Question: The hyperfine interaction in a hydrogen atom
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 * 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.
From Fig. 41.18:
From Example 41.7:
Transcribed Image Text:
41.18 An l = 0 level of a single electron is split by interaction of the spin magnetic moment with an external magnetic field. The greater the magnitude B of the magnetic field, the greater the splitting. The quantity 5.795 × 10 eV/T is just (1.00116)µB. E, + (5.795 × 105eV/T)B E Spin up m, = +} Es I Spin down im, = - B = 0- E, - (5.795 × 10-SeV/T)B B increasing EXAMPLE 41.7 AN EFFECTIVE MAGNETIC FIELD To six significant figures, the wavelengths of the two spectral lines between the two 3p levels is equal to the difference in energy that make up the sodium doublet are A = 588.995 nm and A2 = between the two photons of the sodium doublet. We use this rela- 589.592 nm. Calculate the effective magnetic field experienced by tionship and the results of Example 41.6 to determine B. the electron in the 3p levels of the sodium atom. EXECUTE: The energies of the two photons are Ej = hc/A¡ and E2 = hc/A2. Here E1 > E2 because A < d2, so the difference in their energies is SOLUTION IDENTIFY and SET UP: The two lines in the sodium doublet result ar ---() hc hc from transitions from the two 3p levels, which are split by spin-orbit coupling, to the 3s level, which is not split because it has L = 0. We picture the spin-orbit coupling as an interaction between the electron spin magnetic moment and an effective mag- netic field due to the Nucleus. This example is like Example 41.6 in reverse: There we were given B and found the difference between the energies of the two spin states, while here we use the energy difference to find the target variable B. The difference in energy ΔΕ hc --- = 12 (4.136 x 10 15 eV · s)(2.998 × 10% m/s) (589.592 × 109 m) – (588.995 × 10-9 m) (589.592 x 10 m)(588.995 × 10ºm) %3D 0.00213 eV = 3.41 × 10-22 J UTION This equals the energy difference between the two 3p levels. The EVALUATE: The electron experiences a very strong effective mag- spin-orbit interaction raises one level by 1.70 x 10 22 J (one- netic field. To produce a steady, macroscopic field of this magni- half of 3.41 x 10-22 J) and lowers the other by 1.70 x 10-22 J. tude in the laboratory requires state-of-the-art electromagnets. From Example 41.6, the amount each state is raised or lowered is JU| = (1.00116)µyB, so 1.70 x 10-22 J 9.28 x 10-24 J/T U B = 18.0 T (1.00116)B
> The central-field approximation is more accurate for alkali metals than for transition metals such as iron, nickel, or copper. Why?
> For the H2 molecule the equilibrium spacing of the two protons is 0.074 nm. The mass of a hydrogen atom is 1.67 * 10-27 kg. Calculate the wavelength of the photon emitted in the rotational transition l = 2 to l = 1.
> At a temperature of 290 K, a certain p-n junction has a saturation current IS = 0.500 mA. a. Find the current at this temperature when the voltage is i. 1.00 mV, ii. -1.00 mV, iii. 100 mV, and iv. -100 mV. b. Is there a region of applied voltage w
> a. Suppose a piece of very pure germanium is to be used as a light detector by observing, through the absorption of photons, the increase in conductivity resulting from generation of electron–hole pairs. If each pair requires 0.67 eV of energy, what is t
> Germanium has a band gap of 0.67 eV. Doping with arsenic adds donor levels in the gap 0.01 eV below the bottom of the conduction band. At a temperature of 300 K, the probability is 4.4 * 10-4 that an electron state is occupied at the bottom of the conduc
> Pure germanium has a band gap of 0.67 eV. The Fermi energy is in the middle of the gap. a. For temperatures of 250 K, 300 K, and 350 K, calculate the probability f(E) that a state at the bottom of the conduction band is occupied. b. For each temperatur
> For a solid metal having a Fermi energy of 8.500 eV, what is the probability, at room temperature, that a state having an energy of 8.520 eV is occupied by an electron?
> At the Fermi temperature TF, EF = kTF (see Exercise 42.22). When T = TF, what is the probability that a state with energy E = 2EF is occupied?
> Silver has a Fermi energy of 5.48 eV. Calculate the electron contribution to the molar heat capacity at constant volume of silver, CV, at 300 K. Express your result a. as a multiple of R and b. as a fraction of the actual value for silver, CV = 25.3 J/
> The Fermi energy of sodium is 3.23 eV. a. Find the average energy Eav of the electrons at absolute zero. b. What is the speed of an electron that has energy Eav? c. At what Kelvin temperature T is kT equal to EF? (This is called the Fermi temperature
> Calculate the density of states g(E) for the freeelectron model of a metal if E = 7.0 eV and V = 1.0 cm3. Express your answer in units of states per electron volt.
> Particle A is described by the wave function Ψ(x, y, z). Particle B is described by the wave function Ψ(x, y, z)eiɸ, where ɸ is a real constant. How does the probability of finding particle A within a volume dV around a certain point in space compare wit
> Calculate vrms for free electrons with average kinetic energy 3/2 kT at a temperature of 300 K. How does your result compare to the speed of an electron with a kinetic energy equal to the Fermi energy of copper, calculated in Example 42.7? Why is there s
> a. Calculate the electric potential energy for a K+ ion and a Br- ion separated by a distance of 0.29 nm, the equilibrium separation in the KBr molecule. Treat the ions as point charges. b. The ionization energy of the potassium atom is 4.3 eV. Atomic b
> The gap between valence and conduction bands in silicon is 1.12 eV. A nickel nucleus in an excited state emits a gamma ray photon with wavelength 9.31 * 10-4 nm. How many electrons can be excited from the top of the valence band to the bottom of the cond
> The gap between valence and conduction bands in diamond is 5.47 eV. a. What is the maximum wavelength of a photon that can excite an electron from the top of the valence band into the conduction band? In what region of the electromagnetic spectrum does
> The maximum wavelength of light that a certain silicon photocell can detect is 1.11 µm. a. What is the energy gap (in electron volts) between the valence and conduction bands for this photocell? b. Explain why pure silicon is opaque.
> Potassium bromide (KBr) has a density of 2.75 * 103 kg/m3 and the same crystal structure as NaCl. The mass of a potassium atom is 6.49 * 10-26 kg, and the mass of a bromine atom is 1.33 * 10-25 kg. a. Calculate the average spacing between adjacent atoms
> The spacing of adjacent atoms in a crystal of sodium chloride is 0.282 nm. The mass of a sodium atom is 3.82 * 10-26 kg, and the mass of a chlorine atom is 5.89 * 10-26 kg. Calculate the density of sodium chloride.
> The vibrational and rotational energies of the CO molecule are given by Eq. (42.9). Calculate the wavelength of the photon absorbed by CO in each of these vibration­rotation transitions: a. n = 0, l = 2 → n = 1, l = 3; b. n
> When a hypothetical diatomic molecule having atoms 0.8860 nm apart undergoes a rotational transition from the l = 2 state to the next lower state, it gives up a photon having energy 8.841 * 10-4 eV. When the molecule undergoes a vibrational transition fr
> If a sodium chloride (NaCl) molecule could undergo an n n - 1 vibrational transition with no change in rotational quantum number, a photon with wavelength 20.0 µm would be emitted. The mass of a sodium atom is 3.82 * 10-26 kg, and the mass of a chlo
> a. A particle in a box has ave function Ψ (x, t) = Ψ2(x)e-iE2t/ħ, where Ψn and En are given by Eqs. (40.35) and (40.31), respectively. If the energy of the particle is measured, what is the result? b. If instead the particle has wave function Ψ(x, t)=(1
> A lithium atom has mass 1.17 * 10-26 kg, and a hydrogen atom has mass 1.67 * 10-27 kg. The equilibrium separation between the two nuclei in the LiH molecule is 0.159 nm. a. What is the difference in energy between the l = 3 and l = 4 rotational levels?
> The average kinetic energy of an idealgas atom or molecule is 3/2 kT, where T is the Kelvin temperature (Chapter 18). The rotational inertia of the H2 molecule is 4.6 * 10-48 kg . m2. What is the value of T for which 3/2 kT equals the energy separation
> If the energy of the H2 covalent bond is -4.48 eV, what wavelength of light is needed to break that molecule apart? In what part of the electromagnetic spectrum does this light lie?
> 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?
> An electron is in the hydrogen atom with n = 5. a. Find the possible values of L and Lz for this electron, in units of ħ. b. For each value of L, find all the possible angles between L and the z-axis. c. What are the maximum and minimum values of the
> 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 ħ and in SI units. c. What is the largest orbital angul
> 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?
> 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 correspond
> 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 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 Kα and Kβ x rays of tungsten.
> Sketch the wave function for the potential-energy well shown in Fig. Q40.26 when E1 is less than U0 and when E3 is greater than U0. Figure Q40.26 U(x) Uo ol A В 8.
> Calculate the frequency, energy (in keV), and wavelength of the Kα x ray for the elements a. calcium (Ca, Z = 20); b. cobalt (Co, Z = 27); c. cadmium (Cd, Z = 48).
> A Kα x ray emitted from a sample has an energy of 7.46 keV. Of which element is the sample made?
> Estimate the energy of the highest-l state for a. the L shell of Be+ and b. the N shell of Ca+.
> a. The energy of the 2s state of lithium is -5.391 eV. Calculate the value of Zeff for this state. b. The energy of the 4s state of potassium is -4.339 eV. Calculate the value of Zeff for this state. c. Compare Zeff for the 2s state of lithium, the 3s
> 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 charg
> The energies of the 4s, 4p, and 4d states of potassium are given in Example 41.10. Calculate Zeff for each state. What trend do your results show? How can you explain this trend? From Example 41.10 EXAMPLE 41.10 ENERGIES FOR A VALENCE ELECTRON The
> The 5s electron in rubidium (Rb) sees an effective charge of 2.771e. Calculate the ionization energy of this electron.
> 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 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?
> 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 th
> Compare the allowed energy levels for the hydrogen atom, the particle in a box, and the harmonic oscillator. What are the values of the quantum number n for the ground level and the second excited level of each system?
> 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 n, l, ml, and ms 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 hydrogen atom in a particular orbital angular momentum state is found to have j quantum numbers 7/2 and 9/2. a. What is the letter that labels the value of l for the state? b. If n = 5, what is the energy difference between the j = 7/2 and j = 9/2 le
> Calculate the energy difference between the ms = 1/2 (“spin up”) and ms = - 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, ms = 1/2 or ms = - 1/2 , has th
> a. If you treat an electron as a classical spherical object with a radius of 1.0 * 10-17 m, what angular speed is necessary to produce a spin angular momentum of magnitude 3/4 ħ? b. Use v = rω and the result of part (a) to calculate the speed v of a
> A hydrogen atom in the n = 1, ms = - 1/2 state is placed in a magnetic field with a magnitude of 1.60 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 d
> 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 ene
> A hydrogen atom undergoes a transition from a 2p state to the 1s 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 z-direction. Ignore spin effects; con
> 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
> In Fig. 40.28, how does the probability of finding a particle in the center half of the region -A From Fig. 40.28: 40.28 Newtonian and quantum- mechanical probability distribution func- tions for a harmonic oscillator for the state n = 10. The Newt
> A hydrogen atom in a 3p state is placed in a uniform external magnetic field B. Consider the interaction of the magnetic field with the atom’s orbital magnetic dipole moment. a. What field magnitude B is required to split the 3p state into multiple leve
> A hydrogen atom is in a d state. In the absence of an external magnetic field, the states with different ml values have (approximately) the same energy. Consider the interaction of the magnetic field with the atom’s orbital magnetic dipole moment. a. Ca
> Show that Φ(ϕ)= eimlϕ = Φ(ϕ + 2π) (that is, show that Φ(ϕ) is periodic with period 2π) if and only if ml is restricted to the values 0, ±1, ±2,……. (Hint: Euler’s formula states that eif = cos ϕ + i sin ϕ.)
> a. What is the probability that an electron in the 1s state of a hydrogen atom will be found at a distance less than a/2 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
> a. How many different 5g states does hydrogen have? b. Which of the states in part (a) has the largest angle between L and the z-axis, and what is that angle? c. Which of the states in part (a) has the smallest angle between L and the z-axis, and what
> a. Make a chart showing all possible sets of quantum numbers l and ml for the states of the electron in the hydrogen atom when n = 4. How many combinations are there? b. What are the energies of these states?
> Calculate, in units of ħ, 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 nħ postulated in the Bohr model. What trend do
> A hydrogen atom is in a state that has Lz = 2ħ. In the semiclassical vector model, the angular momentum vector L for this state makes an angle θL = 63.4° with the +z-axis. a. What is the l quantum number for this state? b. What is the smallest possible
> In a particular state of the hydrogen atom, the angle between the angular momentum vector L and the z-axis is θ = 26.6°. If this is the smallest angle for this particular value of the orbital quantum number l, what is l?
> Consider states with angular momentum quantum number l = 2. a. In units of ħ, what is the largest possible value of Lz? b. In units of ħ, what is the value of L? Which is larger: L or the maximum possible Lz? c. For each allowed
> The probability distributions for the harmonic-oscillator wave functions (see Figs. 40.27 and 40.28) begin to resemble the classical (Newtonian) probability distribution when the quantum number n becomes large. Would the distributions become the same as
> For a particle in a three-dimensional cubical box, what is the degeneracy (number of different quantum states with the same energy) of the energy levels a. 3π2ħ2/2mL2 and b. 9π2ħ2/2mL2?
> For the sodium atom of Example 40.8, find a. the ground-state energy; b. the wavelength of a photon emitted when the n = 4 to n = 3 transition occurs; c. the energy difference for any ∆n = 1 transition.
> For the ground­level harmonic oscillator wave function ψ(x) given in Eq. (40.47), |ψ|2 has a maximum at x = 0. a. Compute the ratio of |ψ|2 at x = +A to |ψ|2 at x = 0, where A is given by Eq. (40.48) with n = 0 for the ground level
> In Section 40.5 it is shown that for the ground level of a harmonic oscillator, ∆x ∆px = ħ/2. Do a similar analysis for an excited level that has quantum number n. How does the uncertainty product ∆x ∆px depend on n?
> While undergoing a transition from the n = 1 to the n = 2 energy level, a harmonic oscillator absorbs a photon of wavelength 6.50 µm. What is the wavelength of the absorbed photon when this oscillator undergoes a transition a. from the n = 2 to the n =
> The groundstate energy of a harmonic oscillator is 5.60 eV. If the oscillator undergoes a transition from its n = 3 to n = 2 level by emitting a photon, what is the wavelength of the photon?
> A harmonic oscillator absorbs a photon of wavelength 6.35 µm when it undergoes a transition from the ground state to the first excited state. What is the groundstate energy, in electron volts, of the oscillator?
> Chemists use infrared absorption spectra to identify chemicals in a sample. In one sample, a chemist finds that light of wavelength 5.8 µm is absorbed when a molecule makes a transition from its ground harmonic oscillator level to its first excited level
> Show that ψ(x) given by Eq. (40.47) is a solution to Eq. (40.44) with energy E0 = ħω/2. From Eq. (40.47) From Eq. (40.44)
> A wooden block with mass 0.250 kg is oscillating on the end of a spring that has force constant 110 N/m. Calculate the groundlevel energy and the energy separation between adjacent levels. Express your results in joules and in electron volts. Are quantu
> The wave function shown in Fig. 40.20 is nonzero for both x L. Does this mean that the particle splits into two parts when it strikes the barrier, with one part tunneling through the barrier and the other part bouncing off the barrier? Explain. From Fi
> A proton with initial kinetic energy 50.0 eV encounters a barrier of height 70.0 eV. What is the width of the barrier if the probability of tunneling is 8.0 * 10-3? How does this compare with the barrier width for an electron with the same energy tunneli
> An electron is moving past the square barrier shown in Fig. 40.19, but the energy of the electron is greater than the barrier height. If E = 2U0, what is the ratio of the de Broglie wavelength of the electron in the region x > L to the wavelength for
> An electron with initial kinetic energy 5.0 eV encounters a barrier with height U0 and width 0.60 nm. What is the transmission coefficient if a. U0 = 7.0 eV; b. U0 = 9.0 eV; c. U0 = 13.0 eV?
> An electron with initial kinetic energy 6.0 eV encounters a barrier with height 11.0 eV. What is the probability of tunneling if the width of the barrier is a. 0.80 nm and b. 0.40 nm?
> In a simple model for a radioactive nucleus, an alpha particle (m = 6.64 * 10-27 kg) is trapped by a square barrier that has width 2.0 fm and height 30.0 MeV. a. What is the tunneling probability when the alpha particle encounters the barrier if its kin
> a. An electron with initial kinetic energy 32 eV encounters a square barrier with height 41 eV and width 0.25 nm. What is the probability that the electron will tunnel through the barrier? b. A proton with the same kinetic energy encounters the same bar
> An electron is bound in a square well that has a depth equal to six times the groundlevel energy E1-IDW of an infinite well of the same width. The longestwavelength photon that is absorbed by this electron has a wavelength of 582 nm. Determine the widt
> A proton is bound in a square well of width 4.0 fm = 4.0 * 10-15 m. The depth of the well is six times the groundlevel energy E1-IDW of the corresponding infinite well. If the proton makes a transition from the level with energy E1 to the level with ene
> An electron is in the ground state of a square well of width L = 4.00 * 10-10 m. The depth of the well is six times the groundstate energy of an electron in an infinite well of the same width. What is the kinetic energy of this electron after it has abs
> An electron is bound in a square well of width 1.50 nm and depth U0 = 6E1-IDW. If the electron is initially in the ground level and absorbs a photon, what maximum wavelength can the photon have and still liberate the electron from the well?
> Qualitatively, how would you expect the probability for a particle to tunnel through a potential barrier to depend on the height of the barrier? Explain.
> An electron is moving past the square well shown in Fig. 40.13. The electron has energy E = 3U0. What is the ratio of the de Broglie wavelength of the electron in the region x > L to the wavelength for 0 < x < L?
> An electron is bound in a square well of depth U0 = 6E1-IDW. What is the width of the well if its groundstate energy is 2.00 eV?
> When an electron in a onedimensional box makes a transition from the n = 1 energy level to the n = 2 level, it absorbs a photon of wavelength 426 nm. What is the wavelength of that photon when the electron undergoes a transition a. from the n = 2 to th
> An electron is in a box of width 3.0 * 10-10 m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in a. the n = 1 level; b. the n = 2 level; c. the n = 3 level? In each case how does the wavelength compare
> a. Find the excitation energy from the ground level to the third excited level for an electron confined to a box of width 0.360 nm. b. The electron makes a transition from the n = 1 to n = 4 level by absorbing a photon. Calculate the wavelength of this
> Repeat Exercise 40.16 for the particle in the first excited level.
> Recall that |ψ|2dx is the probability of finding the particle that has normalized wave function ψ(x) in the interval x to x + dx. Consider a particle in a box with rigid walls at x = 0 and x = L. Let the particle be in the ground level and use ψn as give
> Consider a particle moving in one dimension, which we shall call the xaxis. a. What does it mean for the wave function of this particle to be normalized? b. Is the wave function ψ(x)= eax , where a is a positive real number, normalized? Could this be
> An electron in a onedimensional box has groundstate energy 2.00 eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?
> A certain atom requires 3.0 eV of energy to excite an electron from the ground level to the first excited level. Model the atom as an electron in a box and find the width L of the box.
> Figure 40.17 shows the scanning tunneling microscope image of 48 iron atoms placed on a copper surface, the pattern indicating the density of electrons on the copper surface. What can you infer about the potential-energy function inside the circle of iro
> A student remarks that the relationship of ray optics to the more general wave picture is analogous to the relationship of Newtonian mechanics, with well-defined particle trajectories, to quantum mechanics. Comment on this remark.
