Solved: An electron in the n 5 5 energy level of hydrogen

(i) What is the principal quantum number of the initial state of an atom as it emits an Mb line in an x-ray spectrum? (a) 1 (b) 2 (c) 3 (d) 4 (e) 5 (ii) What is the principal quantum number of the final state for this transition? Choose from the same possibilities as in part (i).

If an electron in an atom has the quantum numbers n 5 3, , 5 2, m, 5 1, and ms 5 1 2, what state is it in? (a) 3s (b) 3p (c) 3d (d) 4d (e) 3f

An electron in the n 5 5 energy level of hydrogen undergoes a transition to the n 5 3 energy level. What is the wavelength of the photon the atom emits in this process? (a) 2.28 3 1026 m (b) 8.20 3 1027 m (c) 3.64 3 1027 m (d) 1.28 3 1026 m (e) 5.92 3 1025 m

Consider the n 5 3 energy level in a hydrogen atom. How many electrons can be placed in this level? (a) 1 (b) 2 (c) 8 (d) 9 (e) 18

Which of the following is not one of the basic assumptions of the Bohr model of hydrogen? (a) Only certain electron orbits are stable and allowed. (b) The electron moves in circular orbits about the proton under the influence of the Coulomb force. (c) The charge on the electron is quantized. (d) Radiation is emitted by the atom when the electron moves from a higher energy state to a lower energy state. (e) The angular momentum associated with the electrons orbital motion is quantized.

Let 2E represent the energy of a hydrogen atom. (i) What is the kinetic energy of the electron? (a) 2E (b) E (c) 0 (d) 2E (e) 22E (ii) What is the potential energy of the atom? Choose from the same possibilities (a) through (e)

The periodic table is based on which of the following principles? (a) The uncertainty principle. (b) All electrons in an atom must have the same set of quantum numbers. (c) Energy is conserved in all interactions. (d) All electrons in an atom are in orbitals having the same energy. (e) No two electrons in an atom can have the same set of quantum numbers.

(a) Can a hydrogen atom in the ground state absorb a photon of energy less than 13.6 eV? (b) Can this atom absorb a photon of energy greater than 13.6 eV?

Which of the following electronic configurations are not allowed for an atom? Choose all correct answers. (a) 2s 22p6 (b) 3s 23p7 (c) 3d 74s 2 (d) 3d 104s 24p 6 (e) 1s 22s 22d 1

What can be concluded about a hydrogen atom with its electron in the d state? (a) The atom is ionized. (b) The orbital quantum number is , 5 1. (c) The principal quantum number is n 5 2. (d) The atom is in its ground state. (e) The orbital angular momentum of the atom is not zero.

(i) Rank the following transitions for a hydrogen atom from the transition with the greatest gain in energy to that with the greatest loss, showing any cases of equality. (a) ni 5 2; nf 5 5 (b) ni 5 5; nf 5 3 (c) ni 5 7; nf 5 4 (d) ni 5 4; nf 5 7 (ii) Rank the same transitions as in part (i) according to the wavelength of the photon absorbed or emitted by an otherwise isolated atom from greatest wavelength to smallest.

When an atom emits a photon, what happens? (a) One of its electrons leaves the atom. (b) The atom moves to a state of higher energy. (c) The atom moves to a state of lower energy. (d) One of its electrons collides with another particle. (e) None of those events occur.

(a) In the hydrogen atom, can the quantum number n increase without limit? (b) Can the frequency of possible discrete lines in the spectrum of hydrogen increase without limit? (c) Can the wavelength of possible discrete lines in the spectrum of hydrogen increase without limit?

Consider the quantum numbers (a) n, (b) ,, (c) m,, and (d) ms. (i) Which of these quantum numbers are fractional as opposed to being integers? (ii) Which can sometimes attain negative values? (iii) Which can be zero?

When an electron collides with an atom, it can transfer all or some of its energy to the atom. A hydrogen atom is in its ground state. Incident on the atom are several electrons, each having a kinetic energy of 10.5 eV. What is the result? (a) The atom can be excited to a higher allowed state. (b) The atom is ionized. (c) The electrons pass by the atom without interaction.

A monochromatic beam of light is absorbed by a collection of ground-state hydrogen atoms in such a way that six different wavelengths are observed when the hydrogen relaxes back to the ground state. (a) What is the wavelength of the incident beam? Explain the steps in your solution. (b) What is the longest wavelength in the emission spectrum of these atoms? (c) To what portion of the electromagnetic spectrum and (d) to what series does it belong? (e) What is the shortest wavelength? (f) To what portion of the electromagnetic spectrum and (g) to what series does it belong?

A hydrogen atom is in its second excited state, corresponding to n 5 3. Find (a) the radius of the electrons Bohr orbit and (b) the de Broglie wavelength of the electron in this orbit.

A hydrogen atom is in its first excited state (n 5 2). Calculate (a) the radius of the orbit, (b) the linear momentum of the electron, (c) the angular momentum of the electron, (d) the kinetic energy of the electron, (e) the potential energy of the system, and (f) the total energy of the system.

A photon with energy 2.28 eV is absorbed by a hydrogen atom. Find (a) the minimum n for a hydrogen atom that can be ionized by such a photon and (b) the speed of the electron released from the state in part (a) when it is far from the nucleus.

An electron is in the nth Bohr orbit of the hydrogen atom. (a) Show that the period of the electron is T 5 n3t 0 and determine the numerical value of t 0. (b) On average, an electron remains in the n 5 2 orbit for approximately 10 ms before it jumps down to the n 5 1 (ground-state) orbit. How many revolutions does the electron make in the excited state? (c) Define the period of one revolution as an electron year, analogous to an Earth year being the period of the Earths motion around the Sun. Explain whether we should think of the electron in the n 5 2 orbit as living for a long time.

(a) Construct an energy-level diagram for the He1 ion, for which Z 5 2, using the Bohr model. (b) What is the ionization energy for He1?

A general expression for the energy levels of oneelectron atoms and ions is En 5 2 mke 2 q1 2 q2 2 2U2 n2 Here m is the reduced mass of the atom, given by m 5 m1m2/(m1 1 m2), where m1 is the mass of the electron and m2 is the mass of the nucleus; ke is the Coulomb constant; and q1 and q2 are the charges of the electron and the nucleus, respectively. The wavelength for the n 5 3 to n 5 2 transition of the hydrogen atom is 656.3 nm (visible red light). What are the wavelengths for this same transition in (a) positronium, which consists of an electron and a positron, and (b) singly ionized helium? Note: A positron is a positively charged electron

Atoms of the same element but with different numbers of neutrons in the nucleus are called isotopes. Ordinary hydrogen gas is a mixture of two isotopes containing either one- or two-particle nuclei. These isotopes are hydrogen-1, with a proton nucleus, and hydrogen-2, called deuterium, with a deuteron nucleus. A deuteron is one proton and one neutron bound together. Hydrogen-1 and deuterium have identical chemical properties, but they can be separated via an ultracentrifuge or by other methods. Their emission spectra show lines of the same colors at very slightly different wavelengths. (a) Use the equation given in Problem 22 to show that the difference in wavelength between the hydrogen-1 and deuterium spectral lines associated with a particular electron transition is given by lH 2 lD 5 a1 2 mH mD blH (b) Find the wavelength difference for the Balmer alpha line of hydrogen, with wavelength 656.3 nm, emitted by an atom making a transition from an n 5 3 state to an n 5 2 state. Harold Urey observed this wavelength difference in 1931 and so confirmed his discovery of deuterium.

An electron of momentum p is at a distance r from a stationary proton. The electron has kinetic energy K 5 p2/2me. The atom has potential energy U 5 2kee 2 /r and total energy E 5 K 1 U. If the electron is bound to the proton to form a hydrogen atom, its average position is at the proton but the uncertainty in its position is approximately equal to the radius r of its orbit. The electrons average vector momentum is zero, but its average squared momentum is approximately equal to the squared uncertainty in its momentum as given by the uncertainty principle. Treating the atom as a onedimensional system, (a) estimate the uncertainty in the electrons momentum in terms of r. Estimate the electrons (b) kinetic energy and (c) total energy in terms of r. The actual value of r is the one that minimizes the total energy, resulting in a stable atom. Find (d) that value of r and (e) the resulting total energy. (f) State how your answers compare with the predictions of the Bohr theory.

Plot the wave function c1s (r) versus r (see Eq. 42.22) and the radial probability density function P1s (r) versus r (see Eq. 42.25) for hydrogen. Let r range from 0 to 1.5a0, where a0 is the Bohr radius.

For a spherically symmetric state of a hydrogen atom, the Schrdinger equation in spherical coordinates is 2 U2 2me a d 2 c dr 2 1 2 r dc dr b 2 ke e 2 r c 5 E c (a) Show that the 1s wave function for an electron in hydrogen, c1s1r2 5 1 "pa0 3 e 2r/a 0 satisfies the Schrdinger equation. (b) What is the energy of the atom for this state?

The radial function R(r) of the wave function for a hydrogen atom in the 2p state is c2p 5 1 !3 12a02 3/2 r a0 e 2r/2a0 What is the most likely distance from the nucleus to find an electron in the 2p state?

The ground-state wave function for the electron in a hydrogen atom is c1s1r2 5 1 "pa0 3 e 2r/a 0 where r is the radial coordinate of the electron and a0 is the Bohr radius. (a) Show that the wave function as given is normalized. (b) Find the probability of locating the electron between r1 5 a0/2 and r2 5 3a0/2.

In an experiment, a large number of electrons are fired at a sample of neutral hydrogen atoms and observations are made of how the incident particles scatter. The electron in the ground state of a hydrogen atom is found to be momentarily at a distance a 0/2 from the nucleus in 1 000 of the observations. In this set of trials, how many times is the atomic electron observed at a distance 2a 0 from the nucleus?

List the possible sets of quantum numbers for the hydrogen atom associated with (a) the 3d subshell and (b) the 3p subshell.

If a hydrogen atom has orbital angular momentum 4.714 3 10234 J ? s, what is the orbital quantum number for the state of the atom?

Find all possible values of (a) L, (b) Lz, and (c) u for a hydrogen atom in a 3d state.

Calculate the magnitude of the orbital angular momentum for a hydrogen atom in (a) the 4d state and (b) the 6f state.

How many sets of quantum numbers are possible for a hydrogen atom for which (a) n 5 1, (b) n 5 2, (c) n 5 3, (d) n 5 4, and (e) n 5 5?

An electron in a sodium atom is in the N shell. Determine the maximum value the z component of its angular momentum could have.

(a) Find the mass density of a proton, modeling it as a solid sphere of radius 1.00 3 10215 m. (b) What If? Consider a classical model of an electron as a uniform solid sphere with the same density as the proton. Find its radius. (c) Imagine that this electron possesses spin angular momentum Iv 5 "/2 because of classical rotation about the z axis. Determine the speed of a point on the equator of the electron. (d) State how this speed compares with the speed of light.

A hydrogen atom is in its fifth excited state, with principal quantum number 6. The atom emits a photon with a wavelength of 1 090 nm. Determine the maximum possible magnitude of the orbital angular momentum of the atom after emission.

Why is the following situation impossible? A photon of wavelength 88.0 nm strikes a clean aluminum surface, ejecting a photoelectron. The photoelectron then strikes a hydrogen atom in its ground state, transferring energy to it and exciting the atom to a higher quantum state.

The r2 meson has a charge of 2e, a spin quantum number of 1, and a mass 1 507 times that of the electron. The possible values for its spin magnetic quantum number are 21, 0, and 1. What If? Imagine that the electrons in atoms are replaced by r2 mesons. List the possible sets of quantum numbers for r2 mesons in the 3d subshell.

. (a) As we go down the periodic table, which subshell is filled first, the 3d or the 4s subshell? (b) Which electronic configuration has a lower energy, [Ar]3d 44s 2 or [Ar]3d 54s1? Note: The notation [Ar] represents the filled configuration for argon. Suggestion: Which has the greater number of unpaired spins? (c) Identify the element with the electronic configuration in part (b).

(a) Write out the electronic configuration of the ground state for nitrogen (Z 5 7). (b) Write out the values for the possible set of quantum numbers n, ,, m,, and ms for the electrons in nitrogen.

. Devise a table similar to that shown in Figure 42.18 for atoms containing 11 through 19 electrons. Use Hunds rule and educated guesswork.

A certain element has its outermost electron in a 3p subshell. It has valence 13 because it has three more electrons than a certain noble gas. What element is it?

Scanning through Figure 42.19 in order of increasing atomic number, notice that the electrons usually fill the subshells in such a way that those subshells with the lowest values of n 1 , are filled first. If two subshells have the same value of n 1 ,, the one with the lower value of n is generally filled first. Using these two rules, write the order in which the subshells are filled through n 1 , 5 7

Two electrons in the same atom both have n 5 3 and , 5 1. Assume the electrons are distinguishable, so that interchanging them defines a new state. (a) How many states of the atom are possible considering the quantum numbers these two electrons can have? (b) What If? How many states would be possible if the exclusion principle were inoperative?

For a neutral atom of element 110, what would be the probable ground-state electronic configuration?

Review. For an electron with magnetic moment m Ss in a magnetic field B S , Section 29.5 showed the following. The electronfield system can be in a higher energy state with the z component of the electrons magnetic moment opposite the field or a lower energy state with the z component of the magnetic moment in the direction of the field. The difference in energy between the two states is 2mBB. Under high resolution, many spectral lines are observed to be doublets. The most famous doublet is the pair of two yellow lines in the spectrum of sodium (the D lines), with wavelengths of 588.995 nm and 589.592 nm. Their existence was explained in 1925 by Goudsmit and Uhlenbeck, who postulated that an electron has intrinsic spin angular momentum. When the sodium atom is excited with its outermost electron in a 3p state, the orbital motion of the outermost electron creates a magnetic field. The atoms energy is somewhat different depending on whether the electron is itself spin-up or spin-down in this field. Then the photon energy the atom radiates as it falls back into its ground state depends on the energy of the excited state. Calculate the magnitude of the internal magnetic field, mediating this so-called spin-orbit coupling.

In x-ray production, electrons are accelerated through a high voltage DV and then decelerated by striking a target. Show that the shortest wavelength of an x-ray that can be produced is l min 5 1 240nm # V DV

What minimum accelerating voltage would be required to produce an x-ray with a wavelength of 70.0 pm?

. A tungsten target is struck by electrons that have been accelerated from rest through a 40.0-keV potential difference. Find the shortest wavelength of the radiation emitted

A bismuth target is struck by electrons, and x-rays are emitted. Estimate (a) the M- to L-shell transitional energy for bismuth and (b) the wavelength of the x-ray emitted when an electron falls from the M shell to the L shell.

The 3p level of sodium has an energy of 23.0 eV, and the 3d level has an energy of 21.5 eV. (a) Determine Zeff for each of these states. (b) Explain the difference.

(a) Determine the possible values of the quantum numbers , and m, for the He1 ion in the state corresponding to n 5 3. (b) What is the energy of this state?

The K series of the discrete spectrum of tungsten contains wavelengths of 0.018 5 nm, 0.020 9 nm, and 0.021 5 nm. The K-shell ionization energy is 69.5 keV. Determine the ionization energies of the L, M, and N shells

Use the method illustrated in Example 42.5 to calculate the wavelength of the x-ray emitted from a molybdenum target (Z 5 42) when an electron moves from the L shell (n 5 2) to the K shell (n 5 1).

In x-ray production, electrons are accelerated through a high voltage and then decelerated by striking a target. (a) To make possible the production of x-rays of wavelength l, what is the minimum potential difference DV through which the electrons must be accelerated? (b) State in words how the required potential difference depends on the wavelength. (c) Explain whether your result predicts the correct minimum wavelength in Figure 42.22. (d) Does the relationship from part (a) apply to other kinds of electromagnetic radiation besides x-rays? (e) What does the potential difference approach as l goes to zero? (f) What does the potential difference approach as l increases without limit?

When an electron drops from the M shell (n 5 3) to a vacancy in the K shell (n 5 1), the measured wavelength of the emitted x-ray is found to be 0.101 nm. Identify the element

Figure P42.58 shows portions of the energy-level diagrams of the helium and neon atoms. An electrical discharge excites the He atom from its ground state (arbitrarily assigned the energy E1 5 0) to its excited state of 20.61 eV. The excited He atom collides with a Ne atom in its ground state and excites this atom to the state at 20.66 eV. Lasing action takes place for electron transitions from E3* to E2 in the Ne atoms. From the data in the figure, show that the wavelength of the red HeNe laser light is approximately 633 nm.

The carbon dioxide laser is one of the most powerful developed. The energy difference between the two laser levels is 0.117 eV. Determine (a) the frequency and (b) the wavelength of the radiation emitted by this laser. (c) In what portion of the electromagnetic spectrum is this radiation?

Review. A heliumneon laser can produce a green laser beam instead of a red one. Figure P42.60 shows the transitions involved to form the red beam and the green beam. After a population inversion is established, neon atoms make a variety of downward transitions in falling from the state labeled E4* down eventually to level E1 (arbitrarily assigned the energy E1 5 0). The atoms emit both red light with a wavelength of 632.8 nm in a transition E4* 2 E3 and green light with a wavelength of 543 nm in a competing transition E4* 2 E2. (a) What is the energy E2? Assume the atoms are in a cavity between mirrors designed to reflect the green light with high efficiency but to allow the red light to leave the cavity immediately. Then stimulated emission can lead to the buildup of a collimated beam of green light between the mirrors having a greater intensity than that of the red light. To constitute the radiated laser beam, a small fraction of the green light is permitted to escape by transmission through one mirror. The mirrors forming the resonant cavity can be made of layers of silicon dioxide (index of refraction n 5 1.458) and titanium dioxide (index of refraction varies between 1.9 and 2.6). (b) How thick a layer of silicon dioxide, between layers of titanium dioxide, would minimize reflection of the red light? (c)What should be the thickness of a similar but separate layer of silicon dioxide to maximize reflection of the green light?

A ruby laser delivers a 10.0-ns pulse of 1.00-MW average power. If the photons have a wavelength of 694.3 nm, how many are contained in the pulse?

The number N of atoms in a particular state is called the population of that state. This number depends on the energy of that state and the temperature. In thermal equilibrium, the population of atoms in a state of energy En is given by a Boltzmann distribution expression N 5 Ng e2(En2Eg)/kBT where Ng is the population of the ground state of energy Eg , kB is Boltzmanns constant, and T is the absolute temperature. For simplicity, assume each energy level has only one quantum state associated with it. (a) Before the power is switched on, the neon atoms in a laser are in thermal equilibrium at 27.0C. Find the equilibrium ratio of the populations of the states E4* and E3 shown for the red transition in Figure P42.60. Lasers operate by a clever artificial production of a population inversion between the upper and lower atomic energy states involved in the lasing transition. This term means that more atoms are in the upper excited state than in the lower one. Consider the E4*2E3 transition in Figure P42.60. Assume 2% more atoms occur in the upper state than in the lower. (b) To demonstrate how unnatural such a situation is, find the temperature for which the Boltzmann distribution describes a 2.00% population inversion. (c) Why does such a situation not occur naturally?

A neodymiumyttriumaluminum garnet laser used in eye surgery emits a 3.00-mJ pulse in 1.00 ns, focused to a spot 30.0 mm in diameter on the retina. (a) Find (in SI units) the power per unit area at the retina. (In the optics industry, this quantity is called the irradiance.) (b) What energy is delivered by the pulse to an area of molecular size, taken as a circular area 0.600 nm in diameter?

Review. Figure 42.29 represents the light bouncing between two mirrors in a laser cavity as two traveling waves. These traveling waves moving in opposite directions constitute a standing wave. If the reflecting surfaces are metallic films, the electric field has nodes at both ends. The electromagnetic standing wave is analogous to the standing string wave represented in Figure 18.10. (a) Assume that a heliumneon laser has precisely flat and parallel mirrors 35.124 103 cm apart. Assume that the active medium can efficiently amplify only light with wavelengths between 632.808 40 nm and 632.809 80 nm. Find the number of components that constitute the laser light, and the wavelength of each component, precise to eight digits. (b) Find the root-mean-square speed for a neon atom at 120C. (c) Show that at this temperature the Doppler effect for light emission by moving neon atoms should realistically make the bandwidth of the light amplifier larger than the 0.001 40 nm assumed in part (a).

How much energy is required to ionize a hydrogen atom when it is in (a) the n 5 2 state and (b) the n 5 10 state?

The force on a magnetic moment mz in a nonuniform magnetic field Bz is given by Fz 5 mz (dBz/dz). If a beam of silver atoms travels a horizontal distance of 1.00 m through such a field and each atom has a speed of 100 m/s, how strong must be the field gradient dBz/dz to deflect the beam 1.00 mm?

Suppose a hydrogen atom is in the 2s state, with its wave function given by Equation 42.26. Taking r 5 a 0, calculate values for (a) c2s (a 0), (b) uc2s (a 0)u 2, and (c) P2s(a 0).

. Review. (a) How much energy is required to cause an electron in hydrogen to move from the n 5 1 state to the n 5 2 state? (b) Suppose the atom gains this energy through collisions among hydrogen atoms at a high temperature. At what temperature would the average atomic kinetic energy 3 2kBT be great enough to excite the electron? Here kB is Boltzmanns constant.

In the technique known as electron spin resonance (ESR), a sample containing unpaired electrons is placed in a magnetic field. Consider a situation in which a single electron (not contained in an atom) is immersed in a magnetic field. In this simple situation, only two energy states are possible, corresponding to ms 5 61 2. In ESR, the absorption of a photon causes the electrons spin magnetic moment to flip from the lower energy state to the higher energy state. According to Section 29.5, the change in energy is 2mBB. (The lower energy state corresponds to the case in which the z component of the magnetic moment m Sspin is aligned with the magnetic field, and the higher energy state corresponds to the case in which the z component of m Sspin is aligned opposite to the field.) What is the photon frequency required to excite an ESR transition in a 0.350-T magnetic field?

An electron in chromium moves from the n 5 2 state to the n 5 1 state without emitting a photon. Instead, the excess energy is transferred to an outer electron (one in the n 5 4 state), which is then ejected by the atom. In this Auger (pronounced ohjay) process, the ejected electron is referred to as an Auger electron. Use the Bohr theory to find the kinetic energy of the Auger electron.

The states of matter are solid, liquid, gas, and plasma. Plasma can be described as a gas of charged particles or a gas of ionized atoms. Most of the matter in the Solar System is plasma (throughout the interior of the Sun). In fact, most of the matter in the Universe is plasma; so is a candle flame. Use the information in Figure 42.20 to make an order-of-magnitude estimate for the temperature to which a typical chemical element must be raised to turn into plasma by ionizing most of the atoms in a sample. Explain your reasoning.

Show that the wave function for a hydrogen atom in the 2s state c2s 1r2 5 1 4"2p a 1 a 0 b 3/2 a2 2 r a 0 be 2r/2a0 satisfies the spherically symmetric Schrdinger equation given in Problem 26.

Find the average (expectation) value of 1/r in the 1s state of hydrogen. Note that the general expression is given by 81/r 9 5 3 all space 0 c 0 2 11/r2 dV 5 3 ` 0 P 1r2 11/r2 dr Is the result equal to the inverse of the average value of r?

Why is the following situation impossible? An experiment is performed on an atom. Measurements of the atom when it is in a particular excited state show five possible values of the z component of orbital angular momentum, ranging between 3.16 3 10234 kg ? m2/s and 23.16 3 10234 kg ? m2/s.

In the Bohr model of the hydrogen atom, an electron travels in a circular path. Consider another case in which an electron travels in a circular path: a single electron moving perpendicular to a magnetic field B S . Lev Davidovich Landau (19081968) solved the Schrdinger equation for such an electron. The electron can be considered as a model atom without a nucleus or as the irreducible quantum limit of the cyclotron. Landau proved its energy is quantized in uniform steps of e"B/me. In 1999, a single electron was trapped by a Harvard University research team in an evacuated centimeter-size metal can cooled to a temperature of 80 mK. In a magnetic field of magnitude 5.26 T, the electron circulated for hours in its lowest energy level. (a) Evaluate the size of a quantum jump in the electrons energy. (b)For comparison, evaluate kBT as a measure of the energy available to the electron in blackbody radiation from the walls of its container. Microwave radiation was introduced to excite the electron. Calculate (c) the frequency and (d) the wavelength of the photon the electron absorbed as it jumped to its second energy level. Measurement of the resonant absorption frequency verified the theory and permitted precise determination of properties of the electron.

As the Earth moves around the Sun, its orbits are quantized. (a) Follow the steps of Bohrs analysis of the hydrogen atom to show that the allowed radii of the Earths orbit are given by r 5 n2 U2 GMSME 2 where n is an integer quantum number, MS is the mass of the Sun, and ME is the mass of the Earth. (b) Calculate the numerical value of n for the SunEarth system. (c) Find the distance between the orbit for quantum number n and the next orbit out from the Sun corresponding to the quantum number n 1 1. (d) Discuss the significance of your results from parts (b) and (c).

An elementary theorem in statistics states that the root-mean-square uncertainty in a quantity r is given by Dr 5 !8r 2 9 2 8r 9 2 . Determine the uncertainty in the radial position of the electron in the ground state of the hydrogen atom. Use the average value of r found in Example 42.3: 8r 9 5 3a 0/2. The average value of the squared distance between the electron and the proton is given by 8r 2 9 5 3 all space 0 c 0 2 r 2 dV 5 3 ` 0 P 1r2r 2 dr

Example 42.3 calculates the most probable value and the average value for the radial coordinate r of the electron in the ground state of a hydrogen atom. For comparison with these modal and mean values, find the median value of r. Proceed as follows. (a) Derive an expression for the probability, as a function of r, that the electron in the ground state of hydrogen will be found outside a sphere of radius r centered on the nucleus. (b) Make a graph of the probability as a function of r/a 0. Choose values of r/a 0 ranging from 0 to 4.00 in steps of 0.250. (c) Find the value of r for which the probability of finding the electron outside a sphere of radius r is equal to the probability of finding the electron inside this sphere. You must solve a transcendental equation numerically, and your graph is a good starting point.

(a) For a hydrogen atom making a transition from the n 5 4 state to the n 5 2 state, determine the wavelength of the photon created in the process. (b) Assuming the atom was initially at rest, determine the recoil speed of the hydrogen atom when it emits this photon.

Astronomers observe a series of spectral lines in the light from a distant galaxy. On the hypothesis that the lines form the Lyman series for a (new?) one-electron atom, they start to construct the energy-level diagram shown in Figure P42.80, which gives the wavelengths of the first four lines and the short-wavelength limit of this series. Based on this information, calculate (a) the energies of the ground state and first four excited states for this one-electron atom and (b) the wavelengths of the first three lines and the short-wavelength limit in the Balmer series for this atom. (c) Show that the wavelengths of the first four lines and the short-wavelength limit of the Lyman series for the hydrogen atom are all 60.0% of the wavelengths for the Lyman series in the one-electron atom in the distant galaxy. (d) Based on this observation, explain why this atom could be hydrogen.

We wish to show that the most probable radial position for an electron in the 2s state of hydrogen is r 5 5.236a 0. (a) Use Equations 42.24 and 42.26 to find the radial probability density for the 2s state of hydrogen. (b) Calculate the derivative of the radial probability density with respect to r. (c) Set the derivative in part (b) equal to zero and identify three values of r that represent minima in the function. (d) Find two values of r that represent maxima in the function. (e) Identify which of the values in part (c) represents the highest probability.

All atoms have the same size, to an order of magnitude. (a) To demonstrate this fact, estimate the atomic diameters for aluminum (with molar mass 27.0 g/mol and density 2.70 g/cm3) and uranium (molar mass 238 g/mol and density 18.9 g/cm3). (b) What do the results of part (a) imply about the wave functions for inner-shell electrons as we progress to higher and higher atomic mass atoms?

A pulsed ruby laser emits light at 694.3 nm. For a 14.0-ps pulse containing 3.00 J of energy, find (a) the physical length of the pulse as it travels through space and (b) the number of photons in it. (c) The beam has a circular cross section of diameter 0.600 cm. Find the number of photons per cubic millimeter.

A pulsed laser emits light of wavelength l. For a pulse of duration Dt having energy TER, find (a) the physical length of the pulse as it travels through space and (b) the number of photons in it. (c) The beam has a circular cross section having diameter d. Find the number of photons per unit volume.

Assume three identical uncharged particles of mass m and spin 1 2 are contained in a one-dimensional box of length L. What is the ground-state energy of this system?

Suppose the ionization energy of an atom is 4.10 eV. In the spectrum of this same atom, we observe emission lines with wavelengths 310 nm, 400 nm, and 1 377.8 nm. Use this information to construct the energy-level diagram with the fewest levels. Assume the higher levels are closer together

For hydrogen in the 1s state, what is the probability of finding the electron farther than 2.50a 0 from the nucleus?

For hydrogen in the 1s state, what is the probability of finding the electron farther than ba 0 from the nucleus, where b is an arbitrary number?

The positron is the antiparticle to the electron. It has the same mass and a positive electric charge of the same magnitude as that of the electron. Positronium is a hydrogen-like atom consisting of a positron and an electron revolving around each other. Using the Bohr model, find (a) the allowed distances between the two particles and (b) the allowed energies of the system.

Review. Steven Chu, Claude Cohen-Tannoudji, and William Phillips received the 1997 Nobel Prize in Physics for the development of methods to cool and trap atoms with laser light. One part of their work was with a beam of atoms (mass , 10225 kg) that move at a speed on the order of 1 km/s, similar to the speed of molecules in air at room temperature. An intense laser light beam tuned to a visible atomic transition (assume 500 nm) is directed straight into the atomic beam; that is, the atomic beam and the light beam are traveling in opposite directions. An atom in the ground state immediately absorbs a photon. Total system momentum is conserved in the absorption process. After a lifetime on the order of 1028 s, the excited atom radiates by spontaneous emission. It has an equal probability of emitting a photon in any direction. Therefore, the average recoil of the atom is zero over many absorption and emission cycles. (a) Estimate the average deceleration of the atomic beam. (b) What is the order of magnitude of the distance over which the atoms in the beam are brought to a halt?

(a) Use Bohrs model of the hydrogen atom to show that when the electron moves from the n state to the n 2 1 state, the frequency of the emitted light is f 5 a 2p2 me ke 2 e 4 h3 b 2n 2 1 n2 1n 2 12 2 (b) Bohrs correspondence principle claims that quantum results should reduce to classical results in the limit of large quantum numbers. Show that as n S `, this expression varies as 1/n3 and reduces to the classical frequency one expects the atom to emit. Suggestion: To calculate the classical frequency, note that the frequency of revolution is v/2pr, where v is the speed of the electron and r is given by Equation 42.10.