Solution: An electron is trapped in a one-dimensional

An electron in a one-dimensional infinite potential well of length L has ground-state energy E1.The length is changed to L4 so that the new ground-state energy is .What is the ratio L4/L?

What is the ground-state energy of (a) an electron and (b) a proton if each is trapped in a one- dimensional infinite potential well that is 200 pm wide?

The ground-state energy of an electron trapped in a onedimensional infinite potential well is 2.6 eV.What will this quantity be if the width of the potential well is doubled?

An electron, trapped in a one-dimensional infinite potential well 250 pm wide, is in its ground state. How much energy must it absorb if it is to jump up to the state with n ! 4?

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 energy of 4.7 eV?

A proton is confined to a one-dimensional infinite potential well 100 pm wide.What is its ground-state energy?

Consider an atomic nucleus to be equivalent to a onedimensional infinite potential well with L ! 1.4 " 10#14 m, a typical nuclear diameter. What would be the ground-state energy of an electron if it were trapped in such a potential well? (Note: Nuclei do not contain electrons.)

An electron is trapped in a one-dimensional infinite well and is in its first excited state. Figure 39-27 indicates the five longest wavelengths of light that the electron could absorb in transitions from this initial state via a single photon absorption: la ! 80.78 nm, lb ! 33.66 nm, lc ! 19.23 nm, ld ! 12.62 nm, and le ! 8.98 nm. What is the width of the potential well?

Suppose that an electron trapped in a one-dimensional infinite well of width 250 pm is excited from its first excited state to its third excited state. (a) What energy must be transferred to the electron for this quantum jump? The electron then de-excites back to its ground state by emitting light. In the various possible ways 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 ways on an energy- level diagram. If light of wavelength 29.4 nm happens to be emitted, what are the (g) longest and (h) shortest wavelength that can be emitted afterwards?

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 energy difference equal to the energy difference )E43 between the levels n ! 4 and n ! 3? (c) Show that no pair of adjacent levels has an energy difference equal to 2)E43.

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 energy difference equal to the energy of the n ! 5 level? (c) Show that no pair of adjacent levels has an energy difference equal to the energy of the n ! 6 level.

An electron is trapped in a one-dimensional infinite well of width 250 pm and is in its ground state. What are the (a) longest, (b) second longest, and (c) third longest wavelengths of light that can excite the electron from the ground state via a single photon absorption?

A one-dimensional infinite well of length 200 pm contains an electron in its third excited state. We position an electrondetector probe of width 2.00 pm so that it is centered on a point of maximum probability density. (a) What is the probability of detection by the probe? (b) If we insert the probe as described 1000 times, how many times should we expect the electron to materialize on the end of the probe (and thus be detected)?

An electron is in a certain energy state in a one-dimensional, infinite potential well from x ! 0 to x ! L ! 200 pm. The electrons probability density is zero at x ! 0.300L, and x ! 0.400L; it is not zero at intermediate values of x. The electron then jumps to the next lower energy level by emitting light. What is the change in the electrons energy?

An electron is trapped in a one-dimensional infinite potential well that is 100 pm wide; the electron is in its ground state. What is the probability that you can detect the electron in an interval of width )x ! 5.0 pm centered at x ! (a) 25 pm, (b) 50 pm, and (c) 90 pm? (Hint: The interval )x is so narrow that you can take the probability density to be constant within it.)

A particle is confined to the one-dimensional infinite potential well of Fig. 39-2. If the particle is in its ground state, what is its probability of detection between (a) x ! 0 and x ! 0.25L, (b) x ! 0.75L and x ! L, and (c) x ! 0.25L and x ! 0.75L?

An electron in the n ! 2 state in the finite potential well of Fig. 39-7 absorbs 400 eV of energy from an external source. Using the energy-level diagram of Fig. 39-9, determine the electrons kinetic energy after this absorption, assuming that the electron moves to a position for which x . L

Figure 39-9 gives the energy levels for an electron trapped in a finite potential energy well 450 eV deep. If the electron is in the n ! 3 state, what is its kinetic energy?

Figure 39-28a shows the energy-level diagram for a finite, one-dimensional energy well that contains an electron. The nonquantized region begins at E4 ! 450.0 eV. Figure 39-28b gives the absorption spectrum of the electron when it is in the ground stateit can absorb at the indicated wavelengths: la ! 14.588 nm and lb ! 4.8437 nm and for any wavelength less than lc ! 2.9108 nm.What is the energy of the first excited state?

Figure 39-29a shows a thin tube in which a finite potential trap has been set up where V2 0 V.An electron is shown traveling rightward toward the trap, in a region with a voltage of V1 ! #9.00 V, where it has a kinetic energy of 2.00 eV. When the electron enters the trap region, it can become trapped if it gets rid of enough energy by emitting a photon. The energy levels of the electron within the trap are E1 ! 1.0, E2 ! 2.0, and E3 ! 4.0 eV, and the nonquantized region begins at E4 ! 9.0 eV as shown in the energylevel diagram of Fig. 39-29b. What is the smallest energy (eV) such a photon can have?

(a) Show that for the region x . L in the finite potential well of Fig. 39-7, c(x) ! De2kx is a solution of Schrdingers equation in its one-dimensional form, where D is a constant and k is positive. (b) On what basis do we find this mathematically acceptable solution to be physically unacceptable?

An electron is contained in the rectangular corral of Fig. 39-13, with widths Lx 800 pm and Ly 1600 pm. What is the electrons ground-state energy?

An electron is contained in the rectangular box of Fig. 39-14, with widths Lx ! 800 pm, Ly ! 1600 pm, and Lz ! 390 pm. What is the electrons ground-state energy?

Figure 39-30 shows a two-dimensional, infinite-potential well lying in an xy plane that contains an electron. We probe for the electron along a line that bisects Lx and find three points at which the detection probability is maximum. Those points are separated by 2.00 nm. Then we probe along a line that bisects Ly and find five points at which the detection probability is maximum. Those points are separated by 3.00 nm.What is the energy of the electron?

The two-dimensional, infinite corral of Fig. 39-31 is square, with edge length L 150 pm. A square probe is centered at xy coordinates (0.200L, 0.800L) and has an x width of 5.00 pm and a y width of 5.00 pm. What is the probability of detection if the electron is in the E1,3 energy state?

A rectangular corral of widths Lx ! L and Ly ! 2L holds an electron. What multiple of h2 /8mL2 , where m is the electron mass, gives (a) the energy of the electrons ground state, (b) the energy of its first excited state, (c) the energy of its lowest degenerate states, and (d) the difference between the energies of its second and third excited states?

An electron (mass m) is contained in a rectangular corral of widths Lx L and Ly 2L. (a) How many different frequencies of light could the electron emit or absorb if it makes a transition between a pair of the lowest five energy levels? What multiple of h/8mL2 gives the (b) lowest, (c) second lowest, (d) third lowest, (e) highest, (f) second highest, and (g) third highest frequency?

A cubical box of widths Lx ! Ly ! Lz ! L contains an electron. What multiple of h2 /8mL2 , where m is the electron mass, is (a) the energy of the electrons ground state, (b) the energy of its second excited state, and (c) the difference between the energies of its second and third excited states? How many degenerate states have the energy of (d) the first excited state and (e) the fifth excited state?

An electron (mass m) is contained in a cubical box of widths Lx ! Ly ! Lz. (a) How many different frequencies of light could the electron emit or absorb if it makes a transition between a pair of the lowest five energy levels? What multiple of h/8mL2 gives the (b) lowest, (c) second lowest, (d) third lowest, (e) highest, (f) second highest, and (g) third highest frequency

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 400 pm2 that is centered at x ! L/8 and y ! L/8. The probability of detection turns out to be 4.5 " 10 #8 . What is edge length L?

What is the ratio of the shortest wavelength of the Balmer series to the shortest wavelength of the Lyman series?

An atom (not a hydrogen atom) absorbs a photon whose associated wavelength is 375 nm and then immediately emits a photon whose associated wavelength is 580 nm. How much net energy is absorbed by the atom in this process?

What are the (a) energy, (b) magnitude of the momentum, and (c) wavelength of the photon emitted when a hydrogen atom undergoes a transition from a state with n ! 3 to a state with n ! 1?

Calculate the radial probability density P(r) for the hydrogen atom in its ground state at (a) r ! 0, (b) r ! a, and (c) r ! 2a, where a is the Bohr radius

For the hydrogen atom in its ground state, calculate (a) the probability density c2 (r) and (b) the radial probability density P(r) for r ! a, where a is the Bohr radius

(a) What is the energy E of the hydrogen-atom electron whose probability density is represented by the dot plot of Fig. 39- 21? (b) What minimum energy is needed to remove this electron from the atom?

A neutron with a kinetic energy of 6.0 eV collides with a stationary hydrogen atom in its ground state. Explain why the collision must be elasticthat is, why kinetic energy must be conserved. (Hint: Show that the hydrogen atom cannot be excited as a result of the collision.)

An atom (not a hydrogen atom) absorbs a photon whose associated frequency is 6.2 " 1014 Hz. By what amount does the energy of the atom increase?

Verify that Eq. 39-44, the radial probability density for the ground state of the hydrogen atom, is normalized. That is, verify that the following is true:

What are the (a) wavelength range and (b) frequency range of the Lyman series? What are the (c) wavelength range and (d) frequency range of the Balmer series?

What is the probability that an electron in the ground state of the hydrogen atom will be found between two spherical shells whose radii are r and r ' )r, (a) if r ! 0.500a and )r ! 0.010a and (b) if r ! 1.00a and )r ! 0.01a, where a is the Bohr radius? (Hint: )r is small enough to permit the radial probability density to be taken to be constant between r and r ' )r.)

A hydrogen atom, initially 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 hydrogen atom? (Hint:This is similar to the explosions of Chapter 9.)

In the ground state of the hydrogen atom, the electron has a total energy of #13.6 eV. What are (a) its kinetic energy and (b) its potential energy if the electron is one Bohr radius from the central nucleus?

A hydrogen atom in a state having a binding energy (the energy required to remove an electron) of 0.85 eV makes a transition to a state with an excitation energy (the difference between the energy of the state and that of the ground state) of 10.2 eV. (a) What is the energy 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?

The wave functions for the three states with the dot plots shown in Fig. 39-23, which have n 2, 1, and 0, and #1, are in which the subscripts on c(r, u) give the values of the quantum numbers n, , and the angles u and f are defined in Fig. 39-22. Note that the first wave function is real but the others, which involve the imaginary number i, are complex. Find the radial probability density P(r) for (a) c210 and (b) c21'1 (same as for c21#1). (c) Show that each P(r) is consistent with the corresponding dot plot in Fig. 39-23. (d) Add the radial probability densities for c210, c21'1, and c21#1 and then show that the sum is spherically symmetric, depending only on r.

Calculate the probability that the electron in the hydrogen atom, in its ground state, will be found between spherical shells whose radii are a and 2a, where a is the Bohr radius.

For what value of the principal quantum number n would the effective radius, as shown in a probability density dot plot for the hydrogen atom, be 1.0 mm? Assume that has its maximum value of n # 1. (Hint: See Fig. 39-24.)

Light of wavelength 121.6 nm is emitted by a hydrogen 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?

How much work must be done to pull apart the electron and the proton that make up the hydrogen atom if the atom is initially in (a) its ground state and (b) the state with n ! 2?

Light of wavelength 102.6 nm is emitted by a hydrogen 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?

What is the probability that in the ground state of the hydrogen atom, the electron will be found at a radius greater than the Bohr radius?

A hydrogen atom is excited from its ground state to the state with n ! 4. (a) How much energy must be absorbed by the atom? Consider the photon energies that can be emitted by the atom as it de-excites to the ground state in the several possible 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?

Schrdingers equation for states of the hydrogen atom for which the orbital quantum number is zero is Verify that Eq. 39-39, which describes the ground state of the hydrogen atom, is a solution of this equation.

The wave function for the hydrogen-atom quantum state represented by the dot plot shown in Fig. 39- 21, which has n ! 2 and , is in which a is the Bohr radius and the subscript on c(r) gives the values of the quantum numbers n, , . (a) Plot and show that your plot is consistent with the dot plot of Fig. 39- 21. (b) Show analytically that has a maximum at r ! 4a. (c) Find the radial probability density P200(r) for this state. (d) Show that and thus that the expression above for the wave function c200(r) has been properly normalized.

The radial probability density for the ground state of the hydrogen atom is a maximum when r ! a, where a is the Bohr radius. Show that the average value of r, defined as has the value 1.5a. In this expression for ravg, 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 maximum.

Let )Eadj be the energy difference between two adjacent energy levels for an electron trapped in a one-dimensional infinite potential well. Let E be the energy of either of the two levels. (a) Show that the ratio )Eadj/E approaches the value 2/n at large values of the quantum number n. As n : ,, does (b) )Eadj, (c) E, or (d) )Eadj/E approach zero? (e) What do these results mean in terms of the correspondence principle?

An electron is trapped in a one-dimensional infinite potential well. Show that the energy difference )E between its quantum levels n and n ' 2 is (h2 /2mL2 )(n ' 1).

As Fig. 39-8 suggests, the probability density for an electron in the region 0 / x / L for the finite potential well of Fig. 39-7 is sinusoidal, being given by c2 (x) ! B sin2 kx, in which B is a constant. (a) Show that the wave function c(x) that may be found from this equation is a solution of Schrdingers equation in its onedimensional form. (b) Find an expression for k that makes this true

As Fig. 39-8 suggests, the probability density for the region x L in the finite potential well of Fig. 39-7 drops off exponentially according to c2 (x) ! Ce#2kx, where C is a constant. (a) Show that the wave function c(x) that may be found from this equation is a solution of Schrdingers equation in its onedimensional form. (b) Find an expression for k for this to be true.

An electron is confined to a narrow evacuated tube of length 3.0 m; the tube functions as a one- dimensional infinite potential well. (a) What is the energy difference between the electrons ground state and its first excited state? (b) At what quantum number n would the energy difference between adjacent energy levels be 1.0 eVwhich is measurable, unlike the result of (a)? At that quantum number, (c) what multiple of the electrons rest energy would give the electrons total energy and (d) would the electron be relativistic?

(a) Show that the terms in Schrdingers equation (Eq. 39-18) have the same dimensions. (b) What is the common SI unit for each of these terms?

(a) What is the wavelength of light for the least energetic photon emitted in the Balmer series of the hydrogen atom spectrum lines? (b) What is the wavelength of the series limit?

(a) For a given value of the principal quantum number n for a hydrogen atom, how many values of the orbital quantum number are possible? (b) For a given value of , how many values of the orbital magnetic quantum number are possible? (c) For a given value of n, how many values of are possible?

Verify that the combined value of the constants appearing in Eq. 39-33 is 13.6 eV

A diatomic gas molecule consists of two atoms of mass m separated by a fixed distance d rotating about an axis as indicated in Fig. 39-32. Assuming that its angular momentum is quantized as in the Bohr model for the hydrogen atom, find (a) the possible angular velocities and (b) the possible quantized rotational energies.

In atoms there is a finite, though very small, probability that, at some instant, an orbital electron will actually be found inside the nucleus. In fact, some unstable nuclei use this occasional appearance of the electron to decay by electron capture. Assuming that the proton itself is a sphere of radius 1.1 " 10#15 m and that the wave function of the hydrogen atoms electron holds all the way to the protons center, use the ground-state wave function to calculate the probability that the hydrogen atoms electron is inside its nucleus.

(a) What is the separation in energy between the lowest two energy levels for a container 20 cm on a side containing argon atoms? Assume, for simplicity, that the argon atoms are trapped in a one- dimensional well 20 cm wide. The molar mass of argon is 39.9 g/mol. (b) At 300 K, to the nearest power of ten, what is the ratio of the thermal energy of the atoms to this energy separation? (c) At what temperature does the thermal energy equal the energy separation?

A muon of charge #e and mass m ! 207me (where me is the mass of an electron) orbits the nucleus of a singly ionized helium atom (He+). Assuming that the Bohr model of the hydrogen atom can be applied to this muonhelium system, verify that the energy levels of the system are given by

From the energy-level diagram for hydrogen, explain the observation that the frequency of the second Lyman-series line is the sum of the frequencies of the first Lyman-series line and the first Balmer- series line.This is an example of the empirically discovered Ritz combination principle. Use the diagram to find some other valid combinations

A hydrogen atom can be considered as having a central pointlike proton of positive charge e and an electron of negative charge #e that is distributed about the proton according to the volume charge density r ! A exp(#2r/a0). Here A is a constant, a0 ! 0.53 " 10#10 m, and r is the distance from the center of the atom. (a) Using the fact that the hydrogen is electrically neutral, find A. Then find the (b) magnitude and (c) direction of the atoms electric field at a0.

An old model of a hydrogen atom has the charge 'e of the proton uniformly distributed over a sphere of radius a0, with the electron of charge #e and mass m at its center. (a) What would then be the force on the electron if it were displaced from the center by a distance r # a0? (b) What would be the angular frequency of oscillation of the electron about the center of the atom once the electron was released?

In a simple model of a hydrogen atom, the single electron orbits the single proton (the nucleus) in a circular path. Calculate (a) the electric potential set up by the proton at the orbital radius of 52.9 pm, (b) the electric potential energy of the atom, and (c) the kinetic energy of the electron. (d) How much energy is required to ionize the atom (that is, to remove the electron to an infinite distance with no kinetic energy)? Give the energies in electron-volts

Consider a conduction electron in a cubical crystal of a conducting material. Such an electron is free to move throughout the volume of the crystal but cannot escape to the outside. It is trapped in a three- dimensional infinite well.The electron can move in three dimensions, so that its total energy is given by in which n1, n2, and n3 are positive integer values. Calculate the energies of the lowest five distinct states for a conduction electron moving in a cubical crystal of edge length L ! 0.25 mm.