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physics
modern physics

Questions and Answers of Modern Physics

All electron is bound in a square well with a depth equal to six times the ground-level energy E = of an infinite well of the same width. The longest-wavelength photon that is absorbed by the
All electron with initial kinetic energy 6.0eV encounters a barrier with height 11.0eV. What is the probability of tunneling if the width of the barrier is? (a) 0.80 nm and (b) 0.40 run?
All electron with initial kinetic energy 5.0eV encounters a barrier with height U0 and width .060 nm. What is the transmission coefficient if (a) U0 = 7.0eV; (b) U0 = 9.0eV; (c) U0 = 13.0eV?
All electrons are moving past the square barrier shown in Fig. 40.12, 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
A proton with initial kinetic energy 50.0eV encounters a barrier of height 70.0eV. What is the width of the barrier if the probability of tunneling is 3.0 x 10-3? How does this compare with the
(a) All electrons with initial kinetic energy 32eV encounter a square barrier with height 41eV and width 0.2S nm. What is the probability that the electron will tunnel through the barrier? (b) A
Alpha Decay in a simple model for a radioactive nucleus, an alpha particle (m = 6.64 x 10-27 kg) is trapped by a square barrier that has width 2.0 fm and height 30.0MeV.(a) What is the tunneling
A wooden block with mass 0.2S0 kg is oscillating on the end of a spring that has force constant 110 N/m. Calculate the ground-level energy and the energy separation between adjacent levels. Express
Show that ψ(x) given by Eq. (40.24) is a solution to Eq. (40.22) with energy E0 = hw/2.
Chemists use infrared absorption spectra to identify chemicals in a sample. In one sample, a chemist finds that light of wavelength 5.8µm is absorbed. (a) Find the energy of this transition. (b) If
The ground-state energy of a harmonic oscillator is 5.60eV. 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?
In Section 40.4 it is shown that for the ground level of a harmonic oscillator, ∆x∆px = h. Do a similar analysis for an excited level that has quantum number n. How does the uncertainty
For the ground-level harmonic oscillator wave function ψ(x) given in Eq. (40.24), | ψ |2 has a maximum at x = 0. (a) Compute the ratio of | ψ |2 at = +A to | ψ |2 at x = 0,
For the sodium atom of Example 40.6, 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
Show that the wave function ψ(x) = Ae ikx is a solution of Eq. 40.1 for a particle of mass m, in a region where the potential energy is a constant U0 < E. Find an expression for k, and relate it
Wave functions like the one in Problem 40.34 can represent free particles moving with velocity v = p/m in the x-direction. Consider a beam of such particles incident on a potential-energy step U(x) =
Let AE be the energy difference between the adjacent energy levels En and En+1 for a particle in a box. The ratio Rn = ∆En/E. compares the energy of a level to the energy separation of the next
Photon in a Dye Laser all electrons in a long organic molecule used in a dye laser behaves approximately like a particle in a box with width 4.18nm. What is the wavelength of the photon emitted when
(c) How do the results of parts (a) and (b) compare? Explain.(d) Add the probabilities calculated in parts (a) and (b). (e) Are your results in parts (a), (b), and (d) consistent with Fig. 40.5b?
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?(c) Are your
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 |2 dx that the particle will be found in the interval x
Repeat Problem 40.40 for a particle in the first excited level.
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
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 K is a positive constant(a) Graph this
The penetration distance ? 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
(a) For the finite potential well of Fig. 40.6, what relationships among the constants A and B of Eq. (40.17) and C and D of Eq. (40.19) are obtained by applying the boundary condition that ψ,
An electron with initial kinetic energy 5.5eV encounters a square potential barrier with height 10.0eV. What is the thickness of the barrier if the electron has a 0.10% probability of tunneling
An electron with initial kinetic energy 5.5eV encounters a square potential barrier with height 10.0eV. What is the width of the barrier if the electron has a 0.10% probability of tunneling through
A harmonic oscillator consists of a 0.020-kg mass on a spring. Its frequency is 1.50 Hz, and the mass has a speed of 0.360 m/s as it passes the equilibrium position. (a) What is the value of the
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
Some 164.9-nm photons are emitted in a ∆n = 1 transition within a solid-state lattice. The lattice is modeled as electrons in a box having length 0.500nm. What transition corresponds to the
(a) Show by direct substitution in the Schrodinger equation for the one-dimensional harmonic oscillator that the wave function ψ0 (x) = A0e – a2x2/2, where a2 = mw/h, is a solution with energy
(a) Show by direct substitution in the Schrodinger equation for the one-dimensional harmonic oscillator that the wave function ψ1 (x) = A1xe –a2x2/2, where a2 = mw/h, is a solution with energy
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
Three-Dimensional Anisotropic Harmonic Oscillator An oscillator bas the potential-energy function U(x. y. z) = ½ k1(x2 + y2) + ½ k2’ z2 where k1 > k2. This oscillator is
Three-Dimensional Anisotropic Harmonic Oscillator An oscillator bas the potential-energy function U(x. y. z) = ½ k1(x2 + y2) + ½ k2’ z2 where k1 > k2. This oscillator is called anisotropic
Consider a potential well defined as U(x) = ? for x 0 for x > L (Fig. 40.27). Consider a particle with mass m and kinetic energy E (a) The boundary condition at the infinite wall (x = 0) is
Three-Dimensional Anisotropic Harmonic Oscillator An oscillator bas the potential-energy function U(x. y. z) = ½ k1(x2 + y2) + ½ k2’ z2 where k1 > k2. This oscillator is called anisotropic
The WKB approximation (see Challenge Problem 40.57) can be used to calculate the energy levels for a harmonic oscillator. In this approximation, the energy levels are the solutions to the Equation
(a) Determine the classical turning points for the potential U(x) = A| x | and for an energy E.(b) Carry out the above integral and show that the allowed energy levels in the WKB approximation are
An electron is in the hydrogen atom with n = 3. (a) Find the possible values of L and L, for this electron. in units of h. (b) For each value of L. find all the possible angles between L and the
An electron is in the hydrogen atom with n = 5. (a) Find the possible values of L and L, for this electron in units of h. (b) For each value of L. find all the possible angles between L and the z
The orbital angular momentum of an electron has a magnitude of 4.716 x 10-34 kg ∙ m2/s. What is the angular-momentum quantum number I for this electron?
Consider states with angular-momentum quantum number l = 2. (a) In units of h, what is the largest possible value of L?(b) In units of h, what is the value of L? Which is larger: L or the maximum
Calculate, in units of h, 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
(a) Make a chart showing all the possible sets of quantum numbers I and m, for the states of the electron in the hydrogen atom when n = 5. How many combinations are there? (b) What are the energies
Problem-Solving Strategy 41.1 claims that the electric potential energy of a proton and an electron 0.10 nm apart has magnitude 15eV. Verify this claim.
(a) What is the probability that an electron in the is 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 41.3 to
(a) For the wave function ψ(r, θ, Φ) = R(r) O (θ) Φ (Φ) with Φ (Φ) = Ae imΦ show that | ψ |2 is independent of Φ. (b) What value must A have
For ordinary hydrogen, the reduced mass of the electron and proton is mt = 0.99946m, where m is the electron mass (see Section 38.5). For each of the following cases, find the numerical coefficient
Find the numerical value of a in Eq. (41.8) for (a) a hydrogen atom in which the nucleus is taken to be infinitely massive, so mt = m; (b) Positronium (see Section 38.5), for which mt = m/2 exactly;
Show that Φ (Φ = e im,Φ = Φ (Φ + 2π) (that is, show that Φ (Φ) is periodic with period 2π) if and only if m, is restricted to the values 0, ±l, ±2, ....
In Example 41.3 fill in the missing details that show that P = I - 5e-2.
A hydrogen atom is in a d state. In the absence of an external magnetic field the states with different m, values have (approximately) the same energy. Consider the interaction of the magnetic field
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
A hydrogen atom undergoes a transition from a 2p state to the Is 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
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
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
Calculate the energy difference between the ms = ½ ("spin up") and mg = - ½ ("spin down") levels of a hydrogen atom in the Is state when it is placed in a 1.45-T magnetic field in the negative
List the different possible combinations of l and j for a hydrogen atom in the n = 3 level.
A hydrogen atom in a particular orbital angular momentum state is found to have j quantum numbers 7/2 and 9/2. What is the letter that labels the value of l for the state?
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
Classical Electron Spin (a) If you treat an electron as a classical spherical object with a radius of 1.0 x 10-17m, what angular speed is necessary to produce a spin angular momentum of magnitude
For germanium (Ge, Z = 32), make a list of the number of electrons in each sub shell (1s, 2s, 2p, . .). Use the allowed values of the quantum numbers along with the exclusion principle; do not refer
Make a list of the four quantum numbers n, 1, mt and ma for each of the 10 electrons in the ground state of the neon atom. Do not refer to Table 41.2 or 41.3.
The 58 electron in rubidium (Rb) sees an effective charge of 2.771e. Calculate the ionization energy of this electron.
The energies of the 4s, 4p, and 4d states of potassium are given in Example 41.8. Calculate Z eff for each state. What trend do your results show? How can you explain this trend?
(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
(a) The energy of the 28 state of lithium is -5.391eV. Calculate the value of Zeff for this state. (b) The energy of the 4s state of potassium is -4.339eV. Calculate the value of Zeff for this state.
Estimate the energy of the highest-I state for (a) The L shell of Be+ and (b) The N shell of Ca+.
A Ka x ray emitted from a sample has an energy of 7.46keV. Of which element is the sample made?
Calculate the frequency, energy (in keV), and wavelength of the K" x ray for the elements (a) Calcium (Ca, Z = 20); (b) Cobalt (Co, Z = 27); (c) Cadmium (Cd Z = 48).
(c) What are the largest and smallest values of the spin angular momentum (in terms of h) for the electron in part (a)? (d) What are the largest and smallest values of the orbital angular momentum
Consider an electron in hydrogen having total energy -0.5440eV. (a) What are the possible values of its orbital angular momentum (in terms of h)? (b) What wavelength of light would it take to excite
(a) Show all the distinct states for an electron in the N shell of hydrogen. Include all four quantum numbers.(b) For an f-electron in the N shell, what is the largest possible orbital angular
(a) The energy of an electron in the 4s state of sodium is -1.947eV. What is the effective net charge of the nucleus "seen" by this electron? On the average, how many electrons screen the nucleus?
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. (41.7). For a hydrogen atom in the Is ground
Consider a hydrogen atom in the state. (a) For what value of, is the potential energy U(r) equal to the total energy E? Express your answer in terms of a. This value of, is called the classical
Rydberg Atoms Rydberg atoms are atoms whose outermost electron is in an excited state with a very large principal quantum number. Rydberg atoms have been produced in the laboratory and detected in
The wave function for a hydrogen atom in the 2s state is(a) Verify that this function is normalized.(b) In the Bohr model, the distance between the electron and the nucleus in the n = 2 state is
The wave function for a hydrogen atom in the 2s state is(a) Verify that this function is normalized.(b) In the Bohr model, the distance between the electron and the nucleus in the n = 2 state is
(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 (?L)max the
(a) If the value of L. is known, we cannot know either Lx or Ly precisely. But we can know the value of the quantity √Lx2 + Ly2. Write an expression for this quantity in terms of l, mi, and h.
The normalized radial wave function, (41.2), 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
Stem-Gerlach Experiment in a Stern-Gerlach experiment, the deflecting force on the atom is Fz = -µz (dBz/dz), where µz is given by Eq. (41.22) and dBz/dz is the magnetic-field gradient. In a
Consider the transition from a 3d to a 2p state of hydrogen in an external magnetic field. Assume that the effects of electron spin can be ignored (which is not actually the case) so that the
An atom in a 3d state emits a photon of wavelength 475.082 run when it decays to a 2s state. (a) What is the energy (in electron volts) of the photon emitted in this transition? (b) Use the selection
Spectral Analysis while studying the spectrum of a gas cloud in space, an astronomer magnifies a spectral line that results from a transition from a p state to an s state. She finds that the line at
A hydrogen atom makes a transition from an n = 3 state to an n = 2 state (the Ballmer Ha line) while in a magnetic field in the +z-direction and with magnitude 1.40 T. (a) If the magnetic quantum
A large number of hydrogen atoms in Is 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 = 300K.
Effective Magnetic Field An electron in a hydrogen atom is in the 2p state. In a simple model of the atom, assume that the electron circles the proton in an orbit with radius, equal to the Bohr-model
Weird Universe In another universe, the electron is a spin 3/2 rather than a spin- ½ particles, but all oilier physics are the same as in our universe. In this universe, (a) What are the atomic
(a) What is the ground-state energy in electron volts? (b) What is the ionization energy, the energy required to remove the electron from the N6+ ion if it is initially in the ground state? (c) What
A hydrogen atom in an n = 2, l = 1, m1 = -1 state emits a photon when it decays to an n = 1, l = 0, m1 = 0 ground state. (a) In the absence of an external magnetic field, what is the wavelength of
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 2P½, by the spin-orbit
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.
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. Discuss.
(a) Show that the total number of atomic states (including different spin states) in a shell of principal quantum number n is 2n2. (b) Which shell has 50 states?
(a) If the intrinsic spin angular momentum S of the earth had the same limitations as that of the electron, what would be the angular velocity of our planet's spin on its axis? To get a reasonable
Each of 2N electrons (mass m) is free to move along the x axis. The potential-energy function for each electron is U(x) = ½ k'x2, where k' is a positive constant. The electric and magnetic

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