Ouestions

Light Q,uanta-Photons An electromagnetic wave(light) is quantized, and its quanta are called photons. For alight wave of frequency / and wavelength ,1,, the energy E andmomentum magnitude p of a photon are

E : hf (photon energy) (38-Z)

The Wave Function A matter wave is described by itswave function V(r, !, 2,, r), which can be separated into aspace-dependent part Q(x,, y., z) and a time-dependent part,-iat. For a particle of mass m moving in the x direction withconstant total energy E through a region in which its potentialenergy is U(*), {(*) can be found by solving the simplifiedSchrtidinger equation:

and p-+:+ (photonmomentum).

A),: frO - cos @),

d.',! , 8rz m^t-z -i[E- U(*)]Q:0.ax" n'

(38-7)(38-1s)

Photoelectric Effect When light of high enough fre-quency falls on a clean metal surface, electrons are emittedfrom the surface by photon-electron interactions within themetal. The governing relation is

h.f:Krnu*+O, (38-s)in which hf is the photon energy, K-u* is the kinetic energy ofthe most energetic emitted electrons, and O is the work func-tion of the target material-that is, the minimum energy anelectron must have if it is to emerge from the surface of thetarget. It hf is less than O, electrons are not emitted.

Compton Shift When x rays are scattered by looselybound electrons in a target, some of the scattered x rays havea longer wavelength than do the incident x rays. ThisCompton shift (in wavelength) is given by

A matter wave, like a light wave, is a probability wave in thesense that if a particle detector is inserted into the wave, theprobability that the detector will register a pafiicle during anyspecified time interval is proportional to lrltl',a quantity calledthe probability density.

For a free particle -

that is, a particle for which U(*) :0-moving in the x direction, lrl12 has a constant value for allpositions along the.r axls.

Heisenberg's Uncertainty Principle The probabilisticnature of quantum physics places an important limitation ondetecting a particle's position and momentum. That is, it is notpossible to measure the positioni and the momentum f of aparticle simultaneously with unlimited precision. The uncer-tainties in the components of these quantities are given by

L,x' Lp* = FLLy' Lpy > h

L^z' Lp r. > fL.

(38-20)

Barrier Tunneling According to classical physics, an inci-dent particle will be reflected from a potential energy barrierwhose height is greatu than the particle's kinetic energy.According to quantum physics, however, the particle has afinite probability of tunneling through such a barrier. Theprobability that a given particle of mass m and energy E willtunnel through a barrier of height (lb and thickness L is givenby the transmission coeffici ent T:

f: e-zbL, (38-2r)

Q8-n)

(38-11)

in which 0 is the angle at which the x rays are scattered.

Light Waves and Photons When light interacts withmatter, energy and momentum are transferred via photons.When light is in transit, however, we interpret the light waveas a probability waye, in which the probability (pet unit time)that a photon can be detected is proportional to E2^, where ,E-is the amplitude of the oscillating electric fleld of the lightwave at the detector.

Matter Waves A moving particle such as an electron or aproton can be described as a matter waye; its wavelength(called the de Broglie wavelength) is given by r\ : hlp,,wherep is the magnitude of the particle's momentum. where

Brrzm(U6

X According to the figure for Checkpoint 2, ts the maximumkinetic energy of the ejected electrons greater for a targetmade of sodium or of potassium for a given frequency of inci-dent light?g" Photoelectric effect: Figure38-18 gives the stopping voltageI/ versus the wavelength i oflight for three different materi-als. Rank the materials accord-ing to their work function,greatest first.

3 A metal plate is illuminated with light of a certain fre-quency. Which of the following determine whether or not elec-trons are ejected: (a) the intensity of the light, (b) how longthe plate is exposed to the light, (c) the thermal conductivityof the plate, (d) the area of the plate, (e) the material of whichthe plate is made?4 In the photoelectric effect (for a given target and a given fre-quency of the incident light), which of these quantities, if any, de-pend on the intensity of the incident light beam: (a) the maximumkinetic energy of the electrons, (b) the maximum photoelectriccurrent, (c) the stopping potential, (d) the cutoff frequency?F$ffi" SS-t I Question2.

Chapter 3E I Photons and Matter WavesS Photon A has twice the energy of photon B. (u) Is themomentum of A less than, equal to, or greater than that of B? (b)Is the wavelengthof Aless than, equal to, or greater than that of B?6 Figure 38-19 shows an electron moving through several re-gions where uniform electric potentials V have been set up.Rank the three regions according to the de Broglie wave-length of the electron there , greatest first.

123Ftffi. 3S-t I Question 6.

7 (a) If you double the kinetic energy of a nonrelativistic parti-cle, how does its de Broglie wavelength change? (b) What if youdouble the speed of the particle?S Compton scattering. Figure 38-20 gives the Compton shift A), ver-sus scattering angle $ for three dif-ferent stationary target particles.Rank the particles according totheir mass, greatest first.

9 In a Compton-shift experi-ment, light (in the x-ray range) is Ffiffi" 3ffi-ffffi Question 8.scattered in the forward direction, at Q-

0 in Fig. 38-3. What fraction of thelight's energy does the electron acquire?X0 Let K be the kinetic energy that a sta-tionary free electron gains when a photonscatters from it. We can plot K versus theangle Q at which the photon scatters; seecurve 1 in Fig. 38-ZI.If we switch the targetto be a stationary free proton, does the endpoint of the graph shift (a) upward as sug-gested by curve 2, (b) downward as sug-gested by curve 3, or (c) remain the same?1{ The following nonrelativistic particles all have the samekinetic energy. Rank them in order of their de Broglie wave-lengths, greatest first: electron, alpha particle, neutron.

1A Figure 38-22 shows an electron moving (u) opposite anelectric fleld, (b) in the same direction as an electric field, (c) inthe same direction as a magnetic field, (d) perpendicular to amagnetic field. For each situation, is the de Broglie wavelengthof the electron increasing, decreasing, or remaining the same?

(d)(c)(b)(a)Ff,ffi. S8-ft4 Question12.

13 The table gives relative values for three situations for thebarrier tunneling experiment of Figs. 38-14 and 38-15. Rankthe situations according to the probability of the electron tun-neling through the barrier , greatest first.

BarrierThickness

(u)(b)(c)

5E,

17E2E

LL122L

t 4 For three experiments, Fig.38-23 gives the transmission co-efficient Z for electron tunnelingthrough a potential barrier, plot-ted versus barrier thickness L.The de Broglie wavelengths ofthe electrons are identical in thethree experiments. The only dif-ference in the physical setups isthe barrier heights U6. Rank thethree experiments according toU 6,, greztest first. F!ffi. Sffi-43 Question14.1S At the left in Fig. 38-16, why are the minima in the valuesof-l$lz greater than zero?16 An electron and a proton have the same kinetic energy.Which has the greater de Broglie wavelength?

ElectronEnergy

BarrierHeight

EEE

180"

FEffi" $S-e1Question 10.

Tutoring problem available (at instructor's discretion) in WileyPLUS and WebAssignSSM Worked-out solution available in Student Solutions Manual WWW Worked-out solution is ats

-

oss Number of dots indicates level of problem difficulty lLW lnteractive solution is atAdditional information available in The Flying Circus of Physics and at flyingcircusofphysics.com

sffn. 3S-A The Photon, the Ouantum of Lighte{ At whal rate does the Sun emit photons? For simplicity,assume that the Sun's entire emission at the rate of 3.9 x1026 W is at the single wavelength of 550 nm.wp A helium-neon laser emits red light at wavelength ),: 633nm in a beam of diameter 3.5 mm and at an energy-emission rateof 5.0 mW.A detector in the beam's path totally absorbs the beam.At what rate per unit areadoes the detector absorb photons?

*$ The meter was once defined as 1 650 763.73 wavelengthsof the orange light emitted by a source containing krypton-86atoms. What is the photon energy of that light?oS The yellow-colored light from a highway sodium lampis brightest at a wavelength of 589 nm. What is the photonenergy for light at that wavelength?o$ Monochromatic light (that is, light of a single wave-length) is to be absorbed by a sheet of photographic fllm and

thus recorded on the film. Photon absorption will occur if thephoton energy equals or exceeds 0.6 eV, the smallest amountof energy needed to dissociate an AgBr molecule in the film.(u) What is the greatest wavelength of light that can berecorded by the fllm? (b) In what region of the electromag-netic spectrum is this wavelength located?e$ How fast must an electron move to have a kinetic energyequal to the photon energy of sodium light at wavelength590 nm?ssP An ultraviolet lamp emits light of wavelength 400 nm atthe rate of 400 W. An infrared lamp emits light of wavelength700 nffi, also at the rate of 400 W. (a) Which lamp emits pho-,:fi,*rf" greater rate and (b) what is that greater rate?

ssffi A satellite in Earth orbit maintains a panel of solar cellsof area 2.60 m2 perpendicular to the direction of the Sun'slight rays. The intensity of the light at the panel is 1.39 kW/m2.(a) At what rate does solar energy arrive at the panel? (b) Atwhat rate are solar photons absorbed by the panel? Assumethat the solar radiation is monochromatic, with a wavelengthof 550 rffi, and that all the solar radiation striking the panel isabsorbed. (c) How long would it take for a "mole of photons"to be absorbed by the panel?**S A special kind of lightbulb emits monochromatic lightof wavelength 630 nm. Electrical energy is supplied to it at therate of 60 W, and the bulb is 93% efficient at converting thatenergy to light energy. How many photons are emitted by thebulb during its lifetime of 730 h?o * t S Under ideal conditions, a visual sensation can occur inthe human visual system if light of wavelength 550 nm isabsorbed by the eye's retina at a rate as low as L00 photonsper second. What is the corresponding rate at which energyis absorbed by the retina?e o t $ A 100 W sodium lamp (tr : 589 nm) radiates energyuniformly in all directions. (a) At what rate are photons emit-ted by the lamp? (b) At what distance from the lamp will atotally absorbing screen absorb photons at the rate of 1.00 pho-ton/cmz's? (c) What is the photon flux (photons per unit areaper unit time) on a small screen 2.00 m from the lamp?n*Xfr A light detector has an ab-sorbing area of 2.00 X 10-6 n]and absorbs 50"/" of the incidentlight, which is at wavelength 600nrn. The detector faces anisotropic source, 12.0 m from thesource. The energy E emitted bythe source versus time r is given inFig. 3B-24 (E,=7.2nJ,t,=2.0 s).Atwhat rate ate photons absorbedby the detector?

E (nJ)

Problems

forms a circular diffraction pattern whose central disk has aradius R given by 1 .22 fLIld.It can be shown that 84% of theincident energy ends up within this central disk. At whatrale are photons absorbed by the screen in the central disk ofthe diffraction pattern?

sss. 38-3 The Photoelectric Effect*t$ The work function of tungsten is 4.50 eV. Calculate thespeed of the fastest electrons ejected from a tungsten surfacewhen light whose photon energy is 5.80 eV shines on thesurface.*'ES You wish to pick an element for a photocell that willoperate via the photoelectric effect with visible light. Whichof the following are suitable (work functions are in parenthe-ses): tantalum (4.2 eV), tungsten (4.5 eV), alumintm (4.2 eV),barium (2.5 eV),lithium (2.3 eV)?*XV Light strikes a sodium surface, causing photoelectricemission. The stopping potential for the ejected electrons is5.0 V and the work function of sodium is 2.2 eY. What is thewavelength of the incident light? ssM*X & Find the maximum kinetic energy of electrons ejectedfrom a certain material if the material's work function is 2.3 eVand the frequency of the incident radiation is 3.0 x 101s Hz.I s 1 I The stopping potential for electrons emitted froma surface illuminated by light of wavelength 49I nm is 0.710 V.When the incident wavelength is changed to a new value, thestopping potential is I.43 V. (u) What is this new wavelength?(b) What is the work function for the surface?ssftffi The wavelength associated with the cutoff frequencyfor silver is 325 nm. Find the maximum kinetic energy ofelectrons ejected from a silver surface by ultraviolet light ofwavelength 254 nm.se&n Light of wavelength 200 nm shines on an aluminumsurface;4.20 eV is required to eject an electron. What is thekinetic energy of (a) the fastest and (b) the slowest ejectedelectrons? (c) What is the stopping potential for this situa-tion? (d) What is the cutoff wavelength for aluminum? ssM***3 An orbiting satellite can become charged by the pho-toelectric effect when sunlight ejects electrons from its outersurface. Satellites must be designed to minimrze such charg-itrg. Suppose a satellite is coated with platinuffi, o metal with avery large work function (Q: 5.32 eV). Find the longestwavelength of incident sunlight that can eject an electronfrom the platinum.ssAS (a) If the work function for a certain metal is 1.8 eV,what is the stopping potential for electrons ejected from themetal when light of wavelength 400 nm shines on the metal?(b) What is the maximum speed of the ejected electrons?ssft4 Suppose the fractional efficiency of a cesium surface(with work function 1.80eV) is 1.0 x 10-16'that is, on averageone electron is ejected for every 1016 photons that reach the sur-face. What would be the current of electrons ejected from such asurface if it were illuminated with 600 nm light from a2.00 mWlaser and all the ejected electrons took part in the charge flow?seff5 X rays with a wavelength of 7I pm are directed onto agold foil and eject tightly bound electrons from the gold atoms.The ejected electrons then move in circular paths of radius r ina region of uniform magnetic field B. For the fastest of the

E,

otsF*ffi" S&-44 Problem

r (s)

12,

**1$ A light detector (your eye) has an area of 2.00 x 10-6m2 and absorbs 80% of the incident light, which is at wave-length 500 nm. The detector faces an isotropic source, 3.00 mfrom the source. If the detector absorbs photons at the rate ofexactly 4.000 S-1, at what power does the emitter emit light?oe14 The beam emerging from a 1.5 W argon laser (r\:515 nm) has a diameter d of 3.0 mm.The beam is focused by alens system with an effective focal length fr of 2.5 mm. Thefocused beam strikes a totally absorbing screen, where it

fihapten 3ffi I Photons and Matter Waves

ejected electrons, the product Br is equal to 1.88 x I0-4 T.m.Find (a) the maximum kinetic energy of those electrons and(b) the work done in removing them from the gold atoms.**ff6 In a photoelectric experiment using a sodium surface,you flnd a stopping potential of 1.85 V for a wavelength of300 nm and a stopping potential of 0.820 V for a wavelengthof 400 nm. From these data find (a) a value for the Planck con-stant, (b) the work function O for sodium, and (c) the cutoffwavelength ),6 for sodium.

se. S&-4 Photons Have MomentumURT What (a) frequency, (b) photon energy, and (c) photonmomentum magnitude (in keV/c) are associated with x rayshaving wavelength 35.0 pm?*8S (a) In MeV/c, what is the magnitude of the momentumassociated with a photon having an energy equal to the elec-tron rest energy? What are the (b) wavelength and (c) fre-quency of the corresponding radiation?nff$ Light of wavelength 2.40 pm is directed onto a targetcontaining free electrons. (u) Find the wavelength of lightscattered at 30.0" from the incident direction. (b) Do the samefor a scattering angle of I20" . ssMss$ffi X rays of wavelength 0.0100 nm are directed in thepositive direction of an x axis onto a target containing looselybound electrons. For Compton scattering from one of thoseelectrons, at an angle of 180o, what are (a) the Compton shift,(b) the corresponding change in photon energy, (c) the kineticenergy of the recoiling electron, and (d) the angle between thepositive direction of the x axis and the electron's direction ofmotion?seSS Calculate the percentage change in photon energyduring a collision like that in Fig. 38-5 for Q: 90" and forradiation in (a) the microwave range, with i

- 3.0 cm; (b) the

visible range, with I -

500 nm; (c) the x-ray range, with i -

25 pm:' and (d) the gamma-ray range, with a gamma photonenergy of 1.0 MeV. (e) Wh at are your conclusions about thefeasibility of detecting the Compton shift in these variousregions of the electromagnetic spectrum, judging solely bythe criterion of energy loss in a single photon

- electron en-

counter?ssSA Gamma rays of photon energy 0.511 MeV aredirected onto an aluminum target and are scattered in variousdirections by loosely bound electrons there. (a) What is thewavelength of the incident gamma rays? (b) What is the wave-length of gamma rays scattered at 90.0' to the incident beam?(c) What is the photon energy of the rays scattered in thisdirection?e6g3 Calculate the Compton wavelength for (a) an electronand (b) u proton. What is the photon energy for an electro-magnetic wave with a wavelength equal to the Comptonwavelength of (c) the electron and (d) the proton?es$4 Show that when a photon of energy E is scattered froma free electron at rest, the maximum kinetic energy of therecoiling electron is given by

pzK^u*: p * *pr-ssSS What are (a) the Compton shift Al, (b) the fractional

Compton shift AI/I, and (c) the change AE in photon energyfor light of wavelength |

-

590 nm scattering from a free,initially stationary electron if the scattering is at 90' to thedirection of the incident beam? What are (d) A),, (e) Ai/I, and(0 AE for 90' scattering for photon energy 50.0 keV (x-rayrange)?ss36 What is the maximum kinetic energy of electronsknocked out of a thin copper foil by Compton scattering of anincident beam of. 17 .5 keV x rays? Assume the work functionis negligible.8637 What percentage increase in wavelength leads to a 75%loss of photon energy in a photon-free electron collision?eS$ A photon undergoes Compton scattering off a station-ary free electron. The photon scatters at 90.0' from its initialdirection; its initial wavelength is 3.00 x t0-r2 m. What is theelectron's kinetic ener gy?@639 Consider a collision between an x-ray photon of initialenergy 50.0 keV and an electron at rest, in which the photon isscattered backward and the electron is knocked forward.(a) What is the energy of the back-scattered photon? (b) Whatis the kinetic energy of the electron?**4S What is the maximum wavelength shift for a Comptoncollision between a photon and a free proton?ss4$ Through what angle must a 200 keV photon be scat-tered by a free electron so that the photon loses I0% of itsenergy?

$e. SS-& Electrons and Matter Waves*&2 Calculate the de Broglie wavelength of (a) a 1.00 keVelectron,(b) a 1.00 keV photon, and (.) u 1.00 keV neutron.*4S In an old-fashioned television set, electrons are acceler-ated through a potential difference of 25.0 kV. What is thede Broglie wavelength of such electrons? (Relativity is notneeded.) ssilte44 A stream of protons, each with a speed of 0.9900c, aredirected into a two-slit experiment where the slit separation is4.00 x 10-e m. A two-slit interference pattern is built up onthe viewing screen. What is the angle between the center ofthe pattern and the second minimum (to either side of thecenter) ?**4S What is the wavelength of (a) a photon with energy1.00 eV, (b) an electron with energy 1.00 eY (c) a photon ofenergy 1.00 GeY and (d) an electron with energy 1.00 GeV?**46 An electron and a photon each have a wavelength of0.20 nm. What is the momentum (in kg.m/s) of the (a) elec-tron and (b) photon? What is the energy (in eV) of the (r)electron and (d) photon?s s47 Singly charged sodium ions are accelerated through apotential difference of 300 V. (a) What is the momentumacquired by such an ion? (b) What is its de Broglie wave-length? ssM*t4S The existence of the atomic nucleus was discovered inI9II by Ernest Rutherford, who properly interpreted someexperiments in which a beam of alpha particles was scatteredfrom a metal foil of atoms such as gold. (u) If the alpha par-ticles had a kinetic energy of 7.5 MeV what was their deBroglie wavelength? (b) Explain whether the wave nature of

the incident alpha particles should have been taken intoaccount in interpreting these experiments. The mass of analpha particle is 4.00 u (atomic mass units), and its distance ofclosest approach to the nuclear center in these experimentswas about 30 fm. (Th" wave nature of matter was not postu-lated until more than a decade after these crucial experimentswere first performed.)* "49 The wavelength of the yellow spectral emission line of

sodium is 590 nm. At what kinetic energy would an electronhave that wavelength as its de Broglie wavelength? sstrrt*uS0 Electrons accelerated to an energy of 50 GeV have ade Broglie wavelength ,tr small enough for them to probe thestructure within a target nucleus by scattering from the struc-ture. Assume that the energy is so large that the extreme rela-tivistic relation p : Elcbetween momentum magnitude p andenergy E apphes. (In this extreme situation, the kinetic energyof an electron is much greater than its rest energy.) (a) What isi? (b) If the target nucleus has radius R

- 5.0 fm, what is the

ratio Rli?#eS'l A nonrelativistic particle is moving three times as fastas an electron. The ratio of the de Broglie wavelength of theparticle to that of the electron is 1.813 x 10 -+.By calculatingits mass, identify the particle.s6Se What are (a) the energy of a photon corresponding towavelength 1.00 nffi, (b) the kinetic energy of an electron withde Broglie wavelength 1.00 nffi, (c) the energy of a photoncorresponding to wavelength 1.00 fm, and (d) the kineticenergy of an electron with de Broglie wavelength 1.00 fm?so53 The highest achievable resolving power of a micro-scope is limited only by the wavelength used; that is, thesmallest item that can be distinguished has dimensions aboutequal to the wavelength. Suppose one wishes to "see" insidean atorn. Assuming the atom to have a diameter of 100 pffi,this means that one must be able to resolve a width of, say,10 pm. (a) If an electron microscope is used, what minimumelectron energy is required? (b) If a light microscope is used,what minimum photon energy is required? (c) Which rnicro-scope seems more practical? Why?s"54 What accelerating voltage would be required for theelectrons of an electron microscope if the microscope is tohave the same resolving power as could be obtained using100 keV gamma rays? (See Problem 53 and assume classicalphysics holds.)e oo$$ If the de Broglie wavelength of a proton is 100 fm,(u) what is the speed of the proton and (b) through whatelectric potential would the proton have to be accelerated toacquire this speed?

sec" 3S-7 Schriidinger's Equation@056 (a) Write the wave function Q@) displayed in Eq. 38-19in the form Q@) - a + ib, where a and b are real quantities.(Assume that r/o is real.) (b) Write the time-dependent wavefunction V(x, /) that corresponds to r!(x) written in this form.ss57 Show that Eq. 38-17 is indeed a solution of Eq. 38-16by substituting t@) and its second derivative into Eq. 38-16and noting that an identity results.ssSS (a) Let n

- a * ib be a complex number, where a and

b arc real (positive or negative) numbers. Show that the prod-

Problems

uct nn* is always a positive real number. (b) Let m : c -f idbeanother complex number. Show that lnml : Inl lml.eeSS Show that the angular wave number k for a nonrela-tivistic free particle of mass m can be written as

k-

in which K is the particle's kinetic energy. ssM{h s6S Suppose we put A

-

0 in Eq. 38-17 and relabeled B asfo. (a) What would the resulting wave function then describe?(b) How, if at all,would Fig. 3B-I2 be altered?ssST The function Q@) displayed in Eq. 38-19 describes afree particle, for which we assumed that U(*)

-

0 inSchrodinger's equation (Eq. 38-15). Assume now that U(*)

-

Uo : a constant in that equation. Show that Eq. 38-19 is still asolution of Schr6dinger's equation, with

r--uit-K -;\2m(E - uo)

now giving the angular wave number k of the particle. ssMss6ft In Eq.38-18 keep both terms,putting A

-

B - fo.The

equation then describes the superposition of two matterwaves of equal amplitude, traveling in opposite directions.(Recall that this is the condition for a standing wave.)(a) Show that lV(x, r) l2 is then given by

lV(x, t)12 -

2r['oll t cos 2kx].(b) Plot this function, and demonstrate that it describes thesquare of the amplitude of a standing matter wave. (c) Showthat the nodes of this standing wave are located at

x -

(2n + 1)(1i), where n -

0, 1, 2,3

and ,[ is the de Broglie wavelength of the particle. (d) Write a simi-lar expression for the most probable locations of the particle.

sec. 3E-8 Heisenberg's Uncertainty Principles63 The uncertainty in the position of an electron along an xaxis is given as 50 pffi, which is about equal to the radius of a hy-

ff:ftr*:T;f.il:'L'*::?:f ::ilifrlff ?::l;;T:ffi H;es64 You will find in Chapter 39 that electrons cannot movein definite orbits within atoms, like the planets in our solarsystem. To see why, let us try to "observe" such an orbiting elec-tron by using a light microscope to measure the electron's pre-sumed orbital position with a precision of, say, 10 pm (a typicalatom has a radius of about 100 pm).The wavelength of the lightused in the microscope must then be about 10 pm. (a) Whatwould be the photon energy of this light? (b) FIow much energywould such a photon impart to an electron in a head-on colli-sion? (.) What do these results tell you about the possibility of"viewin g" an atomic electron at two or more points along itspresumed orbital path? (Hint: The outer electrons of atoms arebound to the atom by energies of only a few electron-volts.)o*65 Figure 38-12 shows a case in which the momentumcomponent p* of a paftrcle is fixed so that Lp*

-

0; then, fromHeisenberg's uncertainty principle (Eq. 38-20), the position xof the particle is completely unknown. From the same prin-ciple it follows that the opposite is also true; that is, if the

ffihmpten Sffi I Photons and Matter Waves

position of a particle is exactly known (A" -

0), the uncer-tainty in its momentum is infinite.

Consider an intermediate case, in which the position of aparticle is measured, not to infinite precision, but to withina distance of A,lZrr,where,\ is the particle's de Broglie wave-length. Show that the uncertainty in the (simultaneously mea-sured) momentum component is then equal to the componentitself; that is, Lp,

-

p. Under these circumstances, would ameasured momentum of zero surprise you? What about ameasured momentum of 0.5p? Of 2p? Of lZp?

$fr, ffiffi"$ Barrier Tunneling#s#6 (a) Suppose a beam of 5.0 eV protons strikes a poten-tial energy barrier of height 6.0 eV and thickness 0.70 nm, ata rate equivalent to a current of 1000 A. How long wouldyou have to wait-on average-for one proton to be trans-mitted? (b) How long would you have to wait if the beamconsisted of electrons rather than protons?s 6S7 Consider the barrier-tunneling situation in SampleProblem 38-7. What percentage change in the transmissioncoefficient T occurs for a I.0"/o change in (u) the barrierheight, (b) the barrier thickness, and (c) the kinetic energy ofthe incident electron? ssMsff&ffi Consider a potential energy barrier like that of Fig.38-15 but whose height U6 is 6.0 eV and whose thickness L is0.70 nm. What is the energy of an incident electron whosetransmission coefficient is 0.0010?es#W A 3.0 MeV proton is incident on a potential energybarrier of thickness 10 fm and height 10 MeV. What are (u)the transmission coefflcient f, (b) the kinetic energy K, theproton will have on the other side of the barrier if it tunnelsthrough the barrier, and (c) the kinetic energy K, it will haveif it reflects from the barrier? A 3.0 MeV deuteron (the samecharge but twice the mass as a proton) is incident on the samebarrrer. What are (d) T, (e) K,, and (t) K,?Additional Problems?ffi In about I9I6,R. A. Millikan found the following stopping-potential data for lithium in his photoelectric experiments:

433.9 404.7 365.0

0.55 0.73 1.09

is the Boltzmann constant. (c) Can the atoms be treated asparticles under these conditions? Explain.74 (a) The smallest amount of energy needed to eject anelectron from metallic sodium is 2.28 eV. Does sodium showa photoelectric effect for red light, with I

- 680 nm? (b) What

is the cutoff wavelength for photoelectric emission fromsodium? (c) To what color does that wavelength correspond??S A spectral emission line is electromagnetic radiation that isemitted in a wavelength range narrow enough to be taken as asingle wavelength. One such emission line that is important inastronomy has a wavelength of- 2I cm. What is the photon en-ergy in the electromagnetic wave at that wavelength?

7# Using the classical equations for momentum and kineticenergy, show that an electron's de Broglie wavelength in nan-ometers can be written as I

-

I.2261\[ K, in which K is theelectron's kinetic energy in electron-volts.3V A bullet of mass 40 g travels at 1000 m/s. Although thebullet is clearly too large to be treated as a matter wave, deter-mine what Eq. 38-13 predicts for the de Broglie wavelength ofthe bullet at that speed.?S Figure 38-12 shows that because of Heisenberg's uncer-tainty principle, it is not possible to assign an x coordinate tothe position of the electron. (u) Can you assign a y or a zcoordinate? (Hint: The momentum of the electron has no y orz component.) (b) Describe the extent of the matter wave inthree dimensions.?S Imagine playing baseball in a universe (not ours !) wherethe Planck constant is 0.60 J ' s. What would be the uncertaintyin the position of a 0.50 kg baseball that is moving at 20 m/salong an axis if the uncertainty in the speed is 1.0 m/s? ssMSffi A 1500 kg car moving at20 m/s approaches a hill that is24 mhigh and 30 m long. Although the car and hill are clearly toolarge to be treated as matter waves, determine what Eq. 38-21predicts for the transmission coefficient of the car, as if it couldtunnel through the hill as a matter wave. Tleat the hill as a poten-tial energy barrier where the potential energy is gravitational.ffit Show that l{tl2

-

l\P12, with t! and V related as in Eq.38-14. That is, show that the probability density does not de-pend on the time variable.8# Show that LElE,the fractional loss of energy of a photonduring a collision with a particle of mass m,rs given by

LE pf ,r (1 _ cos {),E mct \where E is the energy of the incident photon,f is the frequencyof the scattered photon, and ,f is defined as in Fig. 38-5.

ffiffi Show, by analyzing a collision between a photon and afree electron (using relativistic mechanics), that it is impossi-ble for a photon to transfer all its energy to a free electron(and thus for the photon to vanish).S4 An electron of mass m and speed v "collides" with agamma -ray photon of initial ener gy hfo, as measured in thelaboratory frame. The photon is scattered in the electron'sdirection of travel. Verify that the energy of the scatteredphoton, as measured in the laboratory frame, is

Wavelength (tr-)Stopping

potential (V)

312.5 253.5

t.67 2.57Use these data to make a plot like Fig. 38-2 (which is forsodium) and then use the plot to flnd (a) the Planck constantand (b) the work function for lithium.7'H Derive Eq. 38-11, the equation for the Compton shift,from Eqs. 38-8,38-9, and 38-10 by eliminating v and 0.?ffi Neutrons in thermal equilibrium with matter have anaverage kinetic energy of (3lDkf, where k is the Boltzmannconstant and Z, which may be taken to be 300 K, is the tem-perature of the environment of the neutrons. (a) What is theaverage kinetic energy of such a neutron? (b) What is thecorresponding de Broglie wavelength?Yff Consider a balloon filled with helium gas at room tem-perature and atmospheric pressure. Calculate (a) the averagede Broglie wavelength of the helium atoms and (b) the aver-age distance between atoms under these conditions. Theaverage kinetic energy of an atom is equalto (312)kT,where k