Normalization and Zero-Point Energy The amplitude A 2 in Eq can be found from the normalizing equation, 1106 CHAPTER 39 MORE ABOUT MATTER WAVES

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1 116 CHAPTER 39 MORE ABOUT MATTER WAVES Fig A dot plot of the radial probabilit densit P(r) for the hdrogen atom in a quantum state with a relativel large principal quantum number namel, n 45 and angular momentum quantum number n 1 44.The dots lie close to the plane, the ring of dots suggesting a classical electron orbit. r = n a If we add the volume probabilit densities for the three states for which n and 1, the combined probabilit densit turns out to be sphericall smmetrical, with no unique ais. One can, then, think of the electron as spending one-third of its time in each of the three states of Fig. 39-3, and one can think of the weighted sum of the three independent wave functions as defining a sphericall smmetric subshell specified b the quantum numbers n, 1. The individual states will displa their separate eistence onl if we place the hdrogen atom in an eternal electric or magnetic field. The three states of the n, 1 subshell will then have different energies, and the field direction will establish the necessar smmetr ais. The n, state, whose volume probabilit densit is shown in Fig. 39-1, also has the same energ as each of the three states of Fig We can view all four states whose quantum numbers are listed in Table 39-3 as forming a sphericall smmetric shell specified b the single quantum number n. The importance of shells and subshells will become evident in Chapter 4, where we discuss atoms having more than one electron. To round out our picture of the hdrogen atom, we displa in Fig a dot plot of the radial probabilit densit for a hdrogen atom state with a relativel high quantum number (n 45) and the highest orbital quantum number that the restrictions of Table 39- permit ( n 1 44). The probabilit densit forms a ring that is smmetrical about the z ais and lies ver close to the plane. The mean radius of the ring is n a, where a is the Bohr radius. This mean radius is more than times the effective radius of the hdrogen atom in its ground state. Figure 39-4 suggests the electron orbit of classical phsics it resembles the circular orbit of a planet around a star. Thus, we have another illustration of Bohr s correspondence principle namel, that at large quantum numbers the predictions of quantum mechanics merge smoothl with those of classical phsics. Imagine what a dot plot like that of Figure 39-4 would look like for reall large values of n and sa, n 1 and 999. The Confinement Principle The confinement principle applies to waves of all kinds, including waves on a string and the matter waves of quantum phsics. It states that confinement leads to quantization that is, to the eistence of discrete states with certain energies. States with intermediate values of energ are disallowed. An Electron in an Infinite Potential Well An infinite potential well is a device for confining an electron. From the confinement principle we epect that the matter wave representing a trapped electron can eist onl in a set of discrete states. For a onedimensional infinite potential well, the energies associated with these quantum states are E n h, for n 1,,3,..., (39-4) 8mL n in which L is the width of the well and n is a quantum number. If the electron is to change from one state to another, its energ must change b the amount E E high E low, (39-5) where E high is the higher energ and E low is the lower energ. If the change is done b photon absorption or emission, the energ of the photon must be hf E E high E low. (39-6) The wave functions associated with the quantum states are n() A sin n L, for n 1,,3,... (39-1) The probabilit densit () for an allowed state has the phsical meaning that () d is the probabilit that the electron will be detected in the interval between and d. For an electron in an infinite well, the probabilit densities are () A sin n for n 1,,3,... (39-1) L, At high quantum numbers n, the electron tends toward classical behavior in that it tends to occup all parts of the well with equal probabilit. This transition from quantum to classical phsics is known as the correspondence principle. n n n Normalization and Zero-Point Energ The amplitude A in Eq can be found from the normalizing equation, n () d 1, (39-14) which asserts that the electron must be somewhere within the well because the probabilit 1 implies certaint.

2 QUESTIONS 117 PART 5 From Eq we see that the lowest permitted energ for the electron is not zero but the energ that corresponds to n 1.This lowest energ is called the zero-point energ of the electron well sstem. An Electron in a Finite Potential Well A finite potential well is one for which the potential energ of an electron inside the well is less than that for one outside the well b a finite amount U. The wave function for an electron trapped in such a well etends into the walls of the well. Two- and Three-Dimensional Electron Traps The quantized energies for an electron trapped in a two-dimensional infinite potential well that forms a rectangular corral are E n,n (39-) where n is a quantum number for which the electron s matter wave fits in well width L and n is a quantum number for which the electron s matter wave fits in well width L. Similarl, the energies for an electron trapped in a three-dimensional infinite potential well that forms a rectangular bo are E n,n,nz h (39-1) 8m n L n L n z L z. Here n z is a third quantum number, one for which the matter wave fits in well width L z. The Hdrogen Atom Both the (incorrect) Bohr model of the hdrogen atom and the (correct) application of Schrödinger s equation to this atom give the quantized energ levels of the atom as E n me4 8 h 1 n h for n 1,,3,... 8m n L n L, ev 13.6, n (39-3, 39-33) From this we find that if the atom makes a transition between an two energ levels as a result of having emitted or absorbed light, the wavelength of the light is given b where 1 R 1 1 n low n high, (39-36) R me 4 (39-37) 8 h 3 c m 1 is the Rdberg constant. The radial probabilit densit P(r) for a state of the hdrogen atom is defined so that P(r) dr is the probabilit that the electron will be detected somewhere in the space between two concentric shells of radii r and r dr centered on the atom s nucleus. For the hdrogen atom s ground state, P(r) 4 a 3 r e r/a, (39-44) in which a, the Bohr radius, is a length unit equal to 5.9 pm. Figure is a plot of P(r) for the ground state. Figures 39-1 and 39-3 represent the volume probabilit densities (not the radial probabilit densities) for the four hdrogen atom states with n. The plot of Fig (n,, m ) is sphericall smmetric. The plots of Fig (n, 1, m, 1, 1) are smmetric about the z ais but, when added together, are also sphericall smmetric. All four states with n have the same energ and ma be usefull regarded as constituting a shell, identified as the n shell. The three states of Fig. 39-3, taken together, ma be regarded as constituting the n, 1 subshell. It is not possible to sepa- rate the four n states eperimentall unless the hdrogen atom is placed in an electric or magnetic field, which permits the establishment of a definite smmetr ais. 1 Three electrons are trapped in three different one-dimensional infinite potential wells of widths (a) 5 pm, (b) pm, and (c) 1 pm. Rank the electrons according to their ground-state energies, greatest first. Is the ground-state energ of a proton trapped in a onedimensional infinite potential well greater than, less than, or equal to that of an electron trapped in the same potential well? 3 An electron is trapped in a one-dimensional infinite potential well in a state with n 17. How man points of (a) zero probabilit and (b) maimum probabilit does its matter wave have? 4 Figure 39-5 shows three infinite potential wells, each on an ais. Without written calculation, determine the wave function c for a ground-state electron trapped in each well. U L L/ L/ +L/ (a) (b) (c) Fig Question 4. 5 A proton and an electron are trapped in identical onedimensional infinite potential wells; each particle is in its ground state. At the center of the wells, is the probabilit densit for the proton greater than, less than, or equal to that of the electron? 6 If ou double the width of a one-dimensional infinite potential well, (a) is the energ of the ground state of the trapped electron 1 1 multiplied b 4,,, 4, or some other number? (b) Are the energies of the higher energ states multiplied b this factor or b some other factor, depending on their quantum number? 7 If ou wanted to use the idealized trap of Fig to trap a positron, would ou need to change (a) the geometr of the trap, (b) the electric potential of the central clinder, or (c) the electric potentials of the two semi-infinite end clinders? (A positron has the same mass as an electron but is positivel charged.) 8 An electron is trapped in a finite potential well that is deep enough to allow the electron to eist in a state with n 4. How man points of (a) zero probabilit and (b) maimum probabilit does its matter wave have within the well? 9 An electron that is trapped in a one-dimensional infinite potential well of width L is ecited from the ground state to the first e-

3 118 CHAPTER 39 MORE ABOUT MATTER WAVES cited state. Does the ecitation increase, decrease, or have no effect on the probabilit of detecting the electron in a small length of the ais (a) at the center of the well and (b) near one of the well walls? 1 An electron, trapped in a finite potential energ well such as that of Fig. 39-7, is in its state of lowest energ. Are (a) its de Broglie wavelength, (b) the magnitude of its momentum, and (c) its energ greater than, the same as, or less than the would be if the potential well were infinite, as in Fig. 39-? 11 From a visual inspection of Fig. 39-8, rank the quantum numbers of the three quantum states according to the de Broglie wavelength of the electron, greatest first. 1 You want to modif the finite potential well of Fig to allow its trapped electron to eist in more than four quantum states. Could ou do so b making the well (a) wider or narrower, (b) deeper or shallower? 13 A hdrogen atom is in the third ecited state. To what state (give the quantum number n) should it jump to (a) emit light with the longest possible wavelength, (b) emit light with the shortest possible wavelength, and (c) absorb light with the longest possible wavelength? 14 Figure 39-6 indicates the lowest energ levels (in electronvolts) for five situations in which an electron is trapped in a one-dimensional infinite potential well. In wells B, C, D, and E, the electron is in the ground state. We shall ecite the electron in well A to the fourth ecited state (at 5 ev). The electron can then de-ecite to the ground state b emitting one or more photons, corresponding to one long jump or several short jumps. Which photon emis- sion energies of this de-ecitation match a photon absorption energ (from the ground state) of the other four electrons? Give the n values. Energ (ev) A B C D E Fig Question Table 39-4 lists the quantum numbers for five proposed hdrogen atom states.which of them are not possible? 16 Table 39-4 n (a) 3 (b) 3 1 (c) (d) 5 5 (e) 5 3 m SSM Tutoring problem available (at instructor s discretion) in WilePLUS and WebAssign Worked-out solution available in Student Solutions Manual WWW Worked-out solution is at Number of dots indicates level of problem difficult ILW Interactive solution is at Additional information available in The Fling Circus of Phsics and at flingcircusofphsics.com sec Energies of a Trapped Electron 1 An electron in a one-dimensional infinite potential well of length L has ground-state energ E 1.The length is changed to L so that the new ground-state energ is E 1.5E 1.What is the ratio L/L? What is the ground-state energ of (a) an electron and (b) a proton if each is trapped in a one-dimensional infinite potential well that is pm wide? 3 The ground-state energ of an electron trapped in a onedimensional infinite potential well is.6 ev. What will this quantit be if the width of the potential well is doubled? 4 An electron, trapped in a one-dimensional infinite potential well 5 pm wide, is in its ground state. How much energ must it absorb if it is to jump up to the state with n 4? 5 What must be the width of a one-dimensional infinite potential well if an electron trapped in it in the n 3 state is to have an energ of 4.7 ev? 6 A proton is confined to a one-dimensional infinite potential well 1 pm wide. What is its ground-state energ? 7 Consider an atomic nucleus to be equivalent to a onedimensional infinite potential well with L m, a tpical nuclear diameter. What would be the ground-state energ of an electron if it were trapped in such a potential well? (Note: Nuclei do not contain electrons.) 8 An electron is trapped in a one-dimensional infinite well and is in its first ecited state. Figure 39-7 indicates the five longest wavelengths of light that the electron could absorb in transitions from this initial state via a single photon absorption: l a = 8.78 nm, l b = nm, l c = 19.3 nm, l d = 1.6 nm, and l e = 8.98 nm.what is the width of the potential well? λ e λ d λ c λ b λ a λ (nm) Fig Problem 8. 9 Suppose that an electron trapped in a one-dimensional infinite well of width 5 pm is ecited from its first ecited state to its third ecited state. (a) What energ must be transferred to the electron for this quantum jump? The electron then de-ecites back to its ground

4 PROBLEMS 119 PART 5 state b emitting light. In the various possible was it can do this, what are the (b) shortest, (c) second shortest, (d) longest, and (e) second longest wavelengths that can be emitted? (f) Show the various possible was on an energ-level diagram. If light of wavelength 9.4 nm happens to be emitted, what are the (g) longest and (h) shortest wavelength that can be emitted afterwards? 1 An electron is trapped in a one-dimensional infinite potential well. For what (a) higher quantum number and (b) lower quantum number is the corresponding energ difference equal to the energ difference E 43 between the levels n 4 and n 3? (c) Show that no pair of adjacent levels has an energ difference equal to E An electron is trapped in a one-dimensional infinite potential well. For what (a) higher quantum number and (b) lower quantum number is the corresponding energ difference equal to the energ of the n 5 level? (c) Show that no pair of adjacent levels has an energ difference equal to the energ of the n 6 level. 1 An electron is trapped in a one-dimensional infinite well of width 5 pm and is in its ground state. What are the (a) longest, (b) second longest, and (c) third longest wavelengths of light that can ecite the electron from the ground state via a single photon absorption? sec Wave Functions of a Trapped Electron 13 A one-dimensional infinite well of length pm contains an electron in its third ecited state. We position an electrondetector probe of width. pm so that it is centered on a point of maimum probabilit densit. (a) What is the probabilit of detection b the probe? (b) If we insert the probe as described 1 times, how man times should we epect the electron to materialize on the end of the probe (and thus be detected)? 14 An electron is in a certain energ state in a one-dimensional, infinite potential well from to L pm. The electron s probabilit densit is zero at.3l, and.4l; it is not zero at intermediate values of. The electron then jumps to the net lower energ level b emitting light. What is the change in the electron s energ? 15 SSM WWW An electron is trapped in a one-dimensional infinite potential well that is 1 pm wide; the electron is in its ground state. What is the probabilit that ou can detect the electron in an interval of width 5. pm centered at (a) 5 pm, (b) 5 pm, and (c) 9 pm? (Hint: The interval is so narrow that ou can take the probabilit densit to be constant within it.) 16 A particle is confined to the one-dimensional infinite potential well of Fig If the particle is in its ground state, what is its probabilit of detection between (a) and.5l, (b).75l and L, and (c).5l and.75l? sec An Electron in a Finite Well 17 An electron in the n state in the finite potential well of Fig absorbs 4 ev of energ from an eternal source. Using the energ-level diagram of Fig. 39-9, determine the electron s kinetic energ after this absorption, assuming that the electron moves to a position for which L. 18 Figure 39-9 gives the energ levels for an electron trapped in a finite potential energ well 45 ev deep. If the electron is in the n 3 state, what is its kinetic energ? 19 Figure 39-8a shows the energ-level diagram for a finite, one-dimensional energ well that contains an electron. The non- quantized region begins at E ev. Figure 39-8b gives the absorption spectrum of the electron when it is in the ground state it can absorb at the indicated wavelengths: l a nm and l b nm and for an wavelength less than l c.918 nm.what is the energ of the first ecited state? Energ (a) E 4 E 3 E E 1 Nonquantized λ c λ b Fig Problem 19. Figure 39-9a shows a thin tube in which a finite potential trap has been set up where V V. An electron is shown traveling rightward toward the trap, in a region with a voltage of V 1 9. V, where it has a kinetic energ of. ev. When the electron enters the trap region, it can become trapped if it gets rid of enough energ b emitting a photon. The energ levels of the electron within the trap are E 1 1., E., and E 3 4. ev, and the nonquantized region begins at E 4 9. ev as shown in the energ-level diagram of Fig. 39-9b. What is the smallest energ (ev) such a photon can have? Tube Nonquantized (b) Energ V 1 V V 1 (a) (b) Fig Problem. 1 (a) Show that for the region L in the finite potential well of Fig. 39-7, c() De k is a solution of Schrödinger s equation in its one-dimensional form, where D is a constant and k is positive. (b) On what basis do we find this mathematicall acceptable solution to be phsicall unacceptable? sec Two- and Three-Dimensional Electron Traps An electron is contained in the rectangular corral of Fig , with widths L 8 pm and L 16 pm. What is the electron s ground-state energ? 3 An electron is contained in the rectangular bo of Fig , with widths L 8 L pm, L 16 pm, and L z 39 pm. What is the electron s ground-state energ? L 4 Figure 39-3 shows a two-dimensional, infinite-potential well ling in an Fig plane that contains an electron. We probe Problem 4. λa E 4 E 3 E E 1 λ

5 111 CHAPTER 39 MORE ABOUT MATTER WAVES for the electron along a line that bisects L and find three points at which the detection probabilit is maimum. Those points are separated b. nm. Then we probe along a line that bisects L and find five points at which the detection probabilit is maimum. Those points are separated b 3. nm.what is the energ of the electron? 5 The two-dimensional, infinite corral of Fig is square, with edge length L 15 pm.a square probe is centered at coordinates (.L,.8L) and has an Probe width of 5. pm and a width of 5. pm. What is the probabilit of detection if the electron is in the E 1,3 energ state? Fig Problem 5. 6 A rectangular corral of widths L L and L L contains an electron. What multiple of h /8mL, where m is the electron mass, gives (a) the energ of the electron s ground state, (b) the energ of its first ecited state, (c) the energ of its lowest degenerate states, and (d) the difference between the energies of its second and third ecited states? 7 SSM WWW An electron (mass m) is contained in a rectangular corral of widths L L and L L. (a) How man different frequencies of light could the electron emit or absorb if it makes a transition between a pair of the lowest five energ levels? What multiple of h/8ml gives the (b) lowest, (c) second lowest, (d) third lowest, (e) highest, (f) second highest, and (g) third highest frequenc? 8 A cubical bo of widths L L L z L contains an electron. What multiple of h /8mL, where m is the electron mass, is (a) the energ of the electron s ground state, (b) the energ of its second ecited state, and (c) the difference between the energies of its second and third ecited states? How man degenerate states have the energ of (d) the first ecited state and (e) the fifth ecited state? 9 An electron (mass m) is contained in a cubical bo of widths L L L z. (a) How man different frequencies of light could the electron emit or absorb if it makes a transition between a pair of the lowest five energ levels? What multiple of h/8ml gives the (b) lowest, (c) second lowest, (d) third lowest, (e) highest, (f) second highest, and (g) third highest frequenc? 3 An electron is in the ground state in a two-dimensional, square, infinite potential well with edge lengths L. We will probe for it in a square of area 4 pm that is centered at L/8 and L/8. The probabilit of detection turns out to be What is edge length L? sec Schrödinger s Equation and the Hdrogen Atom 31 SSM What is the ratio of the shortest wavelength of the Balmer series to the shortest wavelength of the Lman series? 3 An atom (not a hdrogen atom) absorbs a photon whose associated wavelength is 375 nm and then immediatel emits a photon whose associated wavelength is 58 nm. How much net energ is absorbed b the atom in this process? 33 What are the (a) energ, (b) magnitude of the momentum, and (c) wavelength of the photon emitted when a hdrogen atom undergoes a transition from a state with n 3 to a state with n 1? 34 Calculate the radial probabilit densit P(r) for the hdrogen atom in its ground state at (a) r, (b) r a, and (c) r a, where a is the Bohr radius. 35 For the hdrogen atom in its ground state, calculate (a) the probabilit densit c (r) and (b) the radial probabilit densit P(r) for r a, where a is the Bohr radius. 36 (a) What is the energ E of the hdrogen-atom electron whose probabilit densit is represented b the dot plot of Fig. 39-1? (b) What minimum energ is needed to remove this electron from the atom? 37 SSM A neutron with a kinetic energ of 6. ev collides with a stationar hdrogen atom in its ground state. Eplain wh the collision must be elastic that is, wh kinetic energ must be conserved. (Hint: Show that the hdrogen atom cannot be ecited as a result of the collision.) 38 An atom (not a hdrogen atom) absorbs a photon whose associated frequenc is Hz. B what amount does the energ of the atom increase? 39 SSM Verif that Eq , the radial probabilit densit for the ground state of the hdrogen atom, is normalized. That is, verif that the following is true: P(r) dr 1 4 What are the (a) wavelength range and (b) frequenc range of the Lman series? What are the (c) wavelength range and (d) frequenc range of the Balmer series? 41 What is the probabilit that an electron in the ground state of the hdrogen atom will be found between two spherical shells whose radii are r and r r, (a) if r.5a and r.1a and (b) if r 1.a and r.1a, where a is the Bohr radius? (Hint: r is small enough to permit the radial probabilit densit to be taken to be constant between r and r r.) 4 A hdrogen atom, initiall at rest in the n 4 quantum state, undergoes a transition to the ground state, emitting a photon in the process. What is the speed of the recoiling hdrogen atom? (Hint: This is similar to the eplosions of Chapter 9.) 43 In the ground state of the hdrogen atom, the electron has a total energ of 13.6 ev. What are (a) its kinetic energ and (b) its potential energ if the electron is one Bohr radius from the central nucleus? 44 A hdrogen atom in a state having a binding energ (the energ required to remove an electron) of.85 ev makes a transition to a state with an ecitation energ (the difference between the energ of the state and that of the ground state) of 1. ev. (a) What is the energ of the photon emitted as a result of the transition? What are the (b) higher quantum number and (c) lower quantum number of the transition producing this emission? 45 SSM The wave functions for the three states with the dot plots shown in Fig. 39-3, which have n, 1, and m, 1, and 1, are 1(r, ) (1/41)(a 3/ )(r/a)e r/a cos, 11(r, ) (1/81)(a 3/ )(r/a)e r/a (sin )e i, 11(r, ) (1/81)(a 3/ )(r/a)e r/a (sin )e i, in which the subscripts on c(r, u) give the values of the quantum numbers n,, m and the angles u and f are defined in Fig Note that the first wave function is real but the others, which

6 PROBLEMS 1111 PART 5 involve the imaginar number i, are comple. Find the radial probabilit densit P(r) for (a) c 1 and (b) c 11 (same as for c 11 ). (c) Show that each P(r) is consistent with the corresponding dot plot in Fig (d) Add the radial probabilit densities for c 1, c 11, and c 11 and then show that the sum is sphericall smmetric, depending onl on r. 46 Calculate the probabilit that the electron in the hdrogen atom, in its ground state, will be found between spherical shells whose radii are a and a, where a is the Bohr radius. 47 For what value of the principal quantum number n would the effective radius, as shown in a probabilit densit dot plot for the hdrogen atom, be 1. mm? Assume that has its maimum value of n 1. (Hint: See Fig ) 48 Light of wavelength 11.6 nm is emitted b a hdrogen atom. What are the (a) higher quantum number and (b) lower quantum number of the transition producing this emission? (c) What is the name of the series that includes the transition? 49 How much work must be done to pull apart the electron and the proton that make up the hdrogen atom if the atom is initiall in (a) its ground state and (b) the state with n? 5 Light of wavelength 1.6 nm is emitted b a hdrogen atom. What are the (a) higher quantum number and (b) lower quantum number of the transition producing this emission? (c) What is the name of the series that includes the transition? 51 What is the probabilit that in the ground state of the hdrogen atom, the electron will be found at a radius greater than the Bohr radius?) 5 A hdrogen atom is ecited from its ground state to the state with n 4. (a) How much energ must be absorbed b the atom? Consider the photon energies that can be emitted b the atom as it de-ecites to the ground state in the several possible was. (b) How man 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? 53 SSM WWW Schrödinger s equation for states of the hdrogen atom for which the orbital quantum number is zero is 1 d d r dr r dr 8 m [E U(r)]. h Verif that Eq , which describes the ground state of the hdrogen atom, is a solution of this equation. 54 The wave function for the hdrogen-atom quantum state represented b the dot plot shown in Fig. 39-1, which has n and m, is (r) 1 41 in which a is the Bohr radius and the subscript on c(r) gives the values of the quantum numbers n,, m. (a) Plot (r) and show that our plot is consistent with the dot plot of Fig (b) Show analticall that (r) has a maimum at r 4a. (c) Find the radial probabilit densit P (r) for this state. (d) Show that P (r) dr 1 a 3/ r a er/a, and thus that the epression above for the wave function c (r) has been properl normalized. 55 The radial probabilit densit for the ground state of the hdrogen atom is a maimum when r a, where a is the Bohr radius. Show that the average value of r, defined as r avg P(r) r dr, has the value 1.5a. In this epression for r avg, each value of P(r) is weighted with the value of r at which it occurs. Note that the average value of r is greater than the value of r for which P(r) is a maimum. Additional Problems 56 Let E adj be the energ difference between two adjacent energ levels for an electron trapped in a one-dimensional infinite potential well. Let E be the energ of either of the two levels. (a) Show that the ratio E adj /E approaches the value /n at large values of the quantum number n. As n :, does (b) E adj, (c) E, or (d) E adj /E approach zero? (e) What do these results mean in terms of the correspondence principle? 57 An electron is trapped in a one-dimensional infinite potential well. Show that the energ difference E between its quantum levels n and n is (h /ml )(n 1). 58 As Fig suggests, the probabilit densit for an electron in the region L for the finite potential well of Fig is sinusoidal, being given b c () B sin k, in which B is a constant. (a) Show that the wave function c() that ma be found from this equation is a solution of Schrödinger s equation in its one-dimensional form. (b) Find an epression for k that makes this true. 59 SSM As Fig suggests, the probabilit densit for the region L in the finite potential well of Fig drops off eponentiall according to c () Ce k, where C is a constant. (a) Show that the wave function c() that ma be found from this equation is a solution of Schrödinger s equation in its one-dimensional form. (b) Find an epression for k for this to be true. 6 An electron is confined to a narrow evacuated tube of length 3. m; the tube functions as a one-dimensional infinite potential well. (a) What is the energ difference between the electron s ground state and its first ecited state? (b) At what quantum number n would the energ difference between adjacent energ levels be 1. ev which is measurable, unlike the result of (a)? At that quantum number, (c) what multiple of the electron s rest energ would give the electron s total energ and (d) would the electron be relativistic? 61 (a) Show that the terms in Schrödinger s equation (Eq ) have the same dimensions. (b) What is the common SI unit for each of these terms? 6 (a) What is the wavelength of light for the least energetic photon emitted in the Balmer series of the hdrogen atom spectrum lines? (b) What is the wavelength of the series limit? 63 (a) For a given value of the principal quantum number n for a hdrogen atom, how man values of the orbital quantum number are possible? (b) For a given value of, how man values of the orbital magnetic quantum number m are possible? (c) For a given value of n, how man values of m are possible? 64 Verif that the combined value of the constants appearing in Eq is 13.6 ev.