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physics
modern physics

Questions and Answers of Modern Physics

(a) Derive an expression for the total shift in photon wavelength after two successive Compton scatterings from electrons at rest. The photon is scattered by an angle θ1, in the first scattering
Nuclear fusion reactions at the center of the sun produce gamma-ray photons with energies of order 1 MeV (106eV). By contrast, what we see emanating from the sun's surface are visible light photons
An x-ray photon is scattered from a free electron (mass m) at rest. The wavelength of the scattered photon is A', and the final speed of the struck electron is v. (a) What was the initial wavelength
A photon with wavelength 0.1100nm collides with a free electron that is initially at rest. After the collision the wavelength is 0.1132 nm. (a) What is the kinetic energy of the electron after the
(a) Calculate the maximum increase in photon wavelength that can occur during Compton scattering. (b) What is the energy (in electron volts) of the lowest-energy x-ray photon for which Compton
(a) Write the Planck distribution law in terms of the frequency f rather than the wavelength A, to obtain I(f) .(b) Show that where I(λ) is the Planck distribution formula of Eq. (38.32). You
An Ideal Blackbody a large cavity with a very small hole and maintained at a temperature T is a good approximation to an ideal radiator or blackbody. Radiation can pass into or out of the cavity only
The motion of an oscillating flywheel is defined by the relation θ = θ0e−3πt cos4πt, where θ is expressed in radians and t in seconds. Knowing that θ0 = 0.5 rad, determine the angular
Consider a beam of monochromatic light with intensity I incident on a perfectly absorbing surface oriented perpendicular to the beam. Use the photon concept to show that the radiation pressure
Consider Compton scattering of a photon by a moving electron. Before the collision the photon has wavelength ? and is moving in the +x-direction, and the electron is moving in the -x-direction with
(a) An electron moves with a speed of 4.70 x 106m/s. What is its de Broglie wavelength? (b) A proton moves with the same speed. Determine its de Broglie wavelength.
For crystal diffraction experiments (discussed in Section 39.2), wavelengths on the order of 0.20 nm are often appropriate. Find the energy in electron volts for a particle with this wavelength if
An electron has a de Broglie wavelength of 2.80 X 10-10 m. Determine (a) The magnitude of its momentum and (b) Its kinetic energy (in joules and in electron volts).
Wavelength of an Alpha Particle an alpha particle (m = 6.64 x 10-27 kg) emitted in the radioactive decay of uranium-238 has an energy of 4.20MeV. What is its de Broglie wavelength?
In the Bohr model of the hydrogen atom, what is the de Broglie wavelength for the electron when it is in (a) The n = 1 level and (b) The n = 4 level? In each case, compare the de Broglie
(a) A non-relativistic free particle with mass m has kinetic energy K. Derive an expression for the de Broglie wavelength of the particle in terms of m and K. (b) What is the de Broglie wavelength of
Why Don't We Diffract? (a) Calculate the de Broglie wavelength of a typical person walking through a doorway. Make reasonable approximations for the necessary quantities. (b) Will the person in part
What is the de Broglie wavelength for an electron with speed (a) v = 0.480c and (b) v = 0.960c?
(a) If a photon and an electron each have the same energy of 20.0eV, find the wavelength of each. (b) If a photon and an electron each have the same wavelength of 250 run, find the energy' of
Hydrogen gas (H2) is at 0oC. The mass of a hydrogen atom is 1.67 x 10-27kg. (a) What is the average de Broglie wavelength of the hydrogen molecules? (b) How fast would an electron have to move to
Wavelength of a Bullet calculates the de Broglie wavelength of a 5.00-g bullet that is moving at 340m/s. Will the bullet exhibit wavelike properties?
Through what potential difference must electrons be accelerated so they will have? (a) The same wavelength as an x ray of wavelength 0.150 nm and (b) The same energy as the x ray in part (a)?
(a) Approximately how fast should an electron move so it has a wavelength that makes it useful to measure the distance between adjacent atoms in typical crystals (about 0.10 nm)? (b) What is the
A beam of electrons is accelerated from rest through a potential difference of 0.100 kV and then passes through a thin slit the diffracted beam shows its first diffraction minima at ± 11.5° from
A beam of neutrons that all have the same energy scatters from the atoms that have a spacing of 0.0910 nm in the surface plane of a crystal. The m = 1 intensity maximum occurs when the angle θ
A beam of 188-eV electrons is directed at normal incidence onto a crystal surface as shown in Fig. 39.4b. The m = 2 intensity maximum occurs at an angle θ = 60.6°. (a) What is the spacing
A beam of electrons is accelerated from rest through a potential difference of 0.100 kV and then passes through a thin slit the diffracted beam shows its first diffraction minima at ± 11.5° from
A pesky 1.5-mg mosquito is annoying you as you attempt to study physics in your room, which is 5.0 m wide and 2.5 m high. You decide to swat the bothersome insect as it flies toward you, but you need
By extremely careful measurement, you determine the x-coordinate of a car's center of mass with an uncertainty of only 1.00µm. The car has a mass of 1200 kg. (a) What is the minimum uncertainty in
A 10.0-g marble is gently placed on a horizontal tabletop that is 1.75 m wide. (a) What is the maximum uncertainty in the horizontal position of the marble? (b) According to the Heisenberg
A scientist has devised a new method of isolating individual particles. He claims that this method enables him to detect simultaneously the position of a particle along an axis with a standard
(a) The x-coordinate of an electron is measured with an uncertainty of 0.20 mm. What is the x-component of the electron's velocity, v" if the minimum percentage uncertainty in a simultaneous
An atom in a meta-stable state has a lifetime of 5.2ms. What is the uncertainty in energy of the meta-stable state?
(a) The uncertainty in the y-component of a proton's position is 2.0 x 10-12m. What is the minimum uncertainty in a simultaneous measurement of the y-component of the proton's velocity? (b) The
The ψ (''psi'') particle has a rest energy of 3097MeV (1 MeV = 106eV). The ψ particle is unstable with a lifetime of 7.6 x 10-21s. Estimate the uncertainty in rest energy of the ψ
Particle Lifetime the unstable W+ particle has a rest energy of 80.41GeV (1GeV = 109eV) and an uncertainty in rest energy of 2.06GeV. Estimate the lifetime of the W+ particle.
You want to study a biological specimen by means of a wavelength of 10.0 nm, and you have a choice of using electromagnetic waves or an electron microscope. (a) Calculate the ratio of the energy of a
(a) In an electron microscope, what accelerating voltage is needed to produce electrons with wavelength 0.0600 nm? (b) If protons are used instead of electrons, what accelerating voltage is needed to
Consider a wave function given by ψ (x) = A sin kx, where k = 2π/λ and A is areal constant. (a) For what values of x is there the highest probability of finding the particle
Compute | ψ |2 for ψ = ψ sin wt, where ψ is time independent and w is a real constant. Is this a wave function for a stationary state? Why or why not?
Normalization of the Wave Function Consider a particle moving in one dimension which we shall call the x-axis. (a) What does it mean for the wave function of this particle to be normalized? (b)
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
Consider the complex-valued function f (x, y) = (x - iy)/ (x + iy). Calculate | f |2.
Particle A is described by the wave function ψ (x, y, z). Particle B is described by the wave function ψ (x, y, z) eiφ, where Φ is a real constant. How does the probability of
A particle moving in one dimension (the x-axis) 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
Linear Combinations of Wave Functions Let ψ1 and ψ2 be two solutions of Eq. (39.18) with the same energy E. Show that ψ = Bψ1 + Cψ2 is also a solution with energy E, for any
Let ψ1 and ψ2 be two solutions of Eq. (39.18) with energies E1 and E2, respectively, where E1 ≠ E2 Is ψ = Aψ1 + Bψ2, where A and B are nonzero constants, a solution to
A beam of 40-eV electrons traveling in the +x-direction passes through a slit that is parallel to the y-axis and 5.0µwide. The diffraction pattern is recorded on a screen 2.5 m from the slit. (a)
(a) What is the energy of a photon that has wavelength 0.10µm? (b) Through approximately what potential difference must electrons be accelerated so that they will exhibit wave nature in passing
Electrons go through a single slit 150 nm wide and strike a screen 24.0cm away. You find that at angles of ±20.0o from the center of the diffraction pattern, no electrons hit the screen hut
A beam of electrons is accelerated from rest and then passes through a pair of identical thin slits that are 1.25 nm apart. You observe that the first double-slit interference dark fringe occurs at
A beam of protons and a beam of alpha particles (of mass 6.64 x 10-27 kg and charge +2e) are accelerated from rest through the same potential difference and pass through identical circular holes in a
An electron beam and a photon beam pass through identical slits. On a distant screen, the first dark fringe occurs at the same angle for both of the beams. The electron speeds are much slower than
Coherent light is passed through two narrow slits whose separation is 40.0µm. The second-order bright fringe in the interference pattern is located at an angle of 0.0300 rad. If electrons are used
What is the de Broglie wavelength of a red blood cell, with mass 1.00 x 10-11 g, that is moving with a speed of 0.400cm/s? Do we need to be concerned with the wave nature of the blood cells when we
High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression λ = h/p still holds, but we must use the relativistic expression for
(a) What is the de Broglie wavelength of an electron accelerated from rest through a potential increase of 125 V? (b) What is the de Broglie wavelength of an alpha particle (q = + 2e, m = 6.64 x
Suppose that the uncertainty of position of an electron is equal to the radius of the n = 1 Bohr orbit for hydrogen. Calculate the simultaneous minimum uncertainty of the corresponding momentum
(a) A particle with mass m has kinetic energy equal to three times its rest energy. What is the de Broglie wavelength of this particle?(b) Determine the numerical value of the kinetic energy (in MeV)
Proton Energy in a Nucleus the radii of atomic nuclei are of the order of 5.0 x l0-l5 m. (a) Estimate the minimum uncertainty in the momentum of a proton if it is confined within a nucleus. (b) Take
Electron Energy in a Nucleus the radii of atomic nuclei are of the order of 5.0 x 10-15 m. (a) Estimate the minimum uncertainty in the momentum of an electron if it is confined within a nucleus. (b)
In a TV picture tube the accelerating voltage is 15.0kV and the electron beam passes through an aperture 0.50 mm in diameter to a screen 0.300 m away. (a) Calculate the uncertainty in the component
The neutral pion (π0) is an unstable particle produced in high-energy particle collisions. Its mass is about 264 times that of the electron, and it exists for an average lifetime of 8.4 x 10-17
Quantum Effects in Daily Life a 1.25-mg insect flies through a 4.00-mm-diameter hole in an ordinary window screen. The thickness of the screen is 0.500 mm. (a) What should be the approximate
Doorway Diffraction if your wavelength were 1.0 m, you would undergo considerable diffraction in moving through a doorway. (a) What must your speed be for you to have this wavelength? (Assume
Doorway Diffraction if your wavelength were 1.0 m, you would undergo considerable diffraction in moving through a doorway. (a) What must your speed be for you to have this wavelength? (Assume that
You intend to use an electron microscope to study the structure of some crystals. For accurate resolution, you want the electron wavelength to be 1.00 nm. (a) Are these electrons relativistic?
For x rays with wavelength 0.0300 nm, the m = 1 intensity maximum for a crystal occurs when the angle θ in Fig. 36.23c is 35.8°. At what angle θ does the m = 1 maximum occur when a beam of
You intend to use an electron microscope to study the structure of some crystals. For accurate resolution, you want the electron wavelength to be 1.00 nm. (a) Are these electrons relativistic? How do
Zero-Point Energy Consider a particle with mass m moving in a potential U = ½ kx2, as in a mass-spring system. The total energy of the particle is E = p2/2m + ½ kx2. Assume that p and x are
A particle with mass m moves in a potential U(x) = A |x|, where A is a positive constant. In a simplified picture, quarks have a potential energy of interaction of approximately this form, where x
The discussion in Section 39.5 shows that the wave function ψ = ψe-iwt is a stationary state, where ψ is time independent and w is a real (not complex) constant. Consider the wave
The Time-Dependent Schrodinger Equation (39.18) is the time-independent Schrodinger equation in one dimension. The time-dependent Schrodinger equation is If ?(x) is a solution to Eq. (39.18) with
Time-Dependent Wave Function for a Free Particle One example of a time-dependent wave function is that of a free particle [one for which U(x) = 0 for all x] of energy E and x-component of momentum p.
(a) What is the uncertainty in the ball's horizontal momentum, in a direction perpendicular to that in which it is being thrown, if the student throwing the ball knows that it is located within a
A particle is described by the normalized wave function ψ(x, y, z) = Axe-ax2 e -βy2 e-yz2', where A, a, β, and γ are all real, positive constants. The probability that the
A particle is described by the normalized wave function ψ(x, y, z) = Ae –a(x2+y2+z2), where A and a are real, positive constants. (a) Determine the probability of finding the particle at a
Consider the wave packet defined byLet 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) bas fallen to half its maximum
(a) Using the integral in Problem 39.68, 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
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
You have entered a contest in which the contestants drop a marble with mass 20.0 g from the roof of a building onto a small target 25.0 m below. From uncertainty considerations, what is the typical
Ground-Level Billiards (a) Find the lowest energy level for a particle in a box if the particle is a billiard ball (m = 0.20kg) and the box has a width of 1.5 m, the size of a billiard table. (Assume
A proton is in a box of width L. What must the width of the box be for the ground-level energy to be 5.0 MeV, a typical value for the energy with which the particles in a nucleus are bound? Compare
Find the width L of a one-dimensional box that would correspond to the absolute value of the ground state of a hydrogen atom.
(b) For a box with the width calculated in part (a), what is the ground-state energy? How does this correspond to the ground-state energy of a hydrogen atom? (c) Do you think a one-dimensional box is
A certain atom requires 3.0eV 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.
(a) For which values of x, if any, in the range from 0 to L is the probability of finding the particle zero? (b) For which values of x is the probability highest? (c) In parts (a) and (b) are your
Repeat Exercise 40.6 for the particle in the first excited level.
(a) Show that ψ = A sin kx is a solution to Eq. (40.3) if k = √2mE/h. (b) Explain why this is an acceptable wave function for a particle in a box with rigid walls at x = 0 and x = L only
(a) Repeat Exercise 40.8 for ψ = A cos kx. (b) Explain why this cannot be an acceptable wave function for a particle in a box with rigid walls at x = 0 and x = L no matter what the value of k.
(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box that has a width of 0.125 nm. (b) The electron makes a transition from the n = 1 to n
An electron is in a box of width 3.0 x l0-10 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 =
Show that the time-dependent wave function given by Eq. (40.15) is a solution to the one-dimensional Schrodinger equation, Eq. (40.1).
(a) Show that ψ = A sin kx, where k is a constant, is not a solution of Eq. (40.1) for U = U0 and E < U0. 9 (b) Is this ψ a solution for E > U0?
An electron is moving past the square well shown in Fig. 40.6. 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
An electron is bound in a square well of depth U0 = 6E∞. What is the width of the well if its ground-state energy is 2.00eV?
An electron is bound in a square well of width 1.50 nm and depth U0 = 6E∞. If the electron is initially in the ground level and absorbs a photon, what maximum wavelength can the photon have and
Calculate d2 ψ/dx2 for the wave function of Eq. (40.17), and show that the function is a solution of Eq. (40.16).
Calculate d2 ψ/dx2 for the wave function of Eq. (40.19), and show that for any values of C and D it is a solution of Eq. (40.1) for x < 0 and x < L when U = U0 and E < U0.
A proton is bound in a square well of width 4.0 fm = 4.0 x 10-15 m. The depth of the well is six times the ground-level energy E= of the corresponding infinite well. If the proton makes a transition

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