(a) Show that the mean value for 1/r of an electron inthe

For n = 7, what values can have?

For n = 6, = 3, what are the possible values of mi and msl

How many different states are possible for an electron whose principal quantum number is n = 5? Write down the quantum numbers for each state.

If a hydrogen atom has mi = -4 , what are the possible values of n, , and msl

A hydrogen atom has = 5. What are the possible values for n, m i, and ms?

Calculate the magnitude of the angular momentum of an electron in the n = 5, = 3 state of hydrogen.

A hydrogen atom is in the Ig state. Determine (a) the principal quantum number, (b) the energy of the state, (c) the orbital angular momentum and its quantum number , and (d) the possible values for the magnetic quantum number.

(a) Show that the number of different states possible for a given value of is equal to 2(2 + 1). (b) What is this number for = 0, 1, 2,3,4,5, and 6?

Show that the number of different electron states possible for a given value of n is 2n2. (See Problem 8.)

An excited H atom is in a 5d state, (a) Name all the states to which the atom is allowed to jump with the emission of a photon, (b) How many different wavelengths are there (ignoring fine structure)?

The magnitude of the orbital angular momentum in an excited state of hydrogen is 6.84 X 10 34 J-s and the z component is 2.11 X 10-34 J-s. What are all the possible values of n, , and mt for this state?

Show that the ground-state wave function, Eq. 39-5, is normalized. [Hint: See Example 39-4.]

For the ground state of hydrogen, what is the value of () i/j, (b) \if/\2, and (c) PT, at r = 1.5 r0?

For the n = 2, = 0 state of hydrogen, what is the value of (a) if/, (b) |j/f|2, and (c) PT, at r = 4/*0?

By what factor is it more likely to find the electron in the ground state of hydrogen at the Bohr radius (r0) than at twice the Bohr radius (2r0)?

(a) Show that the probability of finding the electron in the ground state of hydrogen at less than one Bohr radius from the nucleus is 32%. (b) What is the probability of finding a Is electron between r = r0 and r = 2r0?

Determine the radius r of a sphere centered on the nucleus within which the probability of finding the electron for the ground state of hydrogen is (a) 50%, (b) 90%, (c) 99%.

(a) Estimate the probability of finding an electron, in the ground state of hydrogen, within the nucleus assuming it to be a sphere of radius r = 1.1 fm. (b) Repeat the estimate assuming the electron is replaced with a muon, which is very similar to an electron (Chapter 43) except that its mass is 207 times greater.

Show that the mean value of r for an electron in the ground state of hydrogen is r = | r 0, by calculating

Show that 1A200 as given by Eq. 39-8 is normalized.

Determine the average radial probability distribution Pr for the n = 2, = 1 state in hydrogen by calculating Pr = 4t7T2[||iA2io|2 + I f e l l 2 + ilA2i- i| 2]-

Use the result of Problem 21 to show that the most probable distance r from the nucleus for an electron in the 2p state of hydrogen is r = 4r0, which is just the second Bohr radius (Eq. 37-11, Fig. 37-25).

For the ground state of hydrogen, what is the probability of finding the electron within a spherical shell of inner radius 0.99r0 and outer radius 1.01 r0?

For the n = 2, = 0 state of hydrogen, what is the probability of finding the electron within a spherical shell of inner radius 4.00 r0 and outer radius 5.00/q? [Hint: You might integrate by parts.

Show that i//100 (Eq. 39-5a) satisfies the Schrodinger equation (Eq. 39-1) with the Coulomb potential, for energy E = -m e4/S e lh 2.

Show that the probability of finding the electron within 1 Bohr radius of the nucleus in the hydrogen atom is (a) 3.4% for the n = 2, = 0 state, and (b) 0.37% for the n = 2, = 1 state. (See Problem 21.)

The wave function for the n = 3, = 0 state in hydrogen is 1 / 2r 2r2 \ - j l iff200 = , 1 ------1-------r \e 3ro. \ j 21irr\ V 3r0 27r0/ (a) Determine the radial probability distribution Pr for this state, and (b) draw the curve for it on a graph, (c) Determine the most probable distance from the nucleus for an electron in this state.

List the quantum numbers for each electron in the ground state of oxygen (Z = 8 ).

List the quantum numbers for each electron in the ground state of (a) carbon (Z = 6 ), (b) aluminum (Z = 13).

How many electrons can be in the n = 6, = 4 subshell?

An electron has mt = 2 and is in its lowest possible energy state. What are the values of n and for this electron?

If the principal quantum number n were limited to the range from 1 to 6, how many elements would we find in nature?

What is the full electron configuration for (a) nickel (Ni), (b) silver (Ag), (c ) uranium (U)? [Hint: See the Periodic Table inside the back cover.]

Estimate the binding energy of the third electron in lithium using Bohr theory. [Hint: This electron has n = 2 and sees a net charge of approximately + le .] The measured value is 5.36 eV.

Using the Bohr formula for the radius of an electron orbit, estimate the average distance from the nucleus for an electron in the innermost (n = 1) orbit in uranium (Z = 92). Approximately how much energy would be required to remove this innermost electron?

Let us apply the exclusion principle to an infinitely high square well (Section 38-8). Let there be five electrons confined to this rigid box whose width is . Find the lowest energy state of this system, by placing the electrons in the lowest available levels, consistent with the Pauli exclusion principle.

Show that the total angular momentum is zero for a filled subshell.

If the shortest-wavelength bremsstrahlung X-rays emitted from an X-ray tube have A = 0.027 nm, what is the voltage across the tube?

What are the shortest-wavelength X-rays emitted by electrons striking the face of a 32.5-kV TV picture tube? What are the longest wavelengths?

I) Show that the cutoff wavelength A0 in an X-ray spectrum is given by 1240 A0 = nm, where V is the X-ray tube voltage in volts.

Estimate the wavelength for an n = 2 to n = 1 transition in iron (Z = 26).

Use the result of Example 39-6 to estimate the X-ray wavelength emitted when a cobalt atom (Z = 27) makes a transition from n = 2 to n = 1.

A mixture of iron and an unknown material are bombarded with electrons. The wavelength of the K a lines are 194 pm for iron and 229 pm for the unknown. What is the unknown material?

Use Bohr theory to estimate the wavelength for an n = 3 to n = 1 transition in molybdenum (Z = 42). The measured value is 0.063 nm. Why do we not expect perfect agreement?

Use conservation of energy and momentum to show that a moving electron cannot give off an X-ray photon unless there is a third object present, such as an atom or nucleus.

Verify that the Bohr magneton has the value |xB = 9.27 X 1024 J/T (see Eq. 39-12).

If the quantum state of an electron is specified by (n, , mg, ms), estimate the energy difference between the states ( l, 0 , 0, - 5) and ( l, 0, 0, + 5) of an electron in the Is state of helium in an external magnetic field of 2.5 T.

Silver atoms (spin = ) are placed in a 1.0-T magnetic field which splits the ground state into two close levels. (a) What is the difference in energy between these two levels, and (b) what wavelength photon could cause a transition from the lower level to the upper one? (c) How would your answer differ if the atoms were hydrogen?

In a Stern-Gerlach experiment, Ag atoms exit the oven with an average speed of 780 m/s and pass through a magnetic field gradient dB /d z = 1.8 X 103T/m for a distance of 5.0 cm. (a) What is the separation of the two beams as they emerge from the magnet? (b) What would the separation be if the g-factor were 1 for electron spin?

For an electron in a 5g state, what are all the possible values of j, m j, / , and J{1

What are the possible values of j for an electron in (a) the Ap, (b) the Af, and (c) the 3d state of hydrogen? (<d) What is J in each case?

(a) Write down the quantum numbers for each electron in the gallium atom, (b) Which subshells are filled? (c) The last electron is in the 4p state; what are the possible values of the total angular momentum quantum number, j, for this electron? (d) Explain why the angular momentum of this last electron also represents the total angular momentum for the entire atom (ignoring any angular momentum of the nucleus). (e) How could you use a Stern-Gerlach experiment to determine which value of j the atom has?

The difference between the 2P3/2 and 2P1/2 energy levels in hydrogen is about 5 X IO-5 eV, due to the spinorbit interaction, (a) Taking the electrons (orbital) magnetic moment to be 1 Bohr magneton, estimate the internal magnetic field due to the electrons orbital motion, (b) Estimate the internal magnetic field using a simple model of the nucleus revolving in a circle about the electron.

A laser used to weld detached retinas puts out 23-mslong pulses of 640-nm light which average 0.63-W output during a pulse. How much energy can be deposited per pulse and how many photons does each pulse contain?

Estimate the angular spread of a laser beam due to diffraction if the beam emerges through a 3.6-mm-diameter mirror. Assume that A = 694 nm. What would be the diameter of this beam if it struck (a) a satellite 380 km above the Earth, (b) the Moon? [Hint: See Chapter 35.]

A low-power laser used in a physics lab might have a power of 0.50 mW and a beam diameter of 3.0 mm. Calculate (a) the average light intensity of the laser beam, and (b) compare it to the intensity of a lightbulb emitting 15 W of light viewed from a distance of 2.0 m.

Calculate the wavelength of a He-Ne laser.

Suppose that the energy level system in Fig. 39-20 is not being pumped and is in thermal equilibrium. Determine the fraction of atoms in levels E2 and E1 relative to E0 at T = 300 K.

To what temperature would the system in Fig. 39-20 have to be raised (see Problem 58) so that in thermal equilibrium the level E2 would have half as many atoms as EqI (Note that pumping mechanisms do not maintain thermal equilibrium.)

Show that a population inversion for two levels (as in a pumped laser) corresponds to a negative Kelvin temperature in the Boltzmann distribution. Explain why such a situation does not contradict the idea that negative Kelvin temperatures cannot be reached in the normal sense of temperature.

The ionization (binding) energy of the outermost electron in boron is 8.26 eV. (a) Use the Bohr model to estimate the effective charge, Zeff, seen by this electron. (b ) Estimate the average orbital radius.

How many electrons can there be in an h subshell?

What is the fu ll electron configuration in the ground state for elements with Z equal to (a) 25, (b) 34, (c) 39? [Hint: See the Periodic Table inside the back cover.]

What are the largest and smallest possible values for the angular momentum L of an electron in the n = 6 shell?

Estimate (a) the quantum number for the orbital angular momentum of the Earth about the Sun, and (b) the number of possible orientations for the plane of Earths orbit.

Use the Bohr theory (especially Eq. 37-15) to show that the Moseley plot (Fig. 39-12) can be written - = a(Z - b), where b & 1, and evaluate a.

Determine the most probable distance from the nucleus of an electron in the n = 2, = 0 state of hydrogen.

Show that the diffractive spread of a laser beam, A/D as described in Section 39-9, is precisely what you might expect from the uncertainty principle. [Hint: Since the beams width is constrained by the dimension of the aperture D, the component of the lights momentum perpendicular to the laser axis is uncertain.]

In the so-called vector model of the atom, space quantization of angular momentum (Fig. 39-3) is illustrated as shown in Fig. 39-28. The angular momentum vector of magnitude L = \/ ( + 1) h is thought of as precessing around the z axis (like a spinning top or gyroscope) in such a way that the z component of angular momentum, Lz = m^h, also stays constant. Calculate the possible values for the angle 6 between L and the z axis (a) for = 1, (b) = 2, and (c) = 3. (d) Determine the minimum value of 6 for = 100 and = 106. Is this consistent with the correspondence principle? FIGURE 3 9 -2 8 The vector model for orbital angular momentum. The orbital angular momentum vector L is imagined to precess about the z axis; L and L z remain constant, but Lx and Ly continually change. Problems 69 and 70.

The vector model (Problem 69) gives some insight into the uncertainty principle for angular momentum, which is A.Lz A(f> > h for the z component. Here cf> is the angular position measured in the plane perpendicular to the z axis. Once mt for an atom is known, Lz is known precisely, so AL z = 0. (a) What does this tell us about 0? (b) What can you say about Lx and Ly , which are not quantized (only L and L z are)? (c) Show that although L x and L y are not quantized, nonetheless (L x + Ly)1/2 = [f( f + 1) - m j]1/2 h is.

(a) Show that the mean value for 1/r of an electron in the ground state of hydrogen equals 1 /r 0, and from this conclude that the mean value of the potential energy is (b) Using E = U + K, find a relationship between the average kinetic energy and the average potential energy in the ground state. [Hint. For (a), see Problem 19 or Example 38-9.]

The angular momentum in the hydrogen atom is given both by the Bohr model and by quantum mechanics. Compare the results for n = 2.

For each of the following atomic transitions, state whether the transition is allowed or forbidden, and why: (a) 4p 3p; (b) 3p Is; (c) 4d 3d; (d) 4d -> 3s; (e) 4s -> 2p.

It is possible for atoms to be excited into states with very high values of the principal quantum number. Electrons in these socalled Rydberg states have very small ionization energies and huge orbital radii. This makes them particularly sensitive to external perturbation, as would be the case if the atom were in an electric field. Consider the n = 45 state of the hydrogen atom. Determine the binding energy, the radius of the orbit, and the effective cross-sectional area of this Rydberg state.

Suppose that the spectrum of an unknown element shows a series of lines with one out of every four matching a line from the Lyman series of hydrogen. Assuming that the unknown element is an ion with Z protons and one electron, determine Z and the element in question.

Suppose that the splitting of energy levels shown in Fig. 39-4 was produced by a 1.6-T magnetic field, (a) What is the separation in energy between adjacent mg levels for the same ? (b) How many different wavelengths w ill there be for 3d to 2p transitions, if can change only by 1 or 0? (c) What is the wavelength for each of these transitions?

Populations in the H atom. Use the Boltzmann factor (Eq. 39-16) to estimate the fraction of H atoms in the n = 2 and n = 3 levels (relative to the ground state) for thermal equilibrium at (a) T = 300 K and (b) T = 6000 K. [Note: Since there are eight states with n = 2 and only two with n = 1, multiply your result for n = 2 by = 4; do similarly for n = 3. ] (c) Given 1.0 g of hydrogen, estimate the number of atoms in each state at T = 6000 K. (d) Estimate the number of n = 3 to n = 1 and n = 2 to n = 1 photons that w ill be emitted per second at T = 6000 K. Assume that the lifetime of each excited state is 10_8 s. [Hint: To evaluate a large exponent, you can use base-10 logarithms, Appendix A.]