An isotropic harmonic oscillator has the potentialenergy function U(x, y, z)= 1/2 k′(x2 + y2 + z2). (Isotropic means that the force constant k′ is the same in all three coordinate directions.) a. Show that for this potential, a solution to Eq. (41.5) is given by ψ = ψnx(x)ψny(y)ψnz(z). In this expression, ψnx(x) is a solution to the one-dimensional harmonic-oscillator Schrödinger equation, Eq. (40.44), with energy Enx =(nx + 1/2)ħω. The functions ψny(y) and ψnz(z) are analogous one-dimensional wave functions for oscillations in the y- and z-directions. Find the energy associated with this c. 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 state (one set of quantum numbers nx, ny, and nz) for the ground level but three states for the first excited level.
> A neutral pion (π0) has a mass of 264 times the electron mass and decays with a lifetime of 8.4 * 10-17 s to
> The radiocarbon in our bodies is one of the naturally occurring sources of radiation. Let’s see how large a dose we receive. 14C decays via β- emission, and 18% of our body’s mass is carbon. a. Write out t
> 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
> The polonium isotope 84 210
> The atomic mass of 12 25
> Thorium 90 230
> One of the problems of in-air testing of nuclear weapons (or, even worse, the use of such weapons!) is the danger of radioactive fallout. One of the most problematic nuclides in such fallout is strontium-90 (90Sr), which breaks down by β- decay with a ha
> The isotope 47 110
> a. Calculate the minimum energy required to remove one neutron from the nucleus 8 17
> a. Calculate the minimum energy required to remove one proton from the nucleus 6 12
> a. When gasoline is burned, it releases 1.3 * 108 J of energy per gallon (3.788 L). Given that the density of gasoline is 737 kg/m3, express the quantity of energy released in J/g of fuel. b. During fission, when a neutron is absorbed by a 235U nucleus,
> A p­n junction is part of the control mechanism for a wind turbine that is used to generate electricity. The turbine has been malfunctioning, so you are running diagnostics. You can remotely change the bias voltage V applied to the junction an
> The table gives the occupation probabilities f(E) as a function of the energy E for a solid conductor at a fixed temperature T To determine the Fermi energy of the solid material, you are asked to analyze this information in terms of the Fermiâ&
> 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 elect
> Why must the wave function of a particle be normalized?
> To determine the equilibrium separation of the atoms in the HCl molecule, you measure the rotational spectrum of HCl. You find that the spectrum contains these wavelengths (among others): 60.4 µm, 69.0 µm, 80.4 µm, 96.4 µm, and 120.4 µm. a. Use your mea
> Metallic lithium has a bcc crystal structure. Each unit cell is a cube of side length a = 0.35 nm. a. For a bcc lattice, what is the number of atoms per unit volume? Give your answer in terms of a. (Hint: How many atoms are there per unit cell?) b. Use
> The one­dimensional calculation of Example 42.4 (Section 42.3) can be extended to three dimensions. For the three dimensional fcc NaCl lattice, the result for the potential energy of a pair of Na+ and Cl- ions due to the electrostatic interact
> Compute the Fermi energy of potassium by making the simple approximation that each atom contributes one free electron. The density of potassium is 851 kg/m3, and the mass of a single potassium atom is 6.49 * 10-26 kg.
> Suppose the hydrogen atom in HF (see the Bridging Problem for this chapter) is replaced by an atom of deuterium, an isotope of hydrogen with a mass of 3.34 * 10-27 kg. The force constant is determined by the electron configuration, so it is the same as f
> The hydrogen iodide (HI) molecule has equilibrium separation 0.160 nm and vibrational frequency 6.93 * 1013 Hz. The mass of a hydrogen atom is 1.67 * 10-27 kg, and the mass of an iodine atom is 2.11 * 10-25 kg. a. Calculate the moment of inertia of HI a
> When an OH molecule undergoes a transition from the n = 0 to the n = 1 vibrational level, its internal vibrational energy increases by 0.463 eV. Calculate the frequency of vibration and the force constant for the interatomic force. (The mass of an oxygen
> The force constant for the internuclear force in a hydrogen molecule (H2) is k′ = 576 N / m. A hydrogen atom has mass 1.67 * 10-27 kg. Calculate the zeropoint vibrational energy for H2 (that is, the vibrational energy the molecule has in the n = 0 groun
> Our galaxy contains numerous molecular clouds, regions many light­years in extent in which the density is high enough and the temperature low enough for atoms to form into molecules. Most of the molecules are H2, but a small fraction of the mo
> The equilibrium separation for NaCl is 0.2361 nm. The mass of a sodium atom is 3.8176 * 10-26 kg. Chlorine has two stable isotopes, 35Cl and 37Cl, that have different masses but identical chemical properties. The atomic mass of 35Cl is 5.8068 * 10-26 kg,
> What are the most significant differences between the Bohr model of the hydrogen atom and the Schrödinger analysis? What are the similarities?
> Part (a) of Problem 42.39 gives an equation for the number of diatomic molecules in the lth rotational level to the number in the ground-state rotational level. a. Derive an expression for the value of l for which this ratio is the largest. b. For the
> Consider a gas of diatomic molecules (moment of inertia I) at an absolute temperature T. If Eg is a groundstate energy and Eex is the energy of an excited state, then the Maxwell–Boltzmann distribution (see Section 39.4) predicts that t
> 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.
> 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 indu
> A hydrogen atom in an n = 2, l = 1, ml = -1 state emits a photon when it decays to an n = 1, l = 0, ml = 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 th
> A lithium atom has three electrons, and the 2S1/2 ground-state electron configuration is 1s22s. The 1s22p excited state is split into two closely spaced levels, 2P3/2 and 2P1/2, by the spin-orbit interaction (see Example 41.7 in Section 41.5). A photon w
> In another universe, the electron is a spin-3/2 rather than a spin-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 groundstate electro
> 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 calc
> 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 Sect
> In a Stern–Gerlach experiment, the deflecting force on the atom is Fz = -µz(dBz/dz), where µz is given by Eq. (41.38) and dBz/dz is the magnetic-field gradient. In a particular experiment, the magnetic-field region is 50.0 cm long; assume the magnetic-fi
> What is the “central-field approximation” and why is it only an approximation?
> 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 0.046
> 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 tra
> a. For an excited state of hydrogen, show that the smallest angle that the orbital angular momentum vector L can have with the z-axis is b. What is the corresponding expression for (θL)max, the largest possible angle between L and the z-ax
> 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
> The normalized radial wave function for the 2p state of the hydrogen atom is R2p =(1/ 24a5 )re-r/2a. After we average over the angular variables, the radial probability function becomes P(r) dr =(R2p)2r2dr. At what value of r is P(r) for the 2p state a m
> 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 valu
> 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
> a. What is the lowest possible energy (in electron volts) of an electron in hydrogen if its orbital angular momentum is 20 ħ ? b. What are the largest and smallest values of the z-component of the orbital angular momentum (in terms of ħ) for the elect
> 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)/2.] b. Which shell has 50 states?
> A particle is described by the normalized wave function ψ(x, y, z)= Axe-αx2e-βy2e-γz2, where A, α, β, and γ are all real, positive constants. The probability that the particle will be found in the infinitesimal volume dx dy dz centered at the point (x0,
> 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 n
> An oscillator has the potential-energy function U(x, y, z)= 1/2 k′1(x2 + y2)+ 1/2 k′2z2, where k′1 > k′2. This oscillator is called anisotropic because the force constant is not the same in all three coordinate directions. a. Find a general expression f
> A particle in the three-dimensional cubical box of Section 41.2 is in the ground state, where nX = nY = nZ = 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 part
> An electron is in a three-dimensional box. The x- and z-sides of the box have the same length, but the y-side has a different length. The two lowest energy levels are 2.24 eV and 3.47 eV, and the degeneracy of each of these levels (including the degenera
> While working in a magnetics lab, you conduct an experiment in which a hydrogen atom in the n = 1 state is in a magnetic field of magnitude B. A photon of wavelength λ (in air) is absorbed in a transition from the ms = - 1/2 to the ms = + 1/
> You are studying the absorption of electromagnetic radiation by electrons in a crystal structure. The situation is well described by an electron in a cubical box of side length L. The electron is initially in the ground state. a. You observe that the lo
> In studying electron screening in multielectron atoms, you begin with the alkali metals. You look up experimental data and find the results given in the table. The ionization energy is the minimum energy required to remove the least-bound electron from
> A hydrogen atom initially in an n = 3, l = 1 state makes a transition to the n = 2, l = 0, j = 1/2 state. Find the difference in wavelength between the following two photons: one emitted in a transition that starts in the n = 3, l = 1, j = 3/2 state and
> When low-energy electrons pass through an ionized gas, electrons of certain energies pass through the gas as if the gas atoms weren’t there and thus have transmission coefficients (tunneling probabilities) T equal to unity. The gas ions can be modeled ap
> As an intern at a research lab, you study the transmission of electrons through a potential barrier. You know the height of the barrier, 8.0 eV, but must measure the width L of the barrier. When you measure the tunneling probability T as a function of th
> 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 dif
> In your research on new solid-state devices, you are studying a solid-state structure that can be modeled accurately as an electron in a one-dimensional infinite potential well (box) of width L. In one of your experiments, electromagnetic radiation is ab
> Consider a potential well defined as U(x)= ∞ for x 0 for x > L (Fig. P40.60). Consider a particle with mass m and kinetic energy E a. The boundary condition at the infinite wall (x = 0) is ψ(0)= 0. What must the form of t
> a. The wave nature of particles results in the quantum-mechanical situation that a particle confined in a box can assume only wavelengths that result in standing waves in the box, with nodes at the box walls. Use this to show that an electron confined in
> a. Show by direct substitution in the Schrödinger equation for the one-dimensional harmonic oscillator that the wave function ψ1(x)= A1xe-α2x2/2, where α2 = mω/ħ, is a solution wit
> For small amplitudes of oscillation the motion of a pendulum is simple harmonic. For a pendulum with a period of 0.500 s, find the ground-level energy and the energy difference between adjacent energy levels. Express your results in joules and in electro
> A harmonic oscillator consists of a 0.020-kg mass on a spring. The oscillation frequency is 1.50 Hz, and the mass has a speed of 0.480 m/s as it passes the equilibrium position. a. What is the value of the quantum number n for its energy level? b. What
> a. For the finite potential well of Fig. 40.13, what relationships among the constants A and B of Eq. (40.38) and C and D of Eq. (40.40) are obtained by applying the boundary condition that c be continuous at x = 0 and at x = L? b. What relationships am
> An electron with initial kinetic energy 5.5 eV encounters a square potential barrier of height 10.0 eV. What is the width of the barrier if the electron has a 0.50% probability of tunneling through the barrier?
> A fellow student proposes that a possible wave function for a free particle with mass m (one for which the potential energy function U(x) is zero) is where κ is a positive constant. a. Graph this proposed wave function. b. Show that the p
> The penetration distance η in a finite potential well is the distance at which the wave function has decreased to 1/e of the wave function at the classical turning point: The penetration distance can be shown to be The probability of fi
> A small amount of magnetic-field splitting of spectral lines occurs even when the atoms are not in a magnetic field. What causes this?
> What is the probability of finding a particle in a box of length L in the region between x = L/4 and x = 3L/4 when the particle is in a. the ground level and b. the first excited level? (Hint: Integrate |ψ(x)|2 dx, where ψ is norm
> A particle is confined within a box with perfectly rigid walls at x = 0 and x = L. Although the magnitude of the instantaneous force exerted on the particle by the walls is infinite and the time over which it acts is zero, the impulse (that involves a pr
> Repeat Problem 40.48 for a particle in the first excited level. From Problem 40.48: Consider a particle in a box with rigid walls at x = 0 and x = L. Let the particle be in the ground level. Calculate the probability |ψ|2dx that the particle will be fo
> Consider a particle in a box with rigid walls at x = 0 and x = L. Let the particle be in the ground level. Calculate the probability |ψ|2dx that the particle will be found in the interval x to x + dx for a. x = L/4; b. x = L/2; c. x = 3L/4.
> An electron in a long, organic molecule used in a dye laser behaves approximately like a particle in a box with width 4.18 nm. What is the wavelength of the photon emitted when the electron undergoes a transition a. from the first excited level to the g
> A particle is in the ground level of a box that extends from x = 0 to x = L. a. What is the probability of finding the particle in the region between 0 and L/4? Calculate this by integrating |ψ(x)|2 dx, where ψ is normalized, from
> Consider a beam of free particles that move with velocity v = p/m in the x-direction and are incident on a potential- energy step U(x)= 0, for x < 0, and U(x)= U0 < E, for x > 0. The wave function for x < 0 is ψ(x)= Aeik1x + Be-ik1x, representing inciden
> a. Using the integral in Problem 40.42, determine the wave function ψ(x) for a function B(k) given by This represents an equal combination of all wave numbers between 0 and k0. Thus ψ(x) represents a particle with average wave num
> A particle of mass m in a one-dimensional box has the following wave function in the region x = 0 to x = L: Here ψ1(x) and ψ3(x) are the normalized stationary-state wave functions for the n = 1 and n = 3 levels, and E1 and E3 are
> Consider the wave packet defined by Let B(k)= e-a2k2. a. The function B(k) has its maximum value at k = 0. Let kh be the value of k at which B(k) has fallen to half its maximum value, and define the width of B(k) as wk = kh. In terms of α
> 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.
> 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 gr
> An electron is in a three-dimensional box with side lengths LX = 0.600 nm and LY = LZ = 2LX. What are the quantum numbers nX, nY, and nZ and the energies, in eV, for the four lowest energy levels? What is the degeneracy of each (including the degeneracy
> 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?
> When a NaF molecule makes a transition from the l = 3 to the l = 2 rotational level with no change in vibrational quantum number or electronic state, a photon with wavelength 3.83 mm is emitted. A sodium atom has mass 3.82 * 10-26 kg, and a fluorine atom
> a. For the sodium chloride molecule (NaCl) discussed at the beginning of Section 42.1, what is the maximum separation of the ions for stability if they may be regarded as point charges? That is, what is the largest separation for which the energy of an N
> Which statement best explains the temperature dependence of the current–voltage characteristics that the graph shows? At higher temperatures: a. The band gap is larger, so the electron– hole pairs have more energy, which causes the current at a given vo
> The sensitivity of a diode thermometer depends on how much the voltage changes for a given temperature change, with the current remaining constant. What is the sensitivity for this diode thermometer, operated at 100 mA, for a temperature change from 25°C
> The current–voltage characteristics of a forward­biased p-n junction diode depend strongly on temperature, as shown in the figure. As a result, diodes can be used as temperature sensors. In actual operation, the voltage is ad
> What type of radioactive decay produces 131I from 131Te? a. Alpha decay; b. β- decay; c. β+ decay; d. gamma decay.
> Which reaction produces 131Te in the nuclear reactor? a. 130Te + n → 131Te; b. 130I + n → 131Te; c. 132Te + n → 131Te; d. 132I + n → 131Te.
> Use Table 41.3 to help determine the ground-state electron configuration of the neutral gallium atom (Ga) as well as the ions Ga+ and Ga-. Gallium has an atomic number of 31. From table 41.3: TABLE 41.3 Ground-State Electron Configurations Atomic E
> Why might 123I be preferred for imaging over 131I? a. The atomic mass of 123I is smaller, so the 123I particles travel farther through tissue. b. Because 123I emits only gamma-ray photons, the radiation dose to the body is lower with that isotope. c. T
> In the reaction that produces 123I, is there a minimum kinetic energy the protons need to make the reaction go? a. No, because the proton has a smaller mass than the neutron. b. No, because the total initial mass is smaller than the total final mass.
> How many 131I atoms are administered in a typical thyroid cancer treatment? a. 4.2 * 1010; b. 1.0 * 1012; c. 2.5 * 1014; d. 3.7 * 1015.
> In the Bohr model, what is the principal quantum number n at which the excited electron is at a radius of 1 µm? a. 140; b. 400; c. 20; d. 81.
> How many different possible electron states are there in the n = 100, l = 2 subshell? a. 2; b. 100; c. 10,000; d. 10.
> Assume that the researchers place an atom in a state with n = 100, l = 2. What is the magnitude of the orbital angular momentum L associated with this state? a. 2 ħ; b. 6 ħ; c. 200 ħ; d. 10100 ħ.
> Take the size of a Rydberg atom to be the diameter of the orbit of the excited electron. If the researchers want to perform this experiment with the rubidium atoms in a gas, with atoms separated by a distance 10 times their size, the density of atoms per
> One advantage of the quantum dot is that, compared to many other fluorescent materials, excited states have relatively long lifetimes (10 ns). What does this mean for the spread in the energy of the photons emitted by quantum dots? a. Quantum dots emit
> Dots that are the same size but made from different materials are compared. In the same transition, a dot of material 1 emits a photon of longer wavelength than the dot of material 2 does. Based on this model, what is a possible explanation? a. The mass
