What is the “central-field approximation” and why is it

For a particle in a three-dimensional box, what is the degeneracy (number of different quantum states with the same energy) of the following energy levels: (a) and (b) ?

Model a hydrogen atom as an electron in a cubical box with side length L. Set the value of L so that the volume of the box equals the volume of a sphere of radius \(a=5.29 \times 10^{-11}\mathrm{\ m}\), the Bohr radius. Calculate the energy separation between the ground and first excited levels, and compare the result to this energy separation calculated from the Bohr model

A photon is emitted when an electron in a three-dimensional box of side length \(8.00\times10^{-11}\mathrm{\ m}\) makes a transition from the \(n_X=2,\ n_Y=2,\ n_Z=1\) state to the \(n_X=1,\ n_Y=1,\ n_Z=1\) state. What is the wavelength of this photon?

For each of the following states of a particle in a three-dimensional box, at what points is the probability distribution function a maximum: \(\text{ (a) }n_X=1,\ n_Y=1,\ n_Z=1\text{ and (b) }n_X=2,\ n_Y=2,\ n_Z=1?\)

A particle is in the three-dimensional box of Section 41.1. For the state , for what planes (in addition to the walls of the box) is the probability distribution function zero? Compare this number of planes to the corresponding number of planes where is zero for the lower-energy state and for the ground state nZ = 1

What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length , the approximate diameter of a nucleus?

Consider an electron in the N shell. (a) What is the smallest orbital angular momentum it could have? (b) What is the largest orbital angular momentum it could have? Express your answers in terms of and in SI units. (c) What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of and in SI units. (d) What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of and in SI units. (e) For the electron in part (c), what is the ratio of its spin angular momentum in the z-direction to its orbital angular momentum in the z-direction?

An electron is in the hydrogen atom with n = 5. (a) Find the possible values of L and \(L_{z}\) for this electron, in units of \(\hbar\). (b) For each value of L, find all the possible angles between \(\vec{L}\) and the z-axis. (c) What are the maximum and minimum values of the magnitude of the angle between \(\vec{L}\) and the z-axis?

The orbital angular momentum of an electron has a magnitude of \(4.716\times10^{-34}\mathrm{\ kg}\cdot\mathrm{m}^2/\mathrm{s}\). What is the angular-momentum quantum number l for this electron?

Consider states with angular-momentum quantum number (a) In units of what is the largest possible value of (b) In units of what is the value of Which is larger: or the maximum possible (c) For each allowed value of what angle does the vector make with the How does the minimum angle for compare to the minimum angle for calculated in Example 41.3?

Calculate, in units of the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of 2, 20, and 200. Compare each with the value of postulated in the Bohr model. What trend do you see?

(a) Make a chart showing all the possible sets of quantum numbers and for the states of the electron in the hydrogen atom when How many combinations are there? (b) What are the energies of these states?

(a) How many different 5g states does hydrogen have? (b) Which of the states in part (a) has the largest angle between and the z-axis, and what is that angle? (c) Which of the states in part (a) has the smallest angle between and the z-axis, and what is that angle?

(a) What is the probability that an electron in the 1s state of a hydrogen atom will be found at a distance less than from the nucleus? (b) Use the results of part (a) and of Example 41.4 to calculate the probability that the electron will be found at distances between and from the nucleus.

CALC In Example 41.4 fill in the missing details that show that

Show that \(\Phi(\phi)=e^{i m_{\phi} \phi}=\Phi(\phi+2 \pi)\) (that is, show that \(\Phi(\phi)\) is periodic with period \(2 \pi)\)) if and only if \(m_{l}\) is restricted to the values \(0,\ \pm1,\ \pm2,\ldots\) (Hint: Euler’s formula states that \(e^{i \phi}=\cos \phi+i \sin \phi\).)

A hydrogen atom in a 3p state is placed in a uniform external magnetic field . Consider the interaction of the magnetic field with the atoms orbital magnetic dipole moment. (a) What field magnitude is required to split the 3 B p state into multiple levels with an energy difference of between adjacent levels? (b) How many levels will there be?

A hydrogen atom is in a d state. In the absence of an external magnetic field the states with different \(m_{l}\) values have (approximately) the same energy. Consider the interaction of the magnetic field with the atom’s orbital magnetic dipole moment. (a) Calculate the splitting (in electron volts) of the \(m_{l}\) levels when the atom is put in a 0.400-T magnetic field that is in the +z-direction. (b) Which \(m_{l}\) level will have the lowest energy? (c) Draw an energy-level diagram that shows the d levels with and without the external magnetic field.

A hydrogen atom in the 5g state is placed in a magnetic field of 0.600 T that is in the (a) Into how many levels is this state split by the interaction of the atoms orbital magnetic dipole moment with the magnetic field? (b) What is the energy separation between adjacent levels? (c) What is the energy separation between the level of lowest energy and the level of highest energy?

CP A hydrogen atom undergoes a transition from a state to the ground state. In the absence of a magnetic field, the energy of the photon emitted is 122 nm. The atom is then placed in a strong magnetic field in the Ignore spin effects; consider only the interaction of the magnetic field with the atoms orbital magnetic moment. (a) How many different photon wavelengths are observed for the transition? What are the values for the initial and final states for the transition that leads to each photon wavelength? (b) One observed wavelength is exactly the same with the magnetic field as without. What are the initial and final values for the transition that produces a photon of this wavelength? (c) One observed wavelength with the field is longer than the wavelength without the field. What are the initial and final values for the transition that produces a photon of this wavelength? (d) Repeat part (c) for the wavelength that is shorter than the wavelength in the absence of the field.

Classical Electron Spin. (a) If you treat an electron as a classical spherical object with a radius of \(1.0\times10^{-17}\mathrm{\ m}\), what angular speed is necessary to produce a spin angular momentum of magnitude \(\sqrt{\frac{3}{4}} \hbar\)? (b) Use \(v=r \omega\) and the result of part (a) to calculate the speed v of a point at the electron’s equator. What does your result suggest about the validity of this model?

A hydrogen atom in the \(n=1, m_{s}=-\frac{1}{2}\) state is placed in a magnetic field with a magnitude of 0.480 T in the +z-direction (a) Find the magnetic interaction energy (in electron volts) of the electron with the field. (b) Is there any orbital magnetic dipole moment interaction for this state? Explain. Can there be an orbital magnetic dipole moment interaction for \(n \neq 1\)

Calculate the energy difference between the (spin up) and (spin down) levels of a hydrogen atom in the 1s state when it is placed in a 1.45-T magnetic field in the negative Which level, or has the lower energy?

The hyperfine interaction in a hydrogen atom between the magnetic dipole moment of the proton and the spin magnetic dipole moment of the electron splits the ground level into two levels separated by \(5.9\times10^{-6}\mathrm{\ eV}\). (a) Calculate the wavelength and frequency of the photon emitted when the atom makes a transition between these states, and compare your answer to the value given at the end of Section 41.5. In what part of the electro-magnetic spectrum does this lie? Such photons are emitted by cold hydrogen clouds in interstellar space; by detecting these photons, astronomers can learn about the number and density of such clouds. (b) Calculate the effective magnetic field experienced by the electron in these states (see Fig. 41.18). Compare your result to the effective magnetic field due to the spin-orbit coupling calculated in Example 41.7.

A hydrogen atom in a particular orbital angular momentum state is found to have j quantum numbers \(\frac{7}{2}\) and \(\frac{9}{2}\). What is the letter that labels the value of l for the state?

For germanium make a list of the number of electrons in each subshell Use the allowed values of the quantum numbers along with the exclusion principle; do not refer to Table 41.3.

Make a list of the four quantum numbers n, l, \(m_{l}\), and \(m_{s}\) for each of the 10 electrons in the ground state of the neon atom. Do not refer to Table 41.2 or 41.3.

(a) Write out the ground-state electron configuration (1s2, 2s2, ) for the carbon atom. (b) What element of next-larger Z has chemical properties similar to those of carbon? Give the ground-state electron configuration for this element

(a) Write out the ground-state electron configuration (1s2, 2s2, ) for the beryllium atom. (b) What element of next-larger Z has chemical properties similar to those of beryllium? Give the ground-state electron configuration of this element. (c) Use the procedure of part (b) to predict what element of next-larger Z than in (b) will have chemical properties similar to those of the element you found in part (b), and give its ground-state electron configuration

For magnesium, the first ionization potential is 7.6 eV. The second ionization potential (additional energy required to remove a second electron) is almost twice this, 15 eV, and the third ionization potential is much larger, about 80 eV. How can these numbers be understood?

The 5s electron in rubidium (Rb) sees an effective charge of 2.771e. Calculate the ionization energy of this electron.

The energies of the 4s, 4p, and 4d states of potassium are given in Example 41.9. Calculate for each state. What trend do your results show? How can you explain this trend?

(a) The doubly charged ion is formed by removing two electrons from a nitrogen atom. What is the ground-state electron configuration for the ion? (b) Estimate the energy of the least strongly bound level in the shell of (c) The doubly charged ion is formed by removing two electrons from a phosphorus atom. What is the ground-state electron configuration for the ion? (d) Estimate the energy of the least strongly bound level in the shell of

(a) The energy of the 2s state of lithium is Calculate the value of for this state. (b) The energy of the 4s state of potassium is Calculate the value of for this state. (c) Compare for the 2s state of lithium, the 3s state of sodium (see Example 41.8), and the 4s state of potassium. What trend do you see? How can you explain this trend?

Estimate the energy of the highest-l state for (a) the shell of and (b) the shell of

A \(K_{\alpha}\) x ray emitted from a sample has an energy of 7.46 keV. Of which element is the sample made?

Calculate the frequency, energy (in keV), and wavelength of the \(K_{\alpha}\) x ray for the elements (a) calcium (Ca, Z = 20); (b) cobalt (Co, Z = 27); (c) cadmium (Cd, Z = 48).

The energies for an electron in the K, L, and M shells of the tungsten atom are and respectively. Calculate the wavelengths of the and x rays of tungsten.

In terms of the ground-state energy \(E_{1,1,1}\), what is the energy of the highest level occupied by an electron when 10 electrons are placed into a cubical box?

A particle in the three-dimensional box of Section 41.2 is in the ground state, where \(n_{X}=n_{Y}=n_{Z}=1\). (a) Calculate the probability that the particle will be found somewhere between x = 0 and x = L/2. (b) Calculate the probability that the particle will be found somewhere between x = L/4 and x = L/2. Compare your results to the result of Example 41.1 for the probability of finding the particle in the region x = 0 to x = L/4.

A particle is in the three-dimensional box of Section 41.2. (a) Consider the cubical volume defined by \(0 \leq x \leq L / 4,0 \leq y \leq L / 4, \text { and } 0 \leq z \leq L / 4\). What fraction of the total volume of the box is this cubical volume? (b) If the particle is in the ground state \(\left(n_{X}=1, n_{Y}=1, n_{Z}=1\right)\) calculate the probability that the particle will be found in the cubical volume defined in part (a). (c) Repeat the calculation of part (b) when the particle is in the state \(n_{X}=2, n_{Y}=1, n_{Z}=1 .\)

CALC A particle is described by the normalized wave function where A, and are all real, positive constants. The probability that the particle will be found in the infinitesimal volume dx dy dz centered at the point is dx dy dz. (a) At what value of is the particle most likely to be found? (b) Are there values of for which the probability of the particle being found is zero? If so, at what

CALC A particle is described by the normalized wave function where A and are real, positive constants. (a) Determine the probability of finding the particle at a distance between r and from the origin. (Hint: See Problem 41.42. Consider a spherical shell centered on the origin with inner radius r and thickness dr.) (b) For what value of r does the probability in part (a) have its maximum value? Is this the same value of r for which is a maximum? Explain any differences.

CP CALC A Three-Dimensional Isotropic Harmonic Oscillator. An isotropic harmonic oscillator has the potentialenergy function (Isotropic means that the force constant is the same in all three coordinate directions.) (a) Show that for this potential, a solution to Eq. (41.5) is given by In this expression, is a solution to the one-dimensional harmonic oscillator Schrdinger equation, Eq. (40.44), with energy The functions and are analogous one-dimensional wave functions for oscillations in the and Find the energy associated with this (b) From your results in part (a) what are the ground-level and first-excited-level energies of the threedimensional isotropic oscillator? (c) Show that there is only one state (one set of quantum numbers and ) for the ground level but three states for the first excited level.

CP CALC Three-Dimensional Anisotropic Harmonic Oscillator. An oscillator has the potential-energy function where This oscillator is called anisotropic because the force constant is not the same in all three coordinate directions. (a) Find a general expression for the energy levels of the oscillator (see Problem 41.44). (b) From your results in part (a), what are the ground-level and first-excited-level energies of this oscillator? (c) How many states (different sets of quantum numbers and ) are there for the ground level and for the first excited level? Compare to part (c) of Problem 41.44.

An electron in hydrogen is in the 5f state. (a) Find the largest possible value of the z-component of its angular momentum. (b) Show that for the electron in part (a), the corresponding x- and y-components of its angular momentum satisfy the equation \(\sqrt{L_x^{\ 2}+L_y^{\ 2}}=\hbar\sqrt{3}\).

(a) Show that the total number of atomic states (including different spin states) in a shell of principal quantum number n is \(2n^2\). [Hint: The sum of the first N integers \(1+2+3+\cdots+N\) is equal to N(N + 1) / 2.] (b) Which shell has 50 states?

(a) What is the lowest possible energy (in electron volts) of an electron in hydrogen if its orbital angular momentum is (b) What are the largest and smallest values of the of the orbital angular momentum (in terms of ) for the electron in part (a)? (c) What are the largest and smallest values of the spin angular momentum (in terms of ) for the electron in part (a)? (d) What are the largest and smallest values of the orbital angular momentum (in terms of ) for an electron in the shell of hydrogen?

Consider an electron in hydrogen having total energy (a) What are the possible values of its orbital angular momentum (in terms of (b) What wavelength of light would it take to excite this electron to the next higher shell? Is this photon visible to humans?

(a) Show all the distinct states for an electron in the N shell of hydrogen. Include all four quantum numbers. (b) For an f electron in the N shell, what is the largest possible orbital angular momentum and the greatest positive value for the component of this angular momentum along any chosen direction (the z-axis)? What is the magnitude of its spin angular momentum? Express these quantities in units of \(\hbar\). (c) For an electron in the d state of the N shell, what are the maximum and minimum angles between its angular momentum vector and any chosen direction (the z-axis)? (d) What is the largest value of the orbital angular momentum for an f electron in the M shell?

(a) The energy of an electron in the 4s state of sodium is What is the effective net charge of the nucleus seen by this electron? On the average, how many electrons screen the nucleus? (b) For an outer electron in the 4p state of potassium, on the average 17.2 inner electrons screen the nucleus. (i) What is the effective net charge of the nucleus seen by this outer electron? (ii) What is the energy of this outer electron?

For a hydrogen atom, the probability P(r) of finding the electron within a spherical shell with inner radius r and outer radius r + dr is given by Eq. (41.25). For a hydrogen atom in the 1s ground state, at what value of r does P(r) have its maximum value? How does your result compare to the distance between the electron and the nucleus for the n = 1 state in the Bohr model, Eq. (41.26)?

Consider a hydrogen atom in the 1s state. (a) For what value of is the potential energy equal to the total energy Express your answer in terms of This value of is called the classical turning point, since this is where a Newtonian particle would stop its motion and reverse direction. (b) For greater than the classical turning point, Classically, the particle cannot be in this region, since the kinetic energy cannot be negative. Calculate the probability of the electron being found in this classically forbidden region.

Rydberg Atoms. Rydberg atoms are atoms whose outermost electron is in an excited state with a very large principal quantum number. Rydberg atoms have been produced in the labo-ratory and detected in interstellar space. (a) Why do all neutral Rydberg atoms with the same n value have essentially the same ionization energy, independent of the total number of electrons in the atom? (b) What is the ionization energy for a Rydberg atom with a principal quantum number of 350? What is the radius in the Bohr model of the Rydberg electron’s orbit? (c) Repeat part (b) for n = 650.

CALC The wave function for a hydrogen atom in the 2s state is (a) Verify that this function is normalized. (b) In the Bohr model, the distance between the electron and the nucleus in the state is exactly 4a. Calculate the probability that an electron in the 2s state will be found at a distance less than 4a from the nucleus.

The normalized wave function for a hydrogen atom in the 2s state is given in Problem 41.55. (a) For a hydrogen atom in the 2s state, at what value of r is P(r) maximum? How does your result compare to 4a, the distance between the electron and the nucleus in the n = 2 state of the Bohr model? (b) At what value of r (other than r = 0 or \(r=\infty\) ) is P(r) equal to zero, so that the probability of finding the electron at that separation from the nucleus is zero? Compare your result to Fig. 41.9.

(a) For an excited state of hydrogen, show that the smallest angle that the orbital angular momentum vector can have with the z-axis is (b) What is the corresponding expression for the largest possible angle between and the z-axis?

(a) If the value of \(L_{z}\) is known, we cannot know either \(L_{x}\) or \(L_{y}\) precisely. But we can know the value of the quantity \(\sqrt{L_x^{\ 2}+L_y^{\ 2}}\). Write an expression for this quantity in terms of l, \(m_{l}\), and \(\hbar\). (b) What is the meaning of \(\sqrt{L_x^{\ 2}+L_y^{\ 2}}\) (c) For a state of nonzero orbital angular momentum, find the maximum and minimum values of \(\sqrt{L_x^{\ 2}+L_y^{\ 2}}\). Explain your results.

CALC The normalized radial wave function for the state of the hydrogen atom is After we average over the angular variables, the radial probability function becomes At what value of r is P(r) for the state a maximum? Compare your results to the radius of the state in the Bohr model.

Stern–Gerlach Experiment. In a Stern–Gerlach experiment, the deflecting force on the atom is \(F_{z}=-\mu_{z}\left(d B_{z} / d z\right)\), where \(\mu_{2}\) is given by Eq. (41.40) and \(d B_{z} / d z\) is the magnetic-field gradient. In a particular experiment the magnetic-field region is 50.0 cm long; assume the magnetic-field gradient is constant in this region. A beam of silver atoms enters the magnetic field with a speed of 525 m/s. What value of \(d B_{z} / d z\) is required to give a separation of 1.0 mm between the two spin components as they exit the field? (Note: The magnetic dipole moment of silver is the same as that for hydrogen, since its valence electron is in an l = 0 state.)

Consider the transition from a 3d to a 2p state of hydrogen in an external magnetic field. Assume that the effects of electron spin can be ignored (which is not actually the case) so that the magnetic field interacts only with the orbital angular momentum. Identify each allowed transition by the values of the initial and final states. For each of these allowed transitions, determine the shift of the transition energy from the zero-field value and show that there are three different transition energies.

An atom in a 3d state emits a photon of wavelength 475.082 nm when it decays to a 2p state. (a) What is the energy (in electron volts) of the photon emitted in this transition? (b) Use the selection rules described in Section 41.4 to find the allowed transitions if the atom is now in an external magnetic field of 3.500 T. Ignore the effects of the electrons spin. (c) For the case in part (b), if the energy of the 3d state was originally 8.50000 eV with no magnetic field present, what will be the energies of the states into which it splits in the magnetic field? (d) What are the allowed wavelengths of the light emitted during transition in part (b)?

Spectral Analysis. While studying the spectrum of a gas cloud in space, an astronomer magnifies a spectral line that results from a transition from a p state to an s state. She finds that the line at 575.050 nm has actually split into three lines, with adjacent lines 0.0462 nm apart, indicating that the gas is in an external magnetic field. (Ignore effects due to electron spin.) What is the strength of the external magnetic field?

A hydrogen atom makes a transition from an state to an state (the Balmer line) while in a magnetic field in the and with magnitude 1.40 T. (a) If the magnetic quantum number is in the initial state and in the final state, by how much is each energy level shifted from the zero-field value? (b) By how much is the wavelength of the line shifted from the zero-field value? Is the wavelength increased or decreased? Disregard the effect of electron spin. [Hint: Use the result of Problem 39.86(c).]

A large number of hydrogen atoms in 1s states are placed in an external magnetic field that is in the Assume that the atoms are in thermal equilibrium at room temperature, According to the MaxwellBoltzmann distribution (see Section 39.4), what is the ratio of the number of atoms in the state to the number in the state when the magnetic-field magnitude is (a) (approximately the earths field); (b) 0.500 T; (c) 5.00 T?

Effective Magnetic Field. An electron in a hydrogen atom is in the state. In a simple model of the atom, assume that the electron circles the proton in an orbit with radius r equal to the Bohr-model radius for Assume that the speed of the orbiting electron can be calculated by setting and taking L to have the quantum-mechanical value for a state. In the frame of the electron, the proton orbits with radius r and speed Model the orbiting proton as a circular current loop, and calculate the magnetic field it produces at the location of the electron.

Weird Universe. In another universe, the electron is a spin-\(\frac{3}{2}\) rather than a spin-\(\frac{1}{2}\) particle, but all other physics are the same as in our universe. In this universe, (a) what are the atomic numbers of the lightest two inert gases? (b) What is the ground-state electron configuration of sodium?

For an ion with nuclear charge Z and a single electron, the electric potential energy is \(-Z e^{2} / 4 \pi \epsilon_{0} r\) and the expression for the energies of the states and for the normalized wave functions are obtained from those for hydrogen by replacing \(e^{2}\) by \(Ze^{2}\). Consider the \(\mathrm{N}^{6+}\) ion, with seven protons and one electron. (a) What is the ground-state energy in electron volts? (b) What is the ionization energy, the energy required to remove the electron from the \(\mathrm{N}^{6+}\) ion if it is initially in the ground state? (c) What is the distance a [given for hydrogen by Eq. (41.26)] for this ion? (d) What is the wavelength of the photon emitted when the \(\mathrm{N}^{6+}\) ion makes a transition from the n = 2 state to the n = 1 ground state?

A hydrogen atom in an \(n=2, l=1, m_{l}=-1\) state emits a photon when it decays to an \(n=1, l=0, m_{l}=0\) ground state. (a) In the absence of an external magnetic field, what is the wavelength of this photon? (b) If the atom is in a magnetic field in the \(+z \text {-direction }\) and with a magnitude of 2.20 T, what is the shift in the wavelength of the photon from the zero-field value? Does the magnetic field increase or decrease the wavelength? Disregard the effect of electron spin. [Hint: Use the result of Problem 39.86(c).]

A lithium atom has three electrons, and the \({ }^{2} S_{1 / 2}\) ground-state electron configuration is \(1 s^{2} 2 s\). The \(1 s^{2} 2 p\) excited state is split into two closely spaced levels, \({ }^{2} P_{3 / 2}\) and \({ }^{2} P_{1 / 2}\), by the spin-orbit interaction (see Example 41.7 in Section 41.5). A photon with wavelength \(67.09608\ \mu\mathrm{m}\) is emitted in the \({ }^{2} P_{3 / 2} \rightarrow{ }^{2} S_{1 / 2}\) transition, and a photon with wavelength \(67.09761\ \mu\mathrm{m}\) is emitted in the \({ }^{2} P_{1 / 2} \rightarrow{ }^{2} S_{1 / 2}\) transition. Calculate the effective magnetic field seen by the electron in the \(1 s^{2} 2 p\) state of the lithium atom. How does your result compare to that for the 3p level of sodium found in Example 41.7?

Estimate the minimum and maximum wavelengths of the characteristic x rays emitted by (a) vanadium and (b) rhenium Discuss any approximations that you make

CP Electron Spin Resonance. Electrons in the lower of two spin states in a magnetic field can absorb a photon of the right frequency and move to the higher state. (a) Find the magnetic-field magnitude B required for this transition in a hydrogen atom with and to be induced by microwaves with wavelength (b) Calculate the value of B for a wavelength of 3.50 cm.

Each of 2N electrons (mass m) is free to move along the x-axis. The potential-energy function for each electron is \(U(x)=\frac{1}{2} k^{\prime} x^{2}\), where \(k^{\prime}\) is a positive constant. The electric and magnetic interactions between electrons can be ignored. Use the exclusion principle to show that the minimum energy of the system of 2N electrons is \(\hbar N^{2} \sqrt{k^{\prime} / m}\). (Hint: See Section 40.5 and the hint given in Problem 41.47.)

Consider a simple model of the helium atom in which two electrons, each with mass m, move around the nucleus (charge ) in the same circular orbit. Each electron has orbital angular momentum (that is, the orbit is the smallest-radius Bohr orbit), and the two electrons are always on opposite sides of the nucleus. Ignore the effects of spin. (a) Determine the radius of the orbit and the orbital speed of each electron. [Hint: Follow the procedure used in Section 39.3 to derive Eqs. (39.8) and (39.9). Each electron experiences an attractive force from the nucleus and a repulsive force from the other electron.] (b) What is the total kinetic energy of the electrons? (c) What is the potential energy of the system (the nucleus and the two electrons)? (d) In this model, how much energy is required to remove both electrons to infinity? How does this compare to the experimental value of 79.0 eV?

CALC Repeat the calculation of Problem 41.53 for a one-electron ion with nuclear charge Z. (See Problem 41.68.) How does the probability of the electron being found in the classically forbidden region depend on Z?

Problem 5E A particle is in the three-dimensional cubical box of Section 41.1. For the state nX = 2, nY = 2, nZ = 1, for what planes (in addition to the walls of the box) is the probability distribution function zero? Compare this number of planes to the corresponding number of planes where |?|2 is zero for the lower-energy state nX = 2, nY = 1, nZ = 1 and for the ground state nX = 1, nY = 1, nZ = 1.

(a) If two electrons in hydrogen atoms have the same principal quantum number, can they have different orbital angular momenta? How? (b) If two electrons in hydrogen atoms have the same orbital quantum number, can they have different principal quantum numbers? How?

Problem 6E What is the energy difference between the two lowest energy levels for a proton in a cubical box with side length 1.00 × 10-14 m, the approximate diameter of a nucleus?

Problem 7DQ In the Stern–Gerlach experiment, why is it essential for the magnetic field to be inhomogeneous (that is, nonuniform)?

Consider an electron in the N shell. (a) What is the smallest orbital angular momentum it could have? (b) What is the largest orbital angular momentum it could have? Express your answers in terms of ? and in SI units. (c) What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of ? and in SI units. (d) What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of ? and in SI units. (e) For the electron in part (c), what is the ratio of its spin angular momentum in the z-direction to its orbital angular momentum in the z-direction?

In the ground state of the helium atom one electron must have “spin down” and the other “spin up.” Why?

A hydrogen atom in a \(3 p\) state is placed in a uniform external magnetic field \(\vec{B}\). Consider the interaction of the magnetic field with the atom’s orbital magnetic dipole moment. (a) What field magnitude is required to split the \(3 p\) state into multiple levels with an energy difference of \(2.71 \times 10^{-5}\) eV between adjacent levels? (b) How many levels will there be? Equation Transcription: Text Transcription: 3p Vector B 3p 2.71x10^-5

Problem 18DQ The energy required to remove the 3s electron from a sodium atom in its ground state is about 5 eV. Would you expect the energy required to remove an additional electron to be about the same, or more, or less? Why?

Problem 18E A hydrogen atom is in a d state in the absence of an external magnetic field the states with different m1 values have (approximately) the same energy. Consider the interaction of the magnetic field with the atom’s orbital magnetic dipole moment (a) Calculate the splitting (in electron volts) of the m1 levels when the atom is put in a 0.400-T magnetic field that is in the +z-direction. (b) Which m1 level will have the lowest energy? (c) Draw an energy-level diagram that shows the d levels with and without the external magnetic field.

Calculate the energy difference between the \(m_{s}=\frac{1}{2}\) (“spin up”) and \(m_{s}=-\frac{1}{2}\) (“spin down”) levels of a hydrogen atom in the 1s state when it is placed in a 1.45-T magnetic field in the negative z-direction. Which level, \(m_{s}=\frac{1}{2}\) or \(m_{s}=-\frac{1}{2}\) has the lower energy? Equation Transcription: Text Transcription: m_s=1 over 2 m_s=-1 over 2 m_s=1 over 2 m_s=-1 over 2

The hyperfine interaction in a hydrogen atom between the magnetic dipole moment of the proton and the spin magnetic dipole moment of the electron splits the ground level into two levels separated by \(5.9 \times 10^{-6}\)eV. (a) Calculate the wavelength and frequency of the photon emitted when the atom makes a transition between these states, and compare your answer to the value given at the end of Section 41.5. In what part of the electromagnetic spectrum does this lie? Such photons are emitted by cold hydrogen clouds in interstellar space; by detecting these photons, astronomers can learn about the number and density of such clouds. (b) Calculate the effective magnetic field experienced by the electron in these states (see Fig. 41.18). Compare your result to the effective magnetic field due to the spin-orbit coupling calculated in Example 41.7.

A hydrogen atom in a particular orbital angular momentum state is found to have quantum numbers \(\frac{7}{2}\) and \(\frac{9}{2}\). What is the letter that labels the value of for the state? Equation Transcription: Text Transcription: 7 over 2 9 over 2

For germanium (Ge, Z = 32), make a list of the number of electrons in each subshell (1s, 2s, 2p, . . . ). Use the allowed values of the quantum numbers along with the exclusion principle; do not refer to Table 41.3.

Make a list of the four quantum numbers \(m_{l}\), and \(m_{s}\) for each of the 10 electrons in the ground state of the neon atom. Do not refer to Table 41.2 or 41.3. Equation Transcription: Text Transcription: m_l m_s

Problem 28E (a) Write out the ground-state electron configuration (1s2, 2s2, …) for the carbon atom. (b) What element of next-larger Z has chemical properties similar to those of carbon? Give the ground-state electron configuration for this element.

Problem 47P (a) Show that the total number of atomic states (including different spin states) in a shell of principal quantum number n is 2n2. [Hint: The sum of the first N integers 1 + 2 + 3 + …+ N is equal to N(N + 1).] (b) Which shell has 50 states?

(a) What is the lowest possible energy (in electron volts) of an electron in hydrogen if its orbital angular momentum is \(\sqrt{12 \hbar}\)? (b) What are the largest and smallest values of the z-component of the orbital angular momentum (in terms of \(\hbar\)) for the electron in part (a)? (c) What are the largest and smallest values of the spin angular momentum (in terms of \(\hbar\)) for the electron in part (a)? (d) What are the largest and smallest values of the orbital angular momentum (in terms of \(\hbar\)) for an electron in the IM shell of hydrogen? Equation Transcription: h h h Text Transcription: Sqrt 12h Hbar Hbar Hbar

Problem 49P Consider an electron in hydrogen having total energy ?0.5440 eV. (a) What are the possible values of its orbital angular momentum (in terms h)? (b) What wavelength of light would it take to excite this electron to the next higher shell? Is this photon visible to humans?

The normalized radial wave function for the \(2 p\) state of the hydrogen atom is \(R_{2 p}=\left(1 / \sqrt{24 a^{5}}\right) r e^{-r / 2 a}\). After we average over the angular variables, the radial probability function becomes \(P(r) d r=\left(R_{2 p}\right)^{2} r^{2} d r\). At what value of is \(P(r)\) for the \(2 p\) state a maximum? Compare your results to the radius of the \(n=2\) state in the Bohr model. Equation Transcription: Text Transcription: 2p R_2p=(1/sqrt 24a^5)re^-r/2a P(r)dr=(R2p)^2r^2dr P(r) 2p n=2

Problem 62P An atom in a 3d state emits a photon of wavelength 475.082 nm when it decays to a 2p state. (a) What is the energy (in electron volts) of the photon emitted in this transition? (b) Use the selection rules described in Section 41.4 to find the allowed transitions if the atom is now in an external magnetic field of 3.500 T. Ignore the effects of the electron’s spin. (c) For the case in part (b), if the energy of the 3d state was originally -8.50000 eV with no magnetic field present, what will be the energies of the states into which it splits in the magnetic field? (d) What are the allowed wavelengths of the light emitted during transition in part (b)?

Problem 63P CALC Spectral Analysis. While studying the spectrum of a gas cloud in space, an astronomer magnifies a spectral line that results from a transition from a p state to an s state. She finds that the line at 575.050 nm has actually split into three lines, with adjacent lines 0.0462 nm apart, indicating that the gas is in an external magnetic field. (Ignore effects due to electron spin.) What is the strength of the external magnetic field?

A hydrogen atom makes a transition from an n = 3 state to an n = 2 state(the Balmer H? line) while in a magnetic field in the +z-direction and with magnitude 1.40 T. (a) If the magnetic quantum number is mt = 2 in the initial (n = 3) state and mn = 1 in the final (n = 2) state, by how much is each energy level shifted from the zero-field value? (b) By how much is the wavelength of the H? line shifted From the zero-field value? Is the wavelength increased or decreased? Disregard the effect of electron spin.[Hint : Use the result of Problem 39.86(c).]

Problem 1DQ Particle A is described by the wave function ?(x, y, z). Particle B is described by the wave function ?(x, y, z)ei?, where ? is a real constant. How does the probability of finding particle A within a volume dV around a certain point in space compare with the probability of finding particle B within this same volume?

For a particle in a three-dimensional box, what is the degeneracy (number of different quantum states with the same energy) of the following energy levels: (a) \(3 \pi^{2} h^{2} / 2 m L^{2}\) and (b) \(9 \pi^{2} h^{2} / 2 m L^{2} ?\) Equation transcription: Text transcription: 3 \pi^{2} h^{2} / 2 m L^{2} 9 \pi^{2} h^{2} / 2 m L^{2} ?

What are the most significant differences between the Bohr model of the hydrogen atom and the Schrödinger analysis? What are the similarities?

Problem 2E CP Model a hydrogen atom as an electron in a cubical box with side length L. Set the value of L so that the volume of the box equals the volume of a sphere of radius a = 5.29 × 10-11 m, the Bohr radius. Calculate the energy separation between the ground and first excited levels, and compare the result to this energy separation calculated from the Bohr model.

For a body orbiting the sun, such as a planet, comet, or asteroid, is there any restriction on the z-component of its orbital angular momentum such as there is with the z-component of the electron’s orbital angular momentum in hydrogen? Explain.

Problem 3E CP A photon is emitted when an electron in a three-dimensional cubical box of side length 8.00 × 10-11 m makes a transition from the nX = 2, nY = 2, nZ = 1 state to the nX = 1, nY = 1, nZ = 1 state. What is the wavelength of this photon?

Problem 4DQ Why is the analysis of the helium atom much more complex than that of the hydrogen atom, either in a Bohr type of model or using the Schrödinger equation?

Problem 4E For each of the following states of a particle in a three-dimensional cubical box, at what points is the probability distribution function a maximum: (a) nX = 1, nY = 1, nZ = 1 and (b) nX = 2, nY = 2, nZ = 1?

The Stern–Gerlach experiment is always performed with beams of neutral atoms. Wouldn’t it be easier to form beams using ionized atoms? Why won’t this work?

An electron is in the hydrogen atom with . (a) Find the possible values of and \(L_{g}\) for this electron, in units of . (b) For each value of , find all the possible angles between \(\vec{L}\) and the z-axis. (c) What are the maximum and minimum values of the magnitude of the angle between and the -axis? Equation transcription: Text transcription: \vec{L} L{g}

The orbital angular momentum of an electron has a magnitude of \(4.716 \times 10^{-34}kg.m^{2}/s\). What is the angular-momentum quantum number for this electron?

Problem 9E The orbital angular momentum of an electron has a magnitude of 4.716 × 10-34 kg ? m2/s. What is the angular momentum quantum number l for this electron?

The central-field approximation is more accurate for alkali metals than for transition metals such as iron, nickel, or copper. Why?

Consider states with angular-momentum quantum number \(I=2\). (a) In units of , what is the largest possible value of \(L_{g}\)? (b) In units of , what is the value of Which is larger: or the maximum possible \(L_{g}\)? (c) For each allowed value of \(L_{g}\), what angle does the vector \(\vec{L}\) make with the -axis? How does the minimum angle for \(I=2\) compare to the minimum angle for \(I=3\) calculated in Example 41.3? Equation transcription: Text transcription: vec{L} L{g} I=2 I=3

Table 41.3 shows that for the ground state of the potassium atom, the outermost electron is in a 4s state. What does this tell you about the relative energies of the 3d and 4s levels for this atom? Explain.

Calculate, in units of , the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of 2,20 , and 200. Compare each with the value of postulated in the Bohr model. What trend do you see?

Problem 12DQ Do gravitational forces play a significant role in atomic structure? Explain.

Problem 12E (a) Make a chart showing all the possible sets of quantum numbers l and m1 for the states of the electron in the hydrogen atom when n = 5. How many combinations are there? (b) What are the energies of these states?

Why do the transition elements (Z = 21 to 30) all have similar chemical properties?

(a) How many different states does hydrogen have? (b) Which of the states in part (a) has the largest angle between \(\vec{L}\) and the -axis, and what is that angle? (c) Which of the states in part (a) has the smallest angle between \(\vec{L}\) and the -axis, and what is that angle? Equation transcription: Text transcription: \vec{L}

Use Table 41.3 to help determine the ground-state electron configuration of the neutral gallium atom (Ga) as well as the ions \(\mathrm{Ga}^{+}\) and \(\mathrm{Ga}^{-}\). Gallium has an atomic number of 31. Equation transcription: Text transcription: {Ga}^{+} {Ga}^{-}

CALC (a) What is the probability that an electron in the state of a hydrogen atom will be found at a distance less than from the nucleus? (b) Use the results of part (a) and of Example to calculate the probability that the electron will be found at distances between and from the nucleus.

Problem 15DQ On the basis of the Pauli exclusion principle, the structure of the periodic table of the elements shows that there must be a fourth quantum number in addition to n, l, and ml . Explain.

CALC In Example fill in the missing details that show that \(P = 1 - 5e^{-2}\).

A small amount of magnetic-field splitting of spectral lines occurs even when the atoms are not in a magnetic field. What causes this?

Show that \(\Phi(\varphi)=e^{i m, \varphi}=\Phi(\varphi+2 \pi)\) (that is, show that \(\Phi(\varphi)\) is periodic with period \(2 \pi\) ) if and only if \(m_{l}\) is restricted to the values \(0, \pm 1, \pm 2, \ldots\) (Hint: Euler's formula states that \(e^{i \phi}=\cos \varphi+i \sin \varphi\).) Equation Transcription: Text Transcription: Phi(phi) = e^im_l psi = Phi(phi + 2 pi) Phi(phi) 2 pi m_l 0, plus or minus 1, plus or minus 2, … e^i phi = cos phi + isin phi

Problem 17DQ The ionization energies of the alkali metals (that is, the lowest energy required to remove one outer electron when the atom is in its ground state) are about 4 or 5 eV, while those of the noble gases are in the range from 11 to 25 eV. Why is there a difference?

What is the “central-field approximation” and why is it only an approximation?

Problem 19E A hydrogen atom in the 5g state is placed in a magnetic field of 0.600 T that is in the z-direction. (a) Into how many levels is this state split by the interaction of the atom’s orbital magnetic dipole moment with the magnetic field? (b) What is the energy separation between adjacent levels? (c) What is the energy separation between the level of lowest energy and the level of highest energy?

The nucleus of a gold atom contains 79 protons. How does the energy required to remove a 1s electron completely from a gold atom compare with the energy required to remove the electron from the ground level in a hydrogen atom? In what region of the electromagnetic spectrum would a photon with this energy for each of these two atoms lie?

CP A hydrogen atom undergoes a transition from a state to the ground state. In the absence of a magnetic field, the energy of the photon emitted is . The atom is then placed in a strong magnetic field in the -direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. (a) How many different photon wavelengths are observed for the \(2 p \rightarrow 1 s\) transition? What are the \(m_{I}\) values for the initial and final states for the transition that leads to each photon wavelength? (b) One observed wavelength is exactly the same with the magnetic field as without. What are the initial and final \(m_{I}\) values for the transition that produces a photon of this wavelength? (c) One observed wavelength with the field is longer than the wavelength without the field. What are the initial and final \(m_{I}\) values for the transition that produces a photon of this wavelength? (d) Repeat part (c) for the wavelength that is shorter than the wavelength in the absence of the field. Equation transcription: Text transcription: 2 p \rightarrow 1 s m{I}

Problem 21DQ (a) Can you show that the orbital angular momentum of an electron in any given direction (e.g., along the z-axis) is always less than or equal to its total orbital angular momentum? In which cases would the two be equal to each other? (b) Is the result in part (a) true for a classical object, such as a spinning top or planet?

CP Classical Electron Spin. (a) If you treat an electron as a classical spherical object with a radius of \(1.0 \times 10^{-17} \mathrm{~m}\), what angular speed is necessary to produce a spin angular momentum of magnitude \(\sqrt{\frac{3}{4} h ?}\) (b) Use \(v=r \omega\) and the result of part (a) to calculate the speed of a point at the electron's equator. What does your result suggest about the validity of this model? Equation transcription: Text transcription: 1.0 \times 10^{-17} \mathrm{~m} \sqrt{\frac{3}{4} h ?} v=r \omega

An atom in its ground level absorbs a photon with energy equal to the K absorption edge. Does absorbing this photon ionize this atom? Explain.

A hydrogen atom in the \(n=, m_{5}=-\frac{1}{2}\) state is placed in a magnetic field with a magnitude of in the -direction. (a) Find the magnetic interaction energy (in electron volts) of the electron with the field. (b) Is there any orbital magnetic dipole moment interaction for this state? Explain. Can there be an orbital magnetic dipole moment interaction for \(n \neq 1\)? Equation transcription: Text transcription: n=, m{5}=-\frac{1}{2} n \neq 1

Problem 23DQ Can a hydrogen atom emit x rays? If so, how? If not, why not?

Problem 29E (a) Write out the ground-state electron configuration (1s2, 2s2, …) for the beryllium atom. (b) What element of next-larger Z has chemical properties similar to those of beryllium? Give the ground-state electron configuration of this element. (c) Use the procedure of part (b) to predict what element of next-larger Z than in (b) will have chemical properties similar to those of the element you found in part (b), and give its ground-state electron configuration.

Problem 30E For magnesium, the first ionization potential is 7.6 eV. The second ionization potential (additional energy required to remove a second electron) is almost twice this, 15 eV, and the third ionization potential is much larger, about 80 eV. How can these numbers be understood?

Problem 31E The 5s electron in rubidium (Rb) sees an effective charge of 2.771e. Calculate the ionization energy of this electron.

The energies of the \(45,4 p\), and states of potassium are given in Example 41.9. Calculate \(Z_{e f f}\) for each state. What trend do your results show? How can you explain this trend? Equation transcription: Text transcription: 45,4 p Z{e f f}

Problem 33E (a) The doubly charged ion N2+ is formed by removing two electrons from a nitrogen atom. What is the ground-state electron configuration for the N2+ ion? (b) Estimate the energy of the least strongly bound level in the L shell of N2+. (c) The doubly charged ion P2+ is formed by removing two electrons from a phosphorus atom. What is the ground-state electron configuration for the P2+ ion? (d) Estimate the energy of the least strongly bound level in the M shell of P2+.

(a) The energy of the 2s state of lithium is -5.391 eV. Calculate the value of \(Z_{eff}\) for this state. (b) The energy of the 4s state of potassium is -4.339 eV. Calculate the value of \(Z_{eff}\) for this state. (c) Compare \(Z_{eff}\) for the 2s state of lithium, the 3s state of sodium (see Example 41.8), and the 4s state of potassium. What trend do you see? How can you explain this trend?

Estimate the energy of the highest-l state for (a) the L shell of \(Be^+\) and (b) the N shell of \(Ca^+\).

A \(K_{\alpha}\) x ray emitted from a sample has an energy of 7.46 keV. Of which element is the sample made?

Problem 37E Calculate the frequency, energy (in keV), and wavelength of the K? x ray for the elements (a) calcium (Ca, Z = 20); (b) cobalt (Co, Z = 27); (c) cadmium (Cd, Z = 48).

Problem 38E The energies for an electron in the K, L, and M shells of the tungsten atom are -69,500 eV, -12,000 eV, and -2200 eV, respectively. Calculate the wavelengths of the K? and K? x rays of tungsten.

Problem 39P In terms of the ground-state energy E1,1,1, what is the energy of the highest level occupied by an electron when 10 electrons are placed into a cubical box?

CALC A particle in the three-dimensional box of Section 41.2 is in the ground state, where \(n_{\underline{x}}=n_{y}=n_{z}=1\) (a) Calculate the probability that the particle will be found somewhere between \(x=0\) and \(x=L / 2\) (b) Calculate the probability that the particle will be found somewhere between \(x=L 4\) and \(x=L / 2\) Compare your results to the result of Example 41.1 for the probability of finding the particle in the region \(x=0\) to \(x=L 4\). Equation transcription: Text transcription: n{\underline{x}}=n{y}=n{z}=1 x=0 x=L / 2 x=L 4

Problem 41P CALC A particle is in the three-dimensional cubical box of Section 41.2. (a) Consider the cubical volume defined by 0 ? x ? L/4, 0 ? y ? L/4, and 0 ? z ? L/4. What fraction of the total volume of the box is this cubical volume? (b) If the particle is in the ground state (nX = 1, nY = 1, nZ = 1), calculate the probability that the particle will be found in the cubical volume defined in part (a). (c) Repeat the calculation of part (b) when the particle is in the state nX = 2, nY = 1, nZ = 1.

CALC A particle is described by the normalized wave function \(\psi(x, y, z)=A x e^{-\alpha x^{2}} e^{-\beta y^{2}} e^{-\gamma z^{2}}\), where A, \(\alpha, \beta\), and \(\gamma\) are all real, positive constants. The probability that the particle will be found in the infinitesimal volume dx dy dz centered at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) is \(\left|\psi\left(x_{0}, y_{0}, z_{0}\right)\right|^{2}\) dx dy dz. (a) At what value of \(x_0\) is the particle most likely to be found? (b) Are there values of \(x_0\) for which the probability of the particle being found is zero? If so, at what \(x_0\)?

CALC A particle is described by the normalized wave function \(\Psi(x, y, z)=A e^{-\alpha\left(x^{2}+y^{2}+z^{2}\right)}\), where and \(\alpha\) are real, positive constants. (a) Determine the probability of finding the particle at a distance between and from the origin. (Hint: See Problem 41.42. Consider a spherical shell centered on the origin with inner radius and thickness ) (b) For what value of does the probability in part (a) have its maximum value? Is this the same value of for which \(|\Psi(x, y, z)|^{2}\) is a maximum? Explain any differences. Equation transcription: Text transcription: \Psi(x, y, z)=A e^{-\alpha\left(x^{2}+y^{2}+z^{2}\right)} \alpha |\Psi(x, y, z)|^{2}

CP CALC A Three-Dimensional Isotropic Harmonic Oscillator. An isotropic harmonic oscillator has the potential energy function \(U(x, y, z)=\frac{1}{2} k^{*}\left(x^{2}+y^{2}+z^{2}\right)\).( Isotropic means that the force constant \(k^{\prime}\) is the same in all three coordinate directions.) (a) Show that for this potential, a solution to Eq. (41.5) is given by \(\Psi=\Psi_{n x}(x) \Psi_{n y}(y) \Psi_{n z}(z)\) In this expression, \(\Psi_{n}(x)\) is a solution to the one-dimensional harmonic oscillator equation, Eq. (40.44), with energy \(E_{n x}=\left(n_{x}+\frac{1}{2}\right) h \omega\). The functions \(\Psi_{n y}(y)\) and \(\Psi_{n z}(z)\) are analogous one-dimensional wave functions for oscillations in the and -directions. Find the energy associated with this \(\Psi\). (b) From your results in part (a) what are the ground-level and first-excited-level energies of the three dimensional isotropic oscillator? (c) Show that there is only one ate (one set of quantum numbers \(n_{z}, n_{y}\) and \(n_{z}\) ) for the ground level but three states for the first excited level. Equation transcription: Text transcription: k^{\prime} U(x, y, z)=\frac{1}{2} k^{*}\left(x^{2}+y^{2}+z^{2}\right) \Psi=\Psi{n x}(x) \Psi{n y}(y) \Psi{n z}(z) \Psi{n}(x) E{n x}=\left(n_{x}+\frac{1}{2}\right) h \omega Psi{n y}(y) Psi{n z}(z) Psi n{z}, n{y} n{z}

CP CALC Three-Dimensional Anisotropic Harmonic Oscillator. An oscillator has the potential-energy function \(U(x, y, z)=\frac{1}{2} k_{1}^{\prime}\left(x^{2}+y^{2}\right)+\frac{1}{2} k_{2}^{\prime} z^{2}\), where \(k_{1}^{\prime}>k_{2}^{\prime}\). This oscillator is called anisotropic because the force constant is not the same in all three coordinate directions. (a) Find a general expression for the energy levels of the oscillator (see Problem 41.44 ). (b) From your results in part (a), what are the ground-level and first-excited-level energies of this oscillator? (c) How many states: (different sets of quantum numbers \(n_{x}, \ n_{y}\), and \(n_{z}\)) are there for the ground level and for the first level. Compare to part (c) of Problem 41.44.

An electron in hydros the \(5 f\) state. (a) Find the largest possible value of the component of its angular momentum. (b) Show that for the electron in part (a), the corresponding x- and y-components of its angular momentum satisfy the equation \(\sqrt{L_{x}^{2}+L_{y}^{2}=\eta \sqrt{3}}\) Equation transcription: Text transcription: 5 f sqrt{L{x}^{2}+L{y}^{2}=eta sqrt{3}}

Problem 50P (a) Show all the distinct states for an electron in the N shell of hydrogen. Include all four quantum numbers. (b) For an f electron in the N shell, what is the largest possible orbital angular momentum and the greatest positive value for the component of this angular momentum along any chosen direction (the z-axis)? What is the magnitude of its spin angular momentum? Express these quantities in units of h. (c) For an electron in the d state of the N shell, what are the maximum and minimum angles between its angular momentum vector and any chosen direction (the z-axis)? (d) What is the largest value of the orbital angular momentum for an f electron in the M shell?

Problem 51P (a) The energy of an electron in the 4s state of sodium is ?1.947 eV. What is the effective net charge of the nucleus “seen” by this electron? On the average, how many electrons screen the nucleus? (b) For an outer electron in the 4p state of potassium, on the average 17.2 inner electrons screen the nucleus. (i) What is the effective net charge of the nucleus “seen” by this outer electron? (ii) What is the energy of this outer electron?

CALC For a hydrogen atom, the probability P(r) of finding the electron within a spherical shell with inner radius r and outer radius r + dr is given by Eq (41.25). For a hydrogen atom in the 1s ground state, at what value of r does P(r) have its maximum value? How does your result compare to the distance between the electron and the nucleus for the n = 1 state in the Bohr model, Eq- (41.26)?

Problem 53P CALC Consider a hydrogen atom in the 1s state. (a) For what value of r is the potential energy U(r) equal to the total energy E? Express your answer in terms of a. This value of r is called the classical turning point, since this is where a Newtonian particle would stop its motion and reverse direction. (b) For r greater than the classical turning point, U(r) > E. Classically, the particle cannot be in this region, since the kinetic energy cannot be negative. Calculate the probability of the electron being found in this classically forbidden region.

Problem 54P Rydberg atoms. Rydberg atoms are atoms whose outermost electron is in an excited state with a very large principal quantum number. Rydberg atoms have been produced in the laboratory and detected in interstellar space. (a) Why do all neutral Rydberg atoms with the same n value have essentially the same ionization energy, independent of the total number of electrons in the atom? (b) What is the ionization energy for a Rydberg atony with a principal quantum number of 350? What is the radius in the Bohr model of the Rydberg electron’s orbit? (c) Repeat part (b) for n = 650.

CALC The wave function for a hydrogen atom in the 2s state is \(\Psi_{2 s}(r)=\frac{1}{\sqrt{32 \pi a^{3}}}\left(2-\frac{r}{a}\right) e^{-r / 2 a}\) (a) Verify that this function is normalized. (b) In the Bohr model, the distance between the electron and the nucleus in the n = 2 state is exactly 4a. Calculate the probability that an electron in the 2s state will be found at a distance less than 4a from the nucleus.

CALC The normalized wave function for a hydrogen atom in the state is given in Problem 41.55. (a) For a hydrogen atom in the state, at what value of is \(p(r)\) maximum? How does your result compare to , the distance between the electron and the nucleus in the state of the Bohr model? (b) At what value of (other than \(r=0\) or \(r=\infty\) ) is \(p(r)\) equal to zero, so that the probability of finding the electron at that separation from the nucleus is zero? Compare your result to Fig. . Equation transcription: Text transcription: p(r) r=0 r=\infty

(a) For an excited state of hydrogen, show the smallest angle that the orbital angular momentum vector \(\vec{L}\) can have with the z -axis is $$\left(\theta_{L}\right)_{\min }=\arccos \left(\frac{n-1}{\sqrt{n(n-1)}}\right)$$ (b) What is the corresponding expression for \(\left(\theta_{L}\right)_{\max }\), the largest possible angle between \(\vec{L}\) and the z -axis? Equation Transcription: Text Transcription: L^rightarrow (theta_L)_min = arcos (n-1/sqrt n(n-1)) (theta_L)_max

(a) If the value of is known, we cannot know either \(L_{x}) or \(L_{y}\) precisely. But we can know the value of the quantity \(\sqrt{L_{x}^{2}+L_{x}^{2}}\) Write an expression for this quantity in terms of \(I, m_{I}\) and . (b) What is the meaning of \(\sqrt{L_{x}^{2}+L_{x}^{2}}\)? (c) For a state of nonzero orbital angular momentum, find the maximum and minimum values of \(\sqrt{L_{x}^{2}+L_{x}^{2}}\) Explain your results. Equation transcription: Text transcription: L{x} L{y} \sqrt{L{x}^{2}+L{x}^{2}}

CP Stern-Gerlach Experiment. In a Stern-Gerlach experiment, the deflecting force on the atom is \(F_{z}=-\mu_{z}\left(d B_{z} / d z\right)\), where \(\mu_{z}\) is given by Eq. (41.40) and \(d B_{z} / d z\) is the magnetic-field gradient. In a particular experiment the magnetic-field region is long; assume the magnetic-field gradient is constant in this region. beam of silver atoms enters the magnetic field with a speed of . What value of \(d B_{z} / d z\) is required to give a separation of between the two spin components as they exit the field? (Note: The magnetic dipole moment of silver is the same as that for hydrogen, since its valence electron is in an state.) Equation transcription: Equation transcription: F_{z}=-\mu{z}\left(d B{z} / d z\right) \mu{z} d B{z} / d z

Consider the transition from a 3d to a 2p state of hydrogen in an external magnetic field. Assume that the effects of electron spin can be ignored (which is not actually the case) so that the magnetic field interacts only with the orbital angular momentum. Identify each allowed transition by the \(m_l\) values of the initial and final states. For each of these allowed transitions, determine the shift of the transition energy from the zero-field value and show that there are three different transition energies.

CP A large number of hydrogen atoms in 1s states are placed in an external magnetic field that is in the +z-direction. Assume that the atoms are in thermal equilibrium at room temperature, T = 300 K. According to the Maxwell-Boltzmann distribution (see Section 39.4), what is the ratio of the number of atoms in the \(m_{s}=\frac{1}{2}\) state to the number in the \(m_{s}= -\frac{1}{2}\) state when the magnetic-field magnitude is (a) \(5.00 \times 10^{-5} T\) (approximately the earth's field); (b) .500 T; (c) 5.00 T?

Effective Magnetic Field. An electron in a hydrogen atom is in the 2p state. In a simple model of the atom, assume that the electron circles the proton in an orbit with radius r equal to the Bohr-model radius for n = 2. Assume that the speed v of the orbiting electron can be calculated by setting L = mvr and taking L to have the quantum-mechanical value for a 2p state. In the frame of the electron, the proton orbits with radius r and speed v. Model the orbiting proton as a circular current loop, and calculate the magnetic field it produces at the location of the electron.

Weird Universe. In another universe, the electron is a spin- \(\frac{3}{2}\) rather than a spin- \(\frac{1}{2}\) particle, but all other physics are the same as in our universe. In this universe, (a) what are the atomic numbers of the lightest two inert gases? (b) What is the ground-state electron configuration of sodium? Equation transcription: Text transcription: \frac{3}{2} \frac{1}{2}

For an ion with nuclear charge and a single electron, the electric potential energy is \(-Z e^{2} / 4 \pi \epsilon_{0} r\) and the expression for the energies of the states and for the normalized wave functions are obtained from those for hydrogen by replacing \(e^{2}\) by \(Z e^{2}\). Consider the \(\mathrm{N}^{6+}\) ion, with seven protons and one electron. (a) What is the ground-state energy in electron volts? (b) What is the ionization energy, the energy required to remove the electron from the \(\mathrm{N}^{6+}\) ion if it is initially in the ground state? (c) What is the distance [given for hydrogen by Eq. (41.26)] for this ion? (d) What is the wavelength of the photon emitted when the \(\mathrm{N}^{6+}\) ion makes a transition from the state to the ground state? Equation transcription: Text transcription: -Z e^{2} / 4 \pi \epsilon{0} r e^{2} Z e^{2} {N}^{6+}

A hydrogen atom in an \(n=2, I=1 m_{I}=-1\) state emits a photon when it decays to an \(n=1, I=0, m_{I}=0\) ground state. (a) In the absence of an external magnetic field, what is the wavelength of this photon? (b) If the atom is in a magnetic field in the -direction and with a magnitude of , what is the shift in the wavelength of the photon from the zero-field value? Does the magnetic field increase or decrease the wavelength? Disregard the effect of electron spin. [Hint: Use the result of Problem Equation transcription: Text transcription: n=2, I=1 m{I}=-1 n=1, I=0, m{I}=0

A lithium atom has three electrons, and the \({ }^{2} S_{1 / 2}\) ground-state electron configuration is \(1 s^{2} 2 s\) The \(1 s^{2} 2 p\) excited state is split into two closely spaced levels, \({ }^{2} P_{3 / 2}\) and \({ }^{2} P_{1 / 2}\), by the spin-orbit interaction (see Example in Section 41.5). A photon with wavelength \(67.09761 \mu \mathrm{m}\) is emitted in the \({ }^{2} P_{3 / 2} \rightarrow{ }^{2} S_{1 / 2}\) transition, and a photon with wavelength is emitted in the \({ }^{2} P_{1 / 2} \rightarrow{ }^{2} S_{1 / 2}\) transition. Calculate the effective magnetic field seen by the electron in the \(1 s^{2} 2 p\) state of the lithium atom. How does your result compare to that for the \(3 p\) level of sodium found in Example 41.7? Equation transcription: Text transcription: { }^{2} S{1 / 2} 1 s^{2} 2 s 1 s^{2} 2 p ^{2} P{3 / 2} ^{2} P{1 / 2} ^{2} P{1 / 2} rightarrow^{2} S{1 / 2} 67.09761 \mu{m} ^{2} P{3 / 2} \rightarrow{^{2} S{1 / 2} 3 p

Estimate the minimum and maximum wavelengths of the characteristic x rays emitted by (a) vanadium (Z = 23) and (b) rhenium (Z = 45). Discuss any approximations that you make.

Problem 72P Electron Spin Resonance. Electrons in the lower of two spin states in a magnetic field can absorb a photon of the right frequency and move to the higher state. (a) Find the magnetic-field magnitude B required for this transition in a hydrogen atom with n = 1 and l = 0 to be induced by microwaves with wavelength ?. (b) Calculate the value of B for a wavelength of 3.50 cm.

Each of 2N electrons (mass m) is free to move along the x-axis. The potential-energy function for each electron is \(U(x)=\frac{1}{2} k^{\prime} x^{2}\) where \(k^{\prime}\) is a positive constant. The electric and magnetic interactions between electrons can be ignored. Use the exclusion principle to show that the minimum energy of the system of 2N electrons is \(\hbar N^{2} \sqrt{k^{\prime} / m}\). (Hint: See Section 40.5 and the hint given in Problem 41.47.)

CP Consider a simple model of the helium atom in which two electrons, each with mass , move around the nucleus (charge \(+2 e\) ) in the same circular orbit. Each electron has orbital angular momentum (that is, the orbit is the smallest-radius Bohr orbit), and the two electrons are always on opposite sides of the nucleus. Ignore the effects of spin. (a) Determine the radius of the orbit and the orbital speed of each electron. [Hint: Follow the procedure used in Section to derive Eqs. (39.8) and (39.9). Each electron experiences an attractive force from the nucleus and a repulsive force from the other electron.] (b) What is the total kinetic energy of the electrons? (c) What is the potential energy of the system (the nucleus and the two electrons)? (d) In this model, how much energy is required to remove both electrons to infinity? How does this compare to the experimental value of \(79.0 \mathrm{eV} ?\) Equation transcription: Text transcription: +2 e 79.0{eV} ?

CALC Repeat the calculation of Problem for a one-electron ion with nuclear charge Z. (See Problem 41.68.) How does the probability of the electron being found in the classically forbidden region depend on ?