) 1lh, m2)2) arc known as Clebsch-Gordan co- efficients. Although we have called the individual angular momenta spins in this appendix, these angular momenta could be orbital as well as spin angular momen- tum. Thus (B.l7), for example, could result from the determination of the total angular momentum states of a spin-~ particle that has orbital angular momentum I = 1. Compare the specific results of this appendix with the more general results of adding spin ~ and orbital angular momentum l in Section 11.5. These Clebsch- Gordan coefficients are routinely tabulated, so you don't actually need to calculate them each time you need them. once you understand how they are obtained. Page 565 (metric system)

APPENDIX C Dirac Delta Functions The Dirac delta function is actually not a functi on at all but a \"generalized function .\" or a \"distribution,\" that is defined through the relation (C.l) j_: =dx f(x) 8(x - x0) f(x0) for any smooth functi on f (x) . From (C.l) we conclude that 8(x - x0) = 0 x ;6 x0 (C.2) and by setting .f(x) = 1 in (C. l ) that 100 (C.3) d x 8(x - x 0) =1 - 00 Thus the delta function is a \"function\" that vanishes everywhere except at a single point but nonetheless has unit area. You can think of a delta function as the limit of a sharply peaked function of unit area (see Fig. C.l), as it becomes progressively narrower and higher. In this limit, the function .f(x ) in the integral in (C. I ) can be set equal to its value at x0 since tllis is the only region in which the integrand is nonzero, and then the constant f(x0) can be pulled outside the integral. We can derive a number of prope1ties of delta functions, with the understanding that identities involving delta functions make sense only when the delta functions appear within an integral. To derive 8(ax) = I (C .4 ) -8(x) (C.5) IaI first consider the result 1-oo00 00 (~) o(y) = ~ f(O) a>0 a ..!_1dx .f(x)o(ax) = dy f a -oo a 549 Page 566 (metric system)

550 I C. Dirac De lta Functions Figure C.l A sharply peaked function with unit area. The Dirac delta function arises in the limit that the function becomes infinitesimally narrow and xo infinitely high. where in the second step we have made the change of variables y = ax. Note that if a < 0, this same change of variables would switch the limits of integration, leading to 1- oo11-oo (y) 1 (y)oo dx j(x)o(ax) = - 00 a oo dy f - o(y) = -1 d y f - o(y) a - a - oo a = -1f(O) a< 0 (C.6) -a These results can be combined together in the form of (C.4). Note that one of the corollaries of (C.4) is 8(- x) = o(x) (C.7) The delta function is an even function. Another relation that follows from (C.4) is 8(x2 - a2) = - 1 [8(x - a) + o(x + a)] (C.8) 21al Since x 2 - a2 vanishes at both x = a and x = - a, we can write 8(x 2 - a 2) = o[(x - a)(x + a)] = 8[2a(x - a)] + 8[- 2a(x + a)] = - 1 [o(x - a)+ o(x +a)] (C.9) 21al .fMore generally, suppose (x) is a function that has a zero at x0 , that is, f (x0) = 0. Expanding f (x) in a Taylor series about x0: +(elf) (d.f)f(x) = f(xo) clx x=xo (x - x0) + ·· ·= clx x=xo (x - x0) + · ·· (C. lO) Page 567 (metric system)

C. Dirac Delta Functions I 551 and taking advantage of (C.4) again, we o btain o[f(x)]= 1 (C.ll) o(x - xo) ldf f dxlx=xo where we can safely ignore the higher order terms in the Taylor series because the delta function vanishes everywhere except at x = x 0. When the delta fun ction is multiplied by a smooth fu nction within an integral, we can give meaning to the derivative of a delta function: co dx f(x).!:!_o(x) = f(x)8(x)l~co -1co dx df(x) 8(x) 1- co dx -oo dx (C.J 2) where the second step follows from an integration by parts. Also the integral of a delta function satisfies 1x dy 8(y - a)= { 0 x < a (C.13) -co 1 x >a =G(x - a) where B(x -a) is the standard step function. From this result, we also see that d (C.14) - G(x -a)= o(x - a) dx A convenient way to represent a delta function is as the limit of a sequence of regular functions that have unit area but that grow progressively more narrow as some parameter is varied. Some examples: l. The function sin AX frr x is plotted in Fig. C.2. This function is well behaved for any finite value of A. The width of the functio n is of order 1/ A, since the first zero of the sine function occurs when AX = rr. Moreover, the height of the f unction at the origin is A/rr. Thus as Aincreases, the function grows narrower and taller. In fact, the normalization factor of 1/ rr has been chosen so that the function has unit area. Therefore, as A ~ oo, the function behaves as a delta function: 8(x ) = lim -1 -sin-J..x (C. 'IS) .\\.-+co rr x 2. An alternative way ofexpressing the representation (C.15) of the delta function is especially useful. Since 1>,sin Ax = ~ dk eikx (C.l6) X 2 - .\\. Page 568 (metric system)

552 I C. Dirac Delta Functions Figure C.2 The function (sin 'J....x)/rr x, which reprc~ents a delta function in the limit A __. oo. we can write (C.l7) 8(x) = _1 foo dk eikx 2rr - oo 3. Another representation of the delta function is given by (C.l8) as can be verified by using the results of Appendix D on Gaussian integrals. 4. Finally, you can show that (C. I9) Page 569 (metric system)

APPENDIX D Gaussian Integrals We first wish to evaluate the integral dx e-ax2 (0.1) i:I (a)= where Rea > 0. A useful trick is to consider the integral squared, 12(a) = 1 00 dx e-ax2100 dy e-ay2 = 1 00 dx 100 dy e-a(x2+y2) (D.2) -oo - oo -oo -oo which can be easily evaluated by switching from Cartesian to polar coordinates: 12;r00=/ 2(a) 1o o r dr = -de 2 1'{ (0.3) e - ar a Thus 00 ../ (a) = dx e-ax2 = ~J- 1- oo a (D.4) What about integrals such as 00i (a. b)= 2 1-00 (0.5) dx e - ax +bx Here we can convert the integral into one in which we can take advantage of (0.4) by completing the square in the exponenr: 2 ax2 - bx =a ( x - -b ) (0 .6) 2a (0.7) Making the change of variables x' = x - (b j2a), we find 00 = 100I (a, b)= 1- oo dx e-ax2+bx e b2 / 4<1 dx' e - a.,.12 = eb2f4a~- -oo a Page 570 (metric system)553

554 I D. Gaussian Integrals We can also evaluate integrals of the form 1!'(a)= 00 dxe - ax2x 2 (0.8) -00 (0.9) by differentiating (0.1) under the integral sign: 1oo V-;, V!'(a)= _ !!__ dx e- ax2 = _!!__ ~ = ~ {7[ da -oo da 2 --;3 This technique can be easily extended. For example, (D.lO) Finally, we should note that although we have derived (0.4) for Rea> 0, we can extend this result to include a in (D. I) being purely imaginary. This is most easily done through contour integration. First consider the closed integral rJ dz e - az2 (D.I I) in the complex planeforthecontour shown in Fig. D.l, with a real and positive. Since the integrand is analytic within the contour, the closed contour integral vanishes: (0.12) Writing z = reiB = r(cos 8 + i sin 8) (D.B) Figure D.l A closed contour in the complex z = x + iy plane. The contributions on the circular arcs vanish as R--+ oo. Page 571 (metric system)

D. Gaussian Integrals I 555 '\"e see on the circular arcs of radius R that the integrand is given by (D.14) ich goes exponentially to zero as R -+ oo for 0 < e < n/4 and n < e < Sn/4 ' 'l.ce cos 2() > 0 for these angles. Thus the contribution of the circular arcs to the comour integral vanishes as R -+ oo. We can parametrize the diagonal line by z = rt .., -.with r running from oo to - oo. Since z2 = r2eirrf Z = ir2 and dz = dr eirr/ 4, \\\\e obtain (D. IS) C 10 equently (D.l6) which is just the same as (D .4) with a replaced by ia , provided we take .Ji = eirr/4. Page 572 (metric system)

Page 573 (metric system)

APPENDIX E The Lagrangian for a Charge q in a Magnetic Field How do we handle magnetic fields within the framework of a Lagrangian? Purely electric forces are easy. After all, the electric potential cp(r) is introduced in electro- statics as the work done per unit charge to bring the charge to the position r from some reference point, which is often taken as at infinity. Then the potential energy of a charge q is V = qcp and the Lagrangian is given by L = T - V = ~m v2 - qcp (E. I) In terms oftheCartesiancoordinatesx 1 = x, x2 = y, andx3 = z, the Euler- Lagrange equation of motion -oL - -d(a- L) - o (E.2) OX; dt OX; for the Lagrangian (E.l ) is given by - q ocp - ~mx; = o (E.3) OX; dt This equation of motion is simply .. ()cp (E.4) mx; = -q - ih; Since the electric field E is given in electrostatics by E = - Vcp, the equation of motion can be expressed in terms of vectors as the force law ma = F = qE. The full Lorentz force F = qE+q(vf c) x B (E.5) Page 574 (metric system) 557

558 E. The Lagrangian for a Charge q in a Magnetic Field mcludes velocity-dependent magnetic forces, which cannot be obtained from a l:..tgrangian of the form (E. I) that is just the difference of the kinetic and potential e'1ergies. Since the magnetic force always acts at tight angles to the velocity, it o--.esn' t change the magnitude of the velocity and thus does no work. However, we .:.m show that the Lagrangian L = -Imv2 - qrp +q-A· v (E.6) 2c · hich differs from (E.l) by the addition of a velocity-dependent term involving the \\ector potential A, yields the Lorentz force (E.5) for the equations of motion. First, we note that the magnetic field B can always be expressed in the form .B = '11 x A, since the magnetic field satisfies '11 · B = 0 and the gradient of a curl vanishes: v . B = '11 . (V X A) = 0 (E.7) Since for the Lagrangian (E.6) i) L . q -I -'l ' = mxI· + -A (E.8) c (E.9) uX; (E.lO) (E. II ) =the canonical momentum p; oLjfJi; is given in vector form by q p= mv +-A c In order to evaluate !!_ (()L) = mx; + CJ. dAt dr a.;;; c dr notice that A; = A;[x(t) , y(t), z(t ), tJand therefore a a ad-A-~ = -A-~ + ~L,.; -A-~d-x j = -A-~ + v· 'VA; dt ot j=l oxj dt or Usi n g IJL ocp q 8A (E.12) (E.13) - = -q- + -V· - (E.14) dXI OXI c ih·I (E.2) becomes -q -ocp + q ~v · -oA. - mx..; - q~ (8-A. -; + v · n A;) =O at v ax; c o:Ji.; c. or m..x. ;= - q -orp +q~v ·C-JA - q~ ( -fJA; + v· V ) OX; c OX; c dt A; Page 575 (metric system)

E. The Lagrangian for a Charge q in a Magnetic Field I 559 In vector notation, (E.I4) can be expressed in terms of the force F on the particle as ( IJA)F=q + -q[V(v ·A) - (v·V)AJ (E.l5) - Vrp-c-a-r, c or F = qE + Cf..v x (V x A) = qE + iv x B (E.I6) cc as desired. Given the Lagrangian (E.6), we can detennine the Hamiltonian in the usual way: 3 H= LPiXi - L i=l L3 1 (E.l7) = - m.X;i; + qrp i=l 2 At first it appears that the vector potential has disappeared entirely from the Hamilto- nian. However, if we express the Hamiltonian in terms of the canonical momentum (E.9), we obtain H = (p - qA jc) 2 + qrp (E.I8) 2m This suggests a mnemonic for the way to turn on electromagnetic interactions in terms of the Hamiltonian: take the energy for a free particle of charge q p2 (E . l 9) E=- 2m and make the replacements p -+ p - qAjc and E-+ E- qrp to generate (E.18), with the energy E replaced by the symbol for the Hamiltonian. Page 576 (metric system)

-APPENDIX F Values of Physical Constants lA = 0.1 nm = 10-10 m w-J1eV = 1.602176487(40) x 9J 1fm = I0- 15 m 1bam = 10-28 m2 = 1.602176487(40) X J0- 12 erg 1dyne= 10-5 newton (N) 1MeV = l06 eV o-1 gauss (G) = 1 4 tesla (T) 1eV/ c2 = 1.782661758(44) x 10- 36 kg I erg= 10- 7 joule (J) 2.99792458 x I09 esu = 1coulomb (C) 0° C = 273. 15 K Quantity Symbol Value Gaussian SI Speed of light c 2.99792458 1010 cmls 108 mls Planck's constant h 6.62606896(33) w - 34 J s li = h/ 2n 1.054571628(53) w-27 ergs Electron charge e 6.5821 1899(16) w - 34 J s 1.602176487(40) w- 27 ergs 10-16 eV s Electron mass 11l e 4.80320427(12) 9.1 0938215(45) 10- 16 eV s 10-19 c Proton mass mp 0.51 0998910( 13) 1.67262 1637(83) 10- IO esu w - 31 kg Neutron mass 111~~, 938.272013(23) 10- 2s g MeV/c2 Reciprocal 939.565346(23) MeV/c2 w - 27 kg 1/a 137 .03 5999679 (94) 10- 24 g MeV/c2 fine-structure MeV/c2 MeV/c2 constant ao MeV/c2 Bohr radius w-10 m (h/meca) /J-s 0.52917720859(36) w - 8 em Bohr magneton 10-5 evrr (ehj2meC) ka 5.7883817555(79) 10- 9 eY/G Boltzmann NA 1.3806504(24) 10-16 erg/K I0-23 J/K constant Avogadro 6.02214179(30) x 1023 /mol constant Values from J. Phys. G: Nuc. Part. Plrys. 37. 075021 (2010) . Page 577 (metric system) 561

APPENDIX G Answers to Selected Problems 1.1 1.2 x I03 G/cm 2.21 0. 12 2.22 ~Nti/2 4.12 sin\\w0tj2) 6.14 l::!. xl::!.px = 0.57ti 6.17 (a) .j30 / L 5 (b) 960j rr 6 (c) 5ti2 jmL2 6.19 - /i2>..2 j8mb2 7.13 0.16 9.9 1.13 A 10.4 0.24 10.7 0.70 10.11 (a) (2.40)2 ti2f 2Jw2 , (5.52)2112f2f..La2, (8.65?ti2f2f..La2 (b) (2.40)2 h2 / 2f..La2, (3.83)2ti2f 2J.La2 , (5. 14)2 h2f2J.La2 ?,= =11.7 (b) E f.!> (2j5)(e2j a0)(Rja0 E~~ (1/1120)(e2fao)(Rfao)4 12.5 E::::; -/i2),2j 4rrmb2 563 Page 578 (metric system)

Index Accidental degeneracy, 360, 368, 375 Basis states, II, 2fJ Action, 289 for two spin-1 particles, 141-143 Active transformation, 53, 197 Adjoint matrix, 50 Bell,J., 156 Adjoint operator, 35, 66 Bell's inequality, 161 Aharonov, Y., 487 Bethe, H., 408 Aharonov- Bohm effect, 483-487 Birefringence, 63 Alvare7., L., 447 Bohm,D.. 487 Ammonia molecule. 128- 134 Bohr. N., 4. 26 1 Bohr magneton, 26, 177 energy-time uncertainty relation and. 134 Bohr radius, 355 maser, 134 Bonding orbital, 444 orbitals and shape, 336 Born, M., 194 perturbation theory and, 395-398 Bom approximation, 458-462 in static electric field, 131-133, 395- 398 in lime-dependent electric fi eld, 133-134 validity of, 462-463 Angular momentum, 75-1 06 Bom interpretation, J94, 237 addition of, 545-548 Born-Oppenheimer approx imation, 442 Bose- Einstein statistics, 422 spin-orbit, 401-404 Bosons, 422 spin-spin, 147- 150 Bra(e)ket, 12, 22 commutation relations. 79. 104 Bra vector, 12, 22 conservation of, 118. 138 Breit- Wigner formula, 474 cigcnstates and eigenvalues, 82-89 eigenvalue problem Casimir, H., 529 spin-1 , 100-103 Casimir effect, 528-529 sp i n -~, 94-98 Cavity quantum electrodynamics, 526-529 lowering and raising operators, 85- 86 Center-of-mass and relative coordinates, matrix elements of, 90-91 operators, 36, 65, 79 311- 313 orbital eigenfunctions. 331- 337 Central potentials uncertainty relations, 91- 93. 105 Annill.ilation operator, 254, 276. 498 bound states of, 345-376 Antibonding orbital, 444 conservation of orbital angular momentum, Anticom mutation relations, 499, 535 Anticommutaror, 106, 534 317 Antisymmetric states, 422 scattering from. 458-477 Anyons, 42 1 Centrifugal barrier, 347 Aspect, A., 164 Classical path, 300 Clebsch- Gordan coefficients, 548 Balmer series, 353 Coherent states, 262- 269 for photons, 500 Commutation relations angular momentum, 79, 104 Page 579 (metric system) 565

566 I Index Commutation relations (continued) Dipole moment energy-time, 134- 136 electric, 131 position-momenrum, 306 magnetic, 1 Commutator, 79, l 04 Dirac, P., 247 Commuting operators, 80-81 Dirac delta func6on, 194, 235, 549- 552 Completeness, 25, 43 Completeness relation, 43, 68, 193 rcpresenta6ons of, 551- 552 Complete set, 23 in three dimensions, 304 Complete set of commuting observables, Dirac equation, 40 1, 405-407 Direct product, 142, 183 321 Double-slit experiment, 210- 212, 281, Complex numbers 295-297,485-487 and quantum mechanics, 13, 20 Conservation Effective potential, 320, 348 Ehrenfest, P., 3 of angular momentum, 317 Ehrenfest's theorem, 2~0 of energy, !14, 514- 515 .Eigenbra, 255, 265 of linear momentum, 200, 309- 31 :t Eigenfunction of probability, 226, 455 Constant of motion, I 14, 200 energy, 214 Correlations in a spin-singlet state, 152-155 momentum, 203 Correspondence principle, 260, 263, 343 orbital angular momentum, 331-337 Cosmic background radiation, 326 parity, 273- 274 Coulomb gauge, 488 Eigcnkct, 38, 234, 265 Coulomb interaction, 143 Eigenstate, 38, 67 Coulomb potential, 3 13, 348 Eigenvalue, 38, 67 Covalent bond, 441 Einstein, A., 154, 165 Creation operator, 254, 276, 498 Einstein-Podolsky-Rosen paradox, 155 Cross section Elect1ic dipole differential, 456 approximation, 520 partial wave, 469 moment, 13 I Rutherford, 464 selection rule, 507, 524 total, 456 transition, 524 Curie, P., 178 Electromagnetic units, 539- 543 Curie constant, 178, 188 Electron Curie' s law, 178 g factor, 2 Energy Darwin term, 405-407 conservation of, 138, 514-515 de Broglie relation, 204 eigenstates and eigenvalues, 1J.3 Degeneracy, 81 operator (Hamiltonian), 1l3, 137 uncertainty relation, 134-136 accidental, 360, 368, 375 Entanglement, 166- 169, 179- 181 for harmonic oscillator, 373- 375 Evolutionary time, 135 for hydrogen atom, 359- 360 Exchange operator, 420 Degrees offreedom, 10, 191, 199 eigenstates and eigenvalues of, 420-422 Delta function normalization, 194 Exchange term, 431 Delta function potential, 242 Expectation value, 15, 24, 58, 67 Density of states, 516-5 17 lime dependence of, 114 Density operator, 171-18 1 reduced, 179 Fermi-Dirac statistics, 422 Deuterium, 353- 354 Fermi gas model, 365 Deuteron, 360 Fennions, 422 finite spherical well and, 360-364 Fermi's Golden Rule, 518 fusion and, 447 Feynman, R., 7, 281, 408, 532 Diatomic molecules, 321- 326 Feynman diagrams, 530-532 Diffraction, 300 Page 580 (metric system)

Index I 567 Fine structure, 408 ground state, 424-428 Fine-structure constant, 35 1 variational method and, 433-434 Fine structure of hydrogen, 408 Hermite polynomial, 272 Fourier transform, 204, 206, 237 Hermitian operator, 37,67 Free particle matrix representation of, 50 Hidden-variable theory, !54 Hamiltonian, 20& Hydrogen atom, 348-360. 398-410 propagator, 284 fine structure, 408 Functional integral, 289 hyperfine structure, 143- 147 lifetime of 2p state, 521-524 Gauge radiative transitions, 518-526 Coulomb, 488 uncertainty principle and, 313- 314 transformation, 484 Hydrogen maser, 146 Hydrogen molecule, 447-448 Gaussian integrals, 553-555 Hydrogen molecule ion, 442-446 Gaussian units, 540 Hyperfine interaction, 143- 147 Gaussian wave packet, 204-208 Hypcriine structure, 409 minimum uncertainty state, 207- 208 Identical particles, 419-448 time evolution, 208-210 bosons,422 Generator ferrnions, 422 of rotations, 36-37, 65 helium atom and, 424-432 of time translations, 112, 137 hydrogen molecule and, 447-448 of translations, 197-200, 236, 304, 338 Gerlach, W., 1, 4 Identity operator, 4 1, 46, 68, 235 g factor, 2-3 Impact parameter, 465 magnetic resonance and, 124 Inner product, 12, 142 spin precession of muon, 119- 120 Interaction picture, 510-513 Goudsrnit, S., 2 Interference, 9, 45, 210. 281, 300 Green's function, 459 2rr rotations and. 120-122 Hamiltonian, I 13, 137 graviry and, 297-299 Hard-sphere scattering, 469-471 Intrinsic spin, 2 Harmonic oscillator, 245-276, 369- 375 lnvariance, 303 gaugc, 487 coherent states of, 262-269 under inversion, 274 eigenfunctions, 254-257 rotational, 118. 138,314-317,320 in external electric field, 386-389 translational, 200. 309-311 Iarge-n limit, 259-261 Inversion symmetry, 274 lowering operator, 249. 274 Ionic bonding, 440 number operator, 248 Ionization energy of clements, 438 parity and, 273-274 raising operator, 249, 275 Ket vector, 5, 10, 12,22 three dimensional, 369-375 Klein- Gordon equation, 533 Kronecker delta. 23 in Cartesian coordinates, 370-372 degeneracy, 373-375 Lagranf,>ian, 288 in spherical coordinates, 372-373 Laguerre polynomials, 351 time dependence, 261-262, 264-269 Lamb, W., 408 zero-point energy, 257-259 Lamb shift, 408 Hartt·ee, D., 437 Laplacian Hmtree method, 437 Hcaviside-Lorentz units, 543 in cylindrical coordinates, 344 Heisenberg picture, 508-509 in spherical coordinates, 330 Heisenberg uncertainty relation, 200, 210. 236 Large-n limit Helicity, 64 of harmonic oscillator, 259-261 Helium atom, 424-434 excited states, 428-432 Page 581 (metric system)

568 I Index Least action principle, 29 1-296, 300 spin precession of, 11 9- 120 Legendre polynomials, 334, 342 Muon-catalyzed fusion, 446-447 Legendre's equation, 342 Muonic atom, 446 Lifetime, 5 18 Linear operator, 34 Neutron interferometer, 121 Linewidth, 136, 146 2rr rotations and, 120-122 Loren tz, H., 3 gravity and, 297- 299 Lorentz force No-cloning theorem, 166, 169- 170 in Gaussian units, 542 Normalization in SI units, 539 Lowering operator continuum, 235 for angular momentum, 86 discrete, 23 for harmonic oscillator, 249 Number operator, 248 Lyman series, 353 Observable, 22 Magic numbers, 368 Operators, 34, 65 Magneti c dipole adjoint, 35, 50, 66~ momen t, 1- 3 eigenstates and eigenval ues of, 38, 67 transition, 525 Hen11i tian, 37, 67 Magnetic resonance, 124-.128 identity, 41, 46, 68, 70 Maser, 134, l46 linear, 34 Mat1i x element, 47, 69 matrix. elements of, 47 Matrix mechanics, 29- 32,46-51,69- 70 product of, 51 Matrix representation projection, 42, 68 or bras, 30, 69 unitary, 35, 66 or kets, 30, 69 Optical theorem, 469 or operators, 47, 69 Orbital angular momentum Maxwell's equati ons, 539- 543 eigenCunctions, 33 1-337 in Gaussian units, 542 generators or rotations, 31 5 in Heaviside-Lorentz units, 543 operators, 3 15 in SI units, 539 McKellar, A., 326 in position space, 328- 330 Meas urement Orbitals, 336 of position, 191-192 Orthonormal set, 23 of spill, 3- 5 Outer product., 142 lvlinimum uncertainty state, 207, 258,267 Overl ap in tegral, 444 Mixed state, 171, 183 Molecules, 321-326 · Pais, A., 165 covalent bonding, 441-448 Parity, 223 orbitals and, 336- 337 eigenfunctions, 273 rotation of, 323- 325 operator, 273 vibration of, 32 1-323 Partial wave analysis, 465-469 vibration-rotation of, 325 for finite potential well, 471-473 Moment of inertia, 3 19, 323 for hard-sphere scattering, 469-471 Momen tum resonances and, 473-476 conser vation of, 309-31 1 Particle in a box eigenfunction, 203, 307 one-dimensional, 2 19- 224 operaLor, 198 three-dimensional, 365- 368 Paschen-Bach effect, 4 16 in position space, 202, 307 Paschen series, 353 Momentum space, 202- 204 Passive transformation, 53, 197 Multielectron atoms, 437-441 Path integrals, 281- 30\"1 Muon Aharonov- Bohm effect and, 485-487 for free patticle, 289-291 catalysis, 446-447 gravity and, 297- 299 Page 582 (metric system)

Index I 569 phasors and, 293-295 in three dimensions, 304 principle of least action and, 291-296 Potential Path of least action, 292 Pauli exclusion principle, 423, 448 harmonic oscillator, 246 Pauli spin matrices, 96, 105 Poynting vector. 496 Pekeris, C., 434 Precession Pen1.ias, A., 326 Periodic boundary conditions, 490 ofmuon, 119-120 Periodic table, 438 of neutron, 120-122 multielectron atoms and, 437-441 of spin- ~ particle, 115-118 Perturbation theory Probability, 9, 13, 24 degenerate, 389-391 conservation of, lll-112 nondegenerate. 381-386 Probability amplitude, 9, 22 Probability current, 226 harmonic oscillator and, 386-389 Probability density, 454 relativistic perturbations in hydrogen and, in momentum space, 203, 307 in one dimension~ l94 398-407 in three dimensions, 304 time-dependent, 504-506 Projection operators, 42, 68 matrix representation of, 48 harmonic oscillator and, 506-507 Propagator, 283 Perturbing Hamiltonian, 381 free particle, 284 Phase Pure state, 171, 183 overall, 18, 22 Quantization of radiation field , 493-499 relative, 25 Quantum electnJdynam.ics (QED), 408, Phase shift, 467 Phase space, 516 532 Phasors, 293 Quantum teleportation, 165-169, 180-181 Photon,59-64,495-498 Quarks, 2. !50, 452 circularly polarized, 62-64, 498 energy and momentum, 495-498 Rabi's formula, 126 intrinsic spin of, 64, 498 Radial equation, 320 linearly polarized, 60-62 Phown-at.om scattering, 530-532 behavior at origin, 345-347 Picture Radiation Heisenberg, 508-509 interaction, 510-513 blackbody, 492 Schrodinger, 507-508 electric dipole, 520-524 Pion magnetic dipole, 524-526 as a bound state of quarks, 150 spontaneow; emission, 518-528 decay of, 119 stimulated emission, 126, 519 scattering with proton, 475-476 Raising operator Planck's constant, 4 for angular momentum, 86 Plane wave for harmonic oscillator, 249 partial wave expansion, 466 Ramsauer- Townsend effect, 477 Podolsky, B., 154 Reduced density operator. 179 Poisson distribution, 50 I, 535 Reduced mass. 312 Poisson's bright spot, 471 Reflection coefficient, 228 Polarization for step potential, 229 circular, 62 Relative and center-of-mass coordinates, linear, 60 Position eigenstate, 191 311- 313 Position-momentum uncertainty, 200, 207. Resonance 210,236 magnetic, 124-128 Position-space wave function scanering,473-476 Retherford, R., 408 in one dimension, 195 Rigid rotator, 323, 343 Rosen, N., 154 Page 583 (metric system)

570 Index Spherical well finite, 360-364 Rotauon infirlite, 365-368 marrices, 54, 62 ot molecules, 323 Spin, 1- 5 operators, 33-40, 65 measurement, 5 precession, I 15- 118 Rotations b) 1.;r radians, 40, 122 Spinor, 405 generator of. 36, 65 Spin-orbit coupling, 364, 368, 400-405 noncommutativity of, 75 Spin(s) Rutherford scattering, 463-465, 480 addition of, 147-152 Rydberg atom, 527 Spin-spin interaction Rydberg constant, 353 hyperfine splitting of hydrogen, 143- 147, Scanering, 451-478 150 amplitude. 458 a3ymptotic wave function. 454 Spin-statistics theorem, 423 Born approximation, 458-463 Square-well potential differential cross ser.;tion, 456 one-dimensional, 224-231 in one dimension~214-2 19 optical theorem, 469 scattering, 47 1-473 partial waves, 465-469 in three di mensions, 360- 363 photon-atom, 530-532 Standard deviation pion-proton. 475-476 and uncertainty, 16 by potential barrier, 230-23 1 Stark effect, 391-395 resonant, 473-476 Statcoulomb, 540 Ruthe1tord, 45 I, 463-465 State vector by a step potential, 226-230 quantum, I0-14 total cross section, 456 Stationary state, 114 Statistical fluctuations Schrodinger equation, 112. 137 and measurement, 17 in position space, 214 Statistics in three dimensions, 319, 330 Bose- Einstein, 422 time-dependent, 214 Fermi- Dirac, 422 time-independent, 214, 232 Stern, 0., 1, 5 Stern-Gerlach experiment, 3- 5 Schrodinger picture, 507- 508 Stimulated emission, 126,5 19 Schwarz inequality, 92 SupellJOSition, 13, 22,25 Selection rules, 324, 325, 343, 507, 524 Symmetric states, 422 Self-adjoint operator, 37, 67 Symmetry Separation of variables, 371 underexchange,423 SG device, 5 in quantum mecharucs, 339 rotational, 317 modified, 7, 41 translational, 309- 3 I I Shell stn1cture Symmetry operation, 118, 138 of atom, 358, 405 Teleportation. See Quantum teleponation of nucleus, 368 Thomas, L., 401 Silver atom Thomas precession, 40 I and Stem-Gerlach experiment, 3- 5, 26 Time-dependent perturbation theory. 504-506 Singlet spin state, 150, 182 Time derivative Single-valuedness, 329 Solid angle, 332 of expectation val ues, 114 Spectroscopic notation, 359, 404, 425 of operator, I 14 Spherical Bessel equation. 366 Time evolution operator. 111, 137 Spherical Bessel function s, 366, 466 Transformation Spherical coordinates. 316 active, 53, 197 Sphericalharmorucs,331-337 gauge,484 Spherical Neumann functions, 366, 466 Page 584 (metric system)

passive, 53, 197 Index I 571 Transition Urey, H., 354 electric dipole, 524 electric quadrupole, 526 Vacuum state, 495 magnetic dipole. 525 Variational method, 432-433 rate. 517- 518 Transition amplitude, 283 for helium, 433-434 Translational invariance, 309-311 for hydrogen molecule ion. 442-446 Transmission coefficient, 228 Vector operators, 82. 316 for square barrier, 231 Vector potential, 483 for step potential, 229 Aharonov-Bohm cffec.:t and, 484-487 Triplet spin state, 150, 182 operator, 494 Tunneling, 230-234 Vibration of molecule, 321-323 ammonia molecule and. 130 Virial theorem, 378 ethylene and, 251 Two-term recursion rclatitm, 271 Wave function in momentum space, 203 Uhlenbeck, G., 2, 40 I in position spact!. 195 Uncertainty Wave packet definition of, 16, 24 Gaussian, 204-207 Uncertainry principle scattering, 452 spreading, 209 estimaring energies and, 313-314 Uncertainry relations Werner, S. A., 120 Wilson, R., 326 angular momentum, 93 derivation of, 92-93 Yukawa potential, 463 energy-time, 134- 136 Heisenberg, 200, 236 Zeeman, P., 410 position-momentum, 200, 210, 236 Zeeman effect, 410-412 Unitary operator, 35, 66 Zero-point energy, 257-259 Unperturbed Hamil tonian, 38 1 behavior of helium. 258 ofelectromagnetic field, 495 Page 585 (metric system)