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 due to spin)?
> 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
> 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
> 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
> 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
> When a given dot with side length L makes a transition from its first excited state to its ground state, the dot emits green (550 nm) light. If a dot with side length 1.1L is used instead, what wavelength is emitted in the same transition, according to t
> Why do the transition elements (Z = 21 to 30) all have similar chemical properties?
> According to this model, which statement is true about the energy-level spacing of dots of different sizes? a. Smaller dots have equally spaced levels, but larger dots have energy levels that get farther apart as the energy increases. b. Larger dots ha
> Suppose that positron–electron annihilations occur on the line 3 cm from the center of the line connecting two detectors. Will the resultant photons be counted as having arrived at these detectors simultaneously? a. No, because the time difference betwe
> What is the energy of each photon produced by positron– electron annihilation? a. 1/2 mev2, where v is the speed of the emitted positron; b. mev2; c. 1/2 mec2; d. mec2.
> If the annihilation photons come from a part of the body that is separated from the detector by 20 cm of tissue, what percentage of the photons that originally travelled toward the detector remains after they have passed through the tissue? a. 1.4%; b.
> If the voltage rather than the current is kept constant, what happens as the temperature increases from 25°C to 150°C? a. At first the current increases, then it decreases. b. The current increases. c. The current decreases, eventually approaching zer
> A particle moving in one dimension (the x­axis) is described by the wave function where b = 2.00 m-1, A > 0, and the +x-axis points toward the right. a. Determine A so that the wave function is normalized. b. Sketch the graph of the wav
> Let ψ1 and ψ2 be two solutions of Eq. (40.23) with energies E1 and E2, respectively, where E1 ≠ E2 . Is ψ = Aψ1 + Bψ2, where A and B are nonzero constants, a solution to Eq. (40.23)? Explain your answer.
> Compute Ψ 2 for Ψ = Ψ sin ωt, where Ψ is time independent and ω is a real constant. Is this a wave function for a stationary state? Why or why not?
> Consider a wave function given by Ψ(x) = A sin kx, where k = 2π/λ and A is a real constant. a. For what values of x is there the highest probability of finding the particle described by this wave function? Explain. b. For which values of x is the proba
> A particle is described by a wave function Ψ(x) = Ae-ax2, where A and a are real, positive constants. If the value of a is increased, what effect does this have on a. the particle’s uncertainty in position and b. the particle’s uncertainty in momentum?
> Do gravitational forces play a significant role in atomic structure? Explain.
> Consider the free-particle wave function of Example 40.1. Let k2 = 3k1 = 3k. At t = 0 the probability distribution function Ψ (x, t) 2 has a maximum at x = 0. a. What is the smallest positive value of x for which the probability distribution function
> A free particle moving in one dimension has wave function Ψ(x, t) = A[ei(kx-ωt) – e i(2kx-4ωt) where k and v are positive real constants. a. At t = 0 what are the two smallest positive values of x for which the probability function Ψ(x, t) 2 is a maxi
> An electron is moving as a free particle in the –x-direction with momentum that has magnitude 4.50*10-24 kg.m/s. What is the onedimensional timedependent wave function of the electron?
> What particle (a particle, electron, or positron) is emitted in the following radioactive decays? a. 14 27
> The atomic mass of 14C is 14.003242 u. Show that the β- decay of 14C is energetically possible, and calculate the energy released in the decay.
> 238U decays spontaneously by α emission to 234Th. Calculate a. the total energy released by this process and b. the recoil velocity of the 234Th nucleus. The atomic masses are 238.050788 u for 238U and 234.043601 u for 234Th.
> What nuclide is produced in the following radioactive decays? a. α decay of 94 239
> a. Is the decay n p + β- + ve energetically possible? If not, explain why not. If so, calculate the total energy released. b. Is the decay p n + β+ + ve energetically possible? If not, explain why not. If so, calculate the total energy rel
> Use Eq. (43.11) to calculate the binding energy per nucleon for the nuclei 36 86
> Calculate the mass defect, the binding energy (in MeV), and the binding energy per nucleon of a. the nitrogen nucleus, 7 14
> 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. From Table 41.3: TABLE 41.3 Ground-State
> As Eq. (40.21) indicates, the time-dependent wave function for a stationary state is a complex number having a real part and an imaginary part. How can this function have any physical meaning, since part of it is imaginary? From Eq. (40.21): Time-d
> How many protons and how many neutrons are there in a nucleus of the most common isotope of a. silicon, 14 28
> The rotational energy levels of CO are calculated in Example 42.2. If the energy of the rotating molecule is described by the classical expression K = 1/2 Iω2, for the l = 1 level what are a. the angular speed of the rotating molecule; b. th
> Two atoms of cesium (Cs) can form a Cs2 molecule. The equilibrium distance between the nuclei in a Cs2 molecule is 0.447 nm. Calculate the moment of inertia about an axis through the center of mass of the two nuclei and perpendicular to the line joining
> The water molecule has an l = 1 rotational level 1.01 * 10-5 eV above the l = 0 ground level. Calculate the wavelength and frequency of the photon absorbed by water when it undergoes a rotationallevel transition from l = 0 to l = 1. The magnetron oscill
> The H2 molecule has a moment of inertia of 4.6 * 10-48 kg . m2. What is the wavelength l of the photon absorbed when H2 makes a transition from the l = 3 to the l = 4 rotational level?
> A hypothetical NH molecule makes a rotational-level transition from l = 3 to l = 1 and gives off a photon of wavelength 1.780 nm in doing so. What is the separation between the two atoms in this molecule if we model them as point masses? The mass of hydr
> During each of these processes, a photon of light is given up. In each process, what wavelength of light is given up, and in what part of the electromagnetic spectrum is that wavelength? a. A molecule decreases its vibrational energy by 0.198 eV; b. a
> A p-n junction has a saturation current of 6.40 mA. a. At a temperature of 300 K, what voltage is needed to produce a positive current of 40.0 mA? b. For a voltage equal to the negative of the value calculated in part (a), what is the negative current
