sional infinite potential well that forms a rectangular corral are where

Ouestions

Ame'R __Z:I.097373 x 107m 18e(1h'c

LP(r) :

; r2r-2rta

(3e-20)

where nx rs a quantum number for which the electron's matterwave flts in well width L, and ny rs a quantum number forwhich the electron's matter wave fits in well width Lr. Simi-larly, the energies for an electron trapped in a three-dimen-sional infinite potential well that forms a rectangular box are

En*,n,:#(+* ,4),

En*,,y,n,: #(+. +. +)

1 lI I \T: u\

"t"- ,**)'

is the Rydberg constant.The radial probability density P(r) for a state of the

hydrogen atom is deflned so that P(r) dr rs the probability thatthe electron will be detected somewhere in the space betweentwo concentric shells of radii r and r * dr centered on theatom's nucleus. For the hydrogen atom's ground state,

(3e-37)

(3e-44)Here nz rs a third quantum number, one for which the matterwave fits in well width L..

The Hydrogen Atom Both the (incorrect) Bohr modelof the hydrogen atom and the (correct) application of Schro-dinger's equation to this atom give the quantized energylevels of the atom as

L. _ mea r _ 13.6oevnn - Sezghz n2 n2 ' ,,"' (39-32,39-33)

fot n : I,2,3,From this we flnd that if the atom makes a transitionbetween any two energy levels as a result of having emitted orabsorbed light, the wavelength of the light is given by

in which a, the Bohr radius, is a length unit equal to 52.92 pm.Figure 39-20 is a plot of P(r) for the ground state.

Figures 39-22 and 39-24 represent the volume probabilitydensities (not the radial probability densities) for the fourhydrogen atom states with n

- Z.The plot of Fig. 39-22 (,

-

2,( -

0, ffie -

0) is spherically symmetric. The plots of Fig. 39-24(n

-

2,( : t,ffie :0, +I, -1) are symmetric about the z

axis but, when added together, are also spherically symmetric.All four states with n : 2 have the same energy and

may be usefully regarded as constituting a shell, identifled asthe n

- 2 shell. The three states of Fig. 39-24,, taken together,

may be regarded as constituting the n : 2, ( -

1 subshell. It isnot possible to separate the fout n

-

2 states experimentallyunless the hydrogen atom is placed in an electric or magneticfield, which permits the establishment of a deflnite symmetryaxis.

(3e-2r)

(3e-36)

$ An electron is trapped in a one-dimensional infinite po-tential well in a state with n

- 17. How many points of (a)

zero probability and (b) maximum probability does its matterwave have?

ffi Figure 39-26 shows three infinite potential wells, each onan .r axis. Without written calculation, determine the wavefunction {t for a ground-state electron trapped in each well.

S If you wanted to use the ide alued ftap of Fig. 39-I to trapa positron, would you need to change (a) the geometry of the trap,(b) the electric potential of the central cylinder, or (c) the electricpotentials of the two semi-inf,nite end cylinders? (A positron hasthe same mass as an electron but is positively charged.)6 An electron is trapped in a flnite potential well that is deepenough to allow the electron to exist in a state withn

- 4. How many points of (a) zero probability and (b) maxi-

mum probability does its matter wave have within the well?3 A proton and an electron are trapped in identical one-dimensional inflnite potential wells; each particle is in itsground state. At the center of the wells, is the probabilitydensity for the proton greater than, less than, or equal to thatof the electron?

2L(a)

F$ffi. $S-frS

o L/z(b)

Question2.

ix

-L/Z +L/2(c)

S Three electrons are trapped in three different one-dimensional inflnite potential wells of widths (u) 50 pm,(b) 200 pffi, and (c) 100 pm. Rank the electrons according totheir ground-state energies, greatest flrst.& If you double the width of a one-dimensional inflnitepotential well, (u) is the energy of the ground state of thetrapped electron multiplied by 4,2,;,I, ot some other num-ber? (b) Are the energies of the higher energy states multi-plied by this factor or by some other factor, depending ontheir quantum number?

S Is the ground-state energyof a proton trapped in a one-dimensional infinite potentialwell greater than, less than, orequal to that of an electrontrapped in the same potentialwell?g Table 39-4 lists the quantum

ers for flve proposedhydrogen atom states. ch ofthem are not possible?

ITL 4

(u)(b)(.)(d)(")

2-,J.tJ

5.tJ

4.,J

245

5

0

I-4

0

-2

Chapter 39 I More About Matter Wavest 0 You want to modify the finite potential well of Fig. 39-7to allow its trapped electron to exist in more than four quan-tum states. Could you do so by making the well (a) wider ornarrower, (b) deeper or shallower?$ X From a visual inspection of Fig. 39-8, rank the quantumnumbers of the three quantum states according to the deBroglie wavelength of the electron, greatest first.X A An electron, trapped in a flnite potential energy wellsuch as that of Fig. 39-7 ,is in its state of lowest energy. Are (a)its de Broglie wavelength, (b) the magnitude of its momen-tum, and (c) its energy greater than, the same as, or less thanthey would be if the potential well were infinite, &s in Fig.39-2?

13 An electron that is trapped in a one-dimensional inflnitepotential well of width L is excited from the ground state tothe flrst excited state. Does the excitation increase, decrease,or have no effect on the probability of detecting the electronin a small length of the x axis (a) at the center of the well and(b) near one of the well walls?14 Figure 39-27 indicates the lowest energy levels (in elec-tron-volts) for five situations in which an electron is trapped ina one-dimensional infinite potential well. In wells B, C, D,and E, the electron is in the ground state. We shall excite the

electron in well A to the fourth excited state (at 25 eV). Theelectron can then de-excite to the ground state by emittingone or more photons, corresponding to one long ju*p or sev-eral short jumps. Which photon emission energies of this de-excitation match a photon absorption energy (from the groundstate) of the other four electrons? Give the n values.

ABCDEFIffi" S#-47 Question14.

15 A hydrogen atom is in the third excited state. To whatstate (give the quantum number n) should it jump to (a) emitlight with the longest possible wavelength, (b) emit light withthe shortest possible wavelength, and (c) absorb light with thelongest possible wavelength?

La

bJlCJ

14

ffi Tutoring problem available (at instructor's discretion) in WileyPLUSand WebAssign55M Worked-out solution available in Student Solutions Manual WWW Worked-out solution is att

-

c.c Number of dots indicates level of problem difficulty ILW lnteractive solution is atAdditional information available in fhe Flying Circus of Physics and at flyingcircusofphysics.com

sefi. 39-S Energies of a Trapped Electron*{ The ground-state energy of an electron trapped in a one-dimensional infinite potential well is 2.6 eY. What will thisquantity be if the width of the potential well is doubled?tft An electron, trapped in a one-dimensional infinite poten-tial well 250 pm wide, is in its ground state. How much energymust it absorb if it is to jump up to the state with n

-

4?eS What must be the width of a one-dimensional infinitepotential well if an electron trapped in it in the n

- 3 state is

to have an energy of 4.7 eY?oS A proton is confined to a one-dimensional infinite poten-tial well 100 pm wide. What is its ground-state energy?*$ Consider an atomic nucleus to be equivalent to a one-dimensional inflnite potential well with L

- 1.4 x 10 -to

-,

a typical nuclear diameter. What would be the ground-stateenergy of an electron if it were trapped in such a potentialwell? (Note: Nuclei do not contain electrons.)c$ What is the ground-state energy of (a) an electron and(b) a proton if each is trapped in a one-dimensional inflnitepotential well that is 200 pm wide?*p An electron in a one-dimensional infinite potential wellof length L has ground-state ener Ey Er The length is changedto L' so that the new ground-state energy is E', : 0.5004.What is the ratio L'lL?

8e S An electron is trapped in a one-dimensional infinite po-tential well. For what (u) higher quantum number and (b)lower quantum number is the corresponding energy differ-ence equal to the energy difference LE+z between the levelsn

- 4 and n

- 3? (c) Show that no pair of adjacent levels has

an energy difference equal to 2LEa3.oo9 An electron is trapped in a one-dimensional inflnite po-tential well. For what (u) higher quantum number and (b)lower quantum number is the corresponding energy differ-ence equal to the energy of the n

- 5 level? (c) Show that no

pair of adjacent levels has an energy difference equal to theenergy of the n

- 6level.

tt'10 An electron is trapped in a one-dimensional infinitewell of width 250 pm and is in its ground state. What are the(u) longest, (b) second longest, and (.) third longest wave-lengths of light that can excite the electron from the groundstate via a single photon absorption? ffi"oi t Suppose that an electron trapped in a one-dimensionalinfinite well of width 250 pm is excited from its flrst excitedstate to its third excited state. (u) What energy must betransferred to the electron for this quantum jump? The elec-tron then de-excites back to its ground state by emittinglight. In the various possible ways it can do this, what are the(b) shortest, (c) second shortest, (d) longest, and (e) secondlongest wavelengths that can be emitted? (f ) Show the various

possible ways on an energy-level diagram. If light of wave-length 29.4 nm happens to be emitted, what are the (g) longestand (h) shortest wavelength that can be emitted afterwards?ss$P An electron is trapped in a one-dimensional infinitewell and is in its first excited state. Figure 39-28 indicates the fivelongest wavelengths of light that the electron could absorb intransitions from this initial state via a single photon absorption:tro = 80.78 flffi, tru = 33.66 rffi, i. = 19.23 flffi, id -- 12.62 flffi, andtr" : 8.98 nm. What is the width of the potential well?

Ln ha h, Lb LoI + /, (nm)0

F$ffi" S#-trffi Problem12.

s#n" SS-4 Wave Functions of a Trapped Electrone&t$ An electron is trapped in a one-dimensional infinitepotential well that is 100 pm wide; the electron is in its groundstate. What is the probability that you can detect the electronin an interval of width A,x :5.0 pm centered at x

-

(a) 25pm, (b) 50 pm, and (.) 90 pm? (Hint: The interval Ax is sonarrow that you can take the probability density to be constantwithin it.) ssmssT4 A particle is confined to the one-dimensional inflnitepotential well of Fig. 39-2.If the particle is in its ground state,what is its probability of detection between (u) x

-

0 andx-0.?,5L,(b) x-0.75L and x- L,and (c) x-0.25L andx

- 0.75L?

ou15 A one-dimensional infinite well of lengthz}} pm con-tains an electron in its third excited state. We position an elec-tron-detector probe of width 2.00 pm so that it is centered ona point of maximum probability density. (a) What is the prob-ability of detection by the probe? (b) If we insert the probe asdescribed L000 times, how many times should we expect theelectron to materiahze on the end of the probe (and thus bedetected)?s e -16 An electron is in a certain energy state in a one-dimen-sional, inflnite potential well from x

- 0 to x

- L

- 200 pm.

The electron's probability density is zero at x : 0.300 L, andx

- 0.400L; it is not zero at intermediate values of x. The elec-

tron then jumps to the next lower energy level by emittinglight. What is the change in the electron's eneryy?

se" &ry-S An Electron in a Finite Wells g 7 An electron in the n

- 2 state in the finite potential well

of Fig. 39-l absorbs 400 eV of energy from an external source.Using the energy-level diagram of Fig. 39-9, determine theelectron's kinetic energy after this absorption, assuming thatthe electron moves to a position for which x ) L.s $ S Figure 39-9 gives the energy levels for an electrontrapped in a flnite potential energy well 450 eV deep. If theelectron is in the n

-

3 state, what is its kinetic energy?ssl$ (u) Show that for the region x ) L in the finitepotential well of Fig. 39-J, ilx)

-

pszkx is a solution ofSchrodinger's equation in its one-dimensional form, where Dis a constant and k is positive. (b) On what basis do we findthis mathematically acceptable solution to be physically unac-ceptable?ssffffi Figure 39-29a shows a thin tube in which a finite poten-tial trap has been set up where V2

- 0 V. An electron is shown

Problems

traveling rightward toward the trap, in a region with a voltageof V1

- -9.00 V, where it has a kinetic energy of 2.00 eV. Whenthe electron enters the trap region, it can become trapped if itgets rid of enough energy by emitting a photon. The energy lev-els of the electron within the trap are Et : L.0, Ez - 2.0, andEz:4.0 eY and the nonqru;antrzed region begins at E+:9.0 eVas shown in the energy-level diagram of Fig. 39-29b.What is thesmallest energy (eV) such a photon can have?

Nonquantized

* Tube

VT V2 VI(o) (b)

>-bo$iO

Fl

Ff;ffi" S9-tr9 Problem20.

*uff'! Figure 39-30a shows the energy-level diagram for a fi-nite, one-dimensional energy well that contains an electron.The nonquanttzed region begins at Eo : 450.0 eV. Figure 39-30b gives the absorption spectrum of the electron when it is inthe ground state-it can absorb at the indicated wavelengths:tro : 14.588 nm and tru : 4.8437 nm and for any wavelengthless than tr. : 2.9108 nm. What is the energy of the firstexcited state?

Nonquantized

(") (b)FEm" 3ry-3# Problem2I.

se" SS-7 Two- and Three-Dimensional Electron Traps"Hff An electron is contained in the rectangular corral of Fig.39-I3,with widths L* : 800 pm and Ly

-- 1600 pm. What is the

electron's ground-state ener gy?*83 An electron is contained in the rectangular box of Fig.39-14, with widths L":800 pffi, Lr:1600 pm, and Lr:390pm. What is the electron's ground-state energy?s*44 A rectangular corral of widths L*: L and Lr:2Lcontains an electron. What multiple of h2l8mL2, wher e m is theelectron mass, gives (u) the energy of the electron's groundstate, (b) the energy of its flrst excited state, (c) the energy ofits lowest degenerate states, and (d) the difference betweenthe energies of its second and third excited states?esAS An electron (mass m) is contained in a rectangularcorral of widths L,

- L and L, : 2L. (a) How many different

frequencies of light could the electron emit or absorb if itmakes a transition between a pair of the lowest flve energy

)"bh,

>-bn!CJ

FI

Chmpter SS I More About Matter Waves

levels? What multiple of hl8mL2 gives the (b) lowest, (c) sec-ond lowest, (d) third lowest, (e) highest, (f) second highest,and (g) third highest frequency? ssM wwweoffS A cubical box of widths L" : Lr: L, - L contains anelectron. What multiple of h2l8mL2, where m is the electronmass, is (a) the energy of the electron's ground state, (b) theenergy of its second excited state, and (.) the differencebetween the energies of its second and third excited states?How many degenerate states have the energy of (d) the firstexcited state and (e) the fifth excited state?**#7 An electron (mass m) rs contained in a cubical box ofwidths L* : L, : Lr. (u) How many different frequencies oflight could the electron emit or absorb if it makes a transitionbetween a pair of the lowest five energy levels? What multipleof, hl8mL2 gles the (b) lowest, (c) second lowest, (d) thirdlowest, (e) highest, (f) second highest, and (g) third highestfrequency?**ffffi Figure 39-3I shows a two-di-mensional, infinite-potential well lyingin an xy plane that contains an electron.We probe for the electron along a linethat bisects L* and find three points atwhich the detection probability is maxi-mum. Those points are separated by 2.00nm. Then we probe along a line that bi-sects L, and find flve points at which thedetection probability is maximum. Thosepoints are separated by 3.00 nm. What isthe energy of the electron?**frS The two-dimensional, infinite cor-ral of Fig. 39-32 is square, with edge lengthL

-

150 pm. A square probe is centeredat xy coordinates (0.200L, 0.800L) andhas an x width of 5.00 pm and a y width of5.00 pm. What is the probability of detec-tion if the electron is in the Esenergy state?

s3S What are the (a) energy., (b) magnitude of the momen-tum, and (c) wavelength of the photon emitted when a hydro-gen atom undergoes a transition from a state with n

- 3 to a

statewithn-t?*3& An atom (not a hydrogen atom) absorbs a photonwhose associated frequency is 6.2 x 10ra Hz. By what amountdoes the energy of the atom increase?s37 What is the ratio of the shortest wavelength of the Balmerseries to the shortest wavelength of the Lyman series? ssMs3& An atom (not a hydrogen atom) absorbs a photon whoseassociated wavelength is 375 nm and then immediately emitsa photon whose associated wavelength is 580 nm. How muchnet energy is absorbed by the atom in this process?ss$W How much work must be done to pull apart the electronand the proton that make up the hydrogen atom if the atom isinitially in (a) its ground state and (b) the state with n

-

2?

*o4S A hydrogen atom is excited from its ground state to thestate with n

- a. @) How much energy must be absorbed by

the atom? Consider the photon energies that can be emittedby the atom as it de-excites to the ground state in the severalpossible ways. (b) How many different energies are possible;what are the (c) highest, (d) second highest, (e) third highest,(f ) lowest, (g) second lowest, and (h) third lowest energies?*udt What is the probability that in the ground state of thehydrogen atom, the electron will be found at a radius greaterthan the Bohr radius? (Hint: See Sample Problem 39-8.)**4fi Light of wavelength 12I.6 nm is emitted by u hydrogenatom. What are the (a) higher quantum number and (b) lowerquantum number of the transition producing this emission?(c) What is the name of the series that includes the transition?*s4$ Schrodinger's equation for states of the hydrogenatom for which the orbital quantum number ( is zero is

I d I ., d.,l'\ 8 tr2m r7;\r;)+f tE- rru)tv:oVerify that Eq. 39-39,which describes the ground state of thehydrogen atom, is a solution of this equation. ssM www**44 What are the (a) wavelength range and (b) frequencyrange of the Lyman series? What are the (c) wavelength rangeand (d) frequency range of the Balmer series?*od$ In the ground state of the hydrogen atom, the electronhas a total energy of

- 13.6 eV. What are (a) its kinetic energy

and (b) its potential energy if the electron is one Bohr radiusfrom the central nucleus?**4& A hydrogen atom, initially at rest in the n - 4 quan-tum state, undergoes a transition to the ground state, emittinga photon in the process. What is the speed of the recoilinghydrogen atom?o*47 Verify that Eq. 39-44,the radial probability density forthe ground state of the hydrogen atom, is normalized. That is,verify that

is true. SsMo*4# A hydrogen atom in a state having a binding energy(the energy required to remove an electron) of 0.85 eV makes

F$ffi. #W-#1Problem23.

F$ffi. S$-ffitrProblem29.

wo*$ffi An electron is in the ground state in a two-dimen-sional, square, infinite potential well with edge lengths L. Wewill probe for it in a square of area 400 pmz that is centered atx

- Ll8 and y

-

LlS.The probability of detection turns out tobe 0.0450. What is edge length L?

$#c. S9-S Schr6dinger's Equationand the Hydrogen Atom*S$ For the hydrogen atom in its ground state, calculate(a) the probability density Q'?) and (b) the radial probabilitydensity P(r) for r : a,where a is the Bohr radius.*Sff Calculate the radial probability density P(r) for thehydrogen atom in its ground state at (a) r:0, (b) t: a, and(.) r : Za,where a is the Bohr radius.*S$ A neutron with a kinetic energy of 6.0 eV collides witha stationary hydrogen atom in its ground state. Explain whythe collision must be elastic-that is, why kinetic energy mustbe conserved. (Hint: Show that the hydrogen atom cannot beexcited as a result of the collision.) ssMsS4 (a) What is the eneryy E of the hydrogen-atom electronwhose probability density is represented by the dot plot ofFig. 39-22? (b) What minimum energy is needed to removethis electron from the atom?

[ ")dr-r

a transition to a state with an excitation energy (the differencebetween the energy of the state and that of the ground state)of 10.2 eV. (a) What is the energy of the photon emitted asa result of the transition? What are the (b) higher quantumnumber and (.) lower quantum number of the transitionproducing this emission?**49 The wave functions for the three states with the dotplots shown in Fig. 39-24, which have n.:2, ( :1, and tn( :0, +1, and

-L,are

*x/1', 0) _ (l I 4t-2fl (a-zrz1Q I a)e- il2a cos 0,*xnr?, 0) _ (l/B\E) (a-trz\Qla)e-,rzo(sin 0)e+if,Qn-r?, 0) - (l/Br[zr) (a-zrzr(rla)e-'t2o(sin 0)e-i6,

in which the subscripts on t?,0) give the values of the quan-tum numbers n, (, ffie and the angles 0 and Q are defined inFig. 39-23. Note that the flrst wave function is real but theothers, which involve the imaginary number i, are complex.Find the radial probability density P(r) for (a) {no and (b) Qn*t(same as for Qzr-t). (c) Show that each P(r) is consistent with thecorresponding dot plot in Fig. 39-24. (d) Add the radial probabil-ity densities for tlh1s, *zt*t, and Qn-r and then show that the sumis spherically symmetric, depending only on r. ssrtl'6e$ffi Calculate the probability that the electron in thehydrogen atom, in its ground state, will be found betweenspherical shells whose radii are a and 2a, where a is the Bohrradius. (Hint: See Sample Problem 39-8.)ssST What is the probability that an electron in the groundstate of the hydrogen atom will be found between two spheri-cal shells whose radii are r and r I Lr, (a) if r: 0.500a andAr:0.010a and (b) tf r

-

1.00a and L,r:0.07a, where a rsthe Bohr radius? (Hint: Lr is small enough to permit the radialprobability density to be taken to be constant betwe en r andr + Ar.)&@Sffi Light of wavelength 102.6 nm is emitted by a hydrogenatom. What are the (a) higher quantum number and (b) lowerquantum number of the transition producing this emission?(c) What is the name of the series that includes the transition?esSS For what value of the principal quantum number nwould the effective radius, as shown in a probability densitydot plot for the hydrogen atom, be 1.0 mm? Assume that ( hasits maximum value of n

- 1. (Hint: See Fig. 39-25.)

eee$S The wave function for the hydrogen-atom quantumstate represented by the dot plot shown in Fig. 39-22, whichhasn

- 2 and (

-

tn(,:0,is

Problems

has the value I.5a. In this expression for rurs, each value ofP(r) is weighted with the value of r at which it occurs. Notethat the average value of r is greater than the value of r forwhich P(r) is a maximum.Additional Problems56 An electron is conflned to a narrow evacuated tube oflength 3.0 m; the tube functions as a one-dimensional infinitepotential well. (a) What is the energy difference between theelectron's ground state and its flrst excited state? (b) At whatquantum number n would the energy difference between ad-jacent energy levels be 1.0 eV-which is measurable, unlikethe result of (u)? At that quantum number, (c) what multipleof the electron's rest energy would give the electron's totalenergy and (d) would the electron be relativistic?SY As Fig. 39-B suggests, the probability density for theregion x ) L rn the flnite potential well of Fig. 39-7 drops offexponentially according to ,lt'(*)

-

Ce-2k*, where C is a con-stant. (a) Show that the wave function ,lr(*) that may be foundfrom this equation is a solution of Schr6dinger's equation inits one-dimensional form. (b) Find an expression for k f.or thisto be true. ssMS& As Fig. 39-B suggests, the probability density for anelectron in the region 0 < x < L for the flnite potential well ofFig. 39-7 is sinusoidal, being given by ,!'(*)

- B sinZ kx, rn

which B is a constant. (a) Show that the wave function ,[(*)that may be found from this equation is a solution ofSchrodinger's equation in its one-dimensional form. (b) Findan expression for k that makes this true.Sq (a) For a given value of the principal quantum number n,how many values of the orbital quantum number ( are possi-ble? (b) For a given value of (,, how many values of the orbitalmagnetic quantum number lne are possible? (c) For a givenvalue of n,,how many values of ma are possible?&S Let LEaaj be the energy difference between two adja-cent energy levels for an electron trapped in a one-dimen-sional infinite potential well .Let Ebe the energy of either ofthe two levels. (u) Show that the ratio LEuajlE approachesthe value Zln at large values of the quantum number n. Asn + oo, does (b) AEuo;, (c) E, or (d) AE uailE approach zero?(e) What do these results mean in terms of the correspon-dence principle?S$ An electron is trapped in a one-dimensional infinite po-tential well. Show that the energy difference AE between itsquantum levels n and n * 2ts (hzl2mLz)(n + I).6g Verify that the combined value of the constants appear-ing in Eq. 39-32 is 13.6 eV.SS (a) Show that the terms in Schrodinger's equation (Eq.39-18) have the same dimensions. (b) What is the common SIunit for each of these terms?S4 Repeat Sample Problem39-6 for the Balmer series of thehydrogen atom.

@ & we showed that the radialpr state of the hydrogen atomis a is the Bohr radius. Showth as

) r dr,

-

t )' -rt2a

'a/in which a is the Bohr radius and the subscript on Q(r) givesthe values of the quantum numbers n, (, ffit.(a) Plot Qtor?)and show that your plot is consistent with the dot plot ofFig. 39-22. (b) Show analytically that tSoo?) has a maximum atr -

4a. (c) Find the radial probability density Ptoo?) for thisstate. (d) Show that

and thus that the expression above for the wave function*zo6(r) has been properly normahzed.

lo* ,rooe) d.r

- r