Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for University Physics Volume 3
Suppose you use an average of 500 k W·h of electric energy per month in your home. (a) How long would 1.00 g of mass converted to electric energy with an efficiency of 38.0% last you? (b) How many homes could be supplied at the 500 k W· h per month rate for one year by the energy from the described mass conversion?(a) A nuclear power plant converts energy from nuclear fission into electricity with an efficiency of 35.0%. How much mass is destroyed in one year to produce a continuous 1000 MW of electric power? (b) Do you think it would be possible to observe this mass loss if the total mass of the fuel is 104 kg?Nuclear-powered rockets were researched for some years before safety concerns paramount. (a) What fraction of a rocket's mass would have to be destroyed to get it into a low Earth orbit, neglecting the decrease in gravity? (Assume an orbital altitude of 250 km, and calculate both the kinetic energy (classical) and the gravitational potential energy needed.) (b) If the ship has a mass of 1.00×105kg (100 tons), what total yield nuclear explosion in tons of TNT is needed?The sun energy at a rate of 3.85×10 26 W by the fusion of hydrogen. About 0.7% of each kilogram of hydrogen goes into the energy generated by the Sun. (a) How many kilograms of hydrogen undergo fusion each second? (b) If the sun is 90.0% hydrogen and half of this can undergo fusion before the sun changes character, how long could it produce energy at its current rate? (c) How many kilograms of mass is the sun losing per second? (d) What fraction of its mass will it have lost in the time found in part (b)?Show that for a particle is invariant under Lorentz transformations.Check Your Understanding The flame of a peach-scented candle has a yellowish color and the flame of a Bunsen’s burner in a chemistry lab has a bluish color. Which flame has a higher temperature?Check Your Understanding An iron poker is being heated. As its temperature rises, the poker begins to glow-first dull red, then bright red, then orange, and then yellow. Use either the blackbody radiation curve or Wien’s law to explain these changes in the color of the glow.Check Your Understanding Suppose that two stars, and . radiate exactly the same total power. If the radius of star is three times that of star , what is the ratio of the surface temperatures of these stars? Which one is hotter?Check Your Understanding A molecule is vibrating at a frequency of 5.010Hz. What is the smallest spacing between its vibrational energy levels?Check Your Understanding Would the result in Example 6.4 be different if the mass were not 1.0 kg g a tiny mass of 1.0 pg, and the amplitude of vibrations were 0.10 m?Check Your Understanding A yellow 589-nm light is incident on a surface whose work function is 1.20 eV. What is the stopping potential? What is the cut-off wavelength?Check Your Understanding Cut-off frequency for the photoelectric effect in some materials is 8.01013 Hz. When the incident light has a frequency of 121014 Hz. the stopping potential is measured as -0.16 V. Estimate a value of Planck’s constant from these data (in units J • s and eV • s ) and determine the percentage error of your estimation.Check Your Understanding An incident 71-pm X-ray is incident on a calcite target. Find the wavelength of the X-ray scattered at a 60° angle. What is the smallest shift that can be expected in this experiment?Check Your Understanding What are the limits of the Lyman series? Can you see these spectral lines?Check Your Understanding When an election in a hydrogen atom is in the first excited state, what prediction does the Bohr model give about its orbital speed and kinetic energy? What is the magnitude of its orbital angular momentum?Check Your Understanding What is de Broglie’s wavelength of a non-relativistic proton with a kinetic energy of 1.0 eV?Check Your Understanding Find the de Broglie wavelength of an electron in the third excited state of hydrogen.Check Your Understanding Find the de Brogue wavelength and kinetic energy of a free electron that travels at a speed of 0.75c.Check Your Understanding For the situation described in Example 6.15, find the angular position of the fifth-order bright fringe on the viewing screen.Check Your Understanding Suppose that the diameter of the aperture in Example 6.16 is halved. How does it affect the resolving power?Which surface has a higher temperature — the surface of a yellow star or that of a red star?Describe what you would see when looking at a body whose temperature is increased from 1000 K to 1,000,000 K.Explain the color changes in a hot body as its temperature is increased.Speculate as to why UV light causes sunburn, whereas visible light does not.Two cavity radiators are constructed with walls made of different metals. At the same temperature, how would their radiation spectra differ?Discuss why some bodies appear black, other bodies appear red, and still other bodies appear white.If everything radiates electromagnetic energy, why can we not see objects at room temperature in a dark room?How much does the power radiated by a blackbody increase when its temperature (in K) is tripled?For the same monochromatic light source, would the photoelectric effect occur for all metals?In the interpretation of the photoelectric effect, how is it known that an electron does not absorb more than one photon?Explain how you can determine the work function from a plot of the stopping potential versus the frequency of the incident radiation in a photoelectric effect experiment. Can you determine the value of Planck’s constant from this plot?Suppose that in the photoelectric-effect experiment we make a plot of the detected current versus the applied potential difference. What information do we obtain from such a plot? Can we determine from it the value of Planck’s constant? Can we determine the work function of the metal?Speculate how increasing the temperature of a photoelectrode affects the outcomes of the photoelectric effect experiment.Which aspects of the photoelectric effect cannot be explained by classical physics?Is the photoelectric effect a consequence of the wave character of radiation or is it a consequence of the particle character of radiation? Explain briefly.The metals sodium, iron, and molybdenum have work functions 25 eV, 3.9 eV, and 4.2 eV, respectively. Which of these metals will emit photoelectrons when illuminated with 400 nm light?Discuss any similarities and differences between the photoelectric and the Compton effects.Which has a greater momentum: an UV photon or an IR photon?Does changing the intensity of a monochromatic light beam affect the momentum of the individual photons in the beam? Does such a change affect the net momentum of the beam?Can the Compton effect occur with visible light? If so, will it be detectable?Is it possible in the Compton experiment to observe scattered X-rays that have a shorter wavelength than the incident X-ray radiation?Show that the Compton wavelength has the dimension of length.At what scattering angle is the wavelength shift in the Compton effect equal to the Compton wavelength?Explain why the patterns of bright emission spectral lines have an identical spectral position to the pattern of dark absorption spectral lines for a given gaseous element.Do the various spectral lines of the hydrogen atom overlap?The Balmer series for hydrogen was discovered before either the Lyman or the Paschen series. Why?When the absorption spectrum of hydrogen at room temperature is analyzed, absorption lines for the Lyman series are found, but none are found for the Balmer series. What does this tell us about the energy state of most hydrogen atoms at room temperature?Hydrogen accounts for about 75% by mass of the matter at the surfaces of most stars. However, the absorption lines of hydrogen are strongest (of highest intensity) in the spectra of stars with a surface temperature of about 9000 K. They are weaker in the sun spectrum and are essentially nonexistent in very hot (temperatures above 25,000 K) or rather cool (temperatures below 3500 K) stars. Speculate as to why surface temperature affects the hydrogen absorption lines that we observe.Discuss the similarities and differences between Thomson’s model of the hydrogen atom and Bohr’s model of the hydrogen atom.Discuss the way in which Thomson’s model is nonphysical. Support your argument with experimental evidence.If, in a hydrogen atom, an electron moves to an orbit with a larger radius, does the energy of the hydrogen atom increase or decrease?How is the energy conserved when an atom makes a transition from a higher to a lower energy state?Suppose an electron in a hydrogen atom makes a transition from the (n+1) th orbit to the nth orbit. Is the wavelength of the emitted photon longer for larger values of n, or for smaller values of n?Discuss why the allowed energies of the hydrogen atom are negative.Can a hydrogen atom absorb a photon whose energy is greater than 13.6 eV?Why can you see through glass but not through wood?Do gravitational forces have a significant effect on atomic energy levels?Show that Planck’s constant has the dimensions of angular momentum.Which type of radiation is most suitable for the observation of diffraction patterns on crystalline solids; radio waves, visible light, or X-rays? Explain.Speculate as to how the diffraction patterns of a typical crystal would be affected if -rays were used instead of X-rays.If an electron and a proton are traveling at the same speed, which one has the shorter de Brogue wavelength?If a particle is accelerating, how does this affect its de Brogue wavelength?Why is the wave-like nature of matter not observed every day for macroscopic objects?What is the wavelength of a neutron at rest? Explain.Why does the setup of Davisson—Germer experiment need to be enclosed in a vacuum chamber? Discuss what result you expect when the chamber is not evacuated.Give an example of an experiment in which light behaves as waves. Give an example of an experiment in which light behaves as a stream of photons.Discuss: How does the interference of water waves differ from the interference of electrons? How are they analogous?Give at least one argument in support of the matter-wave hypothesis.Give at least one argument in support of the particle-nature of radiation.Explain the importance of the Young double-slit experiment.Does the Heisenberg uncertainty principle allow a particle to be at rest in a designated region in space?Can the de Brogue wavelength of a particle be known exactly?Do the photons of red light produce better resolution in a microscope than blue light photons? Explain.Discuss the main difference between an SEM and a TEM.A 200-W heater emits a 1.5-m radiation. (a) What value of the energy quantum does it emit? (b) Assuming that the specific heat of a 4.0-kg body is 0.83kcaI/kg • K, how many of these photons must be absorbed by the body to increase its temperature by 2 K? (c) How long does the heating process in (b) take, assuming that all radiation emitted by the heater gets absorbed by the body?A 900-W microwave generator in an oven generates energy quanta of frequency 2560 MHz. (a) How many energy quanta does it emit per second? (b) How many energy quanta must be absorbed by a pasta dish placed in the radiation cavity to increase its temperature by 45.0 K? Assume that the dish has a mass of 0.5 kg and that its specific heat is 0.9 kcal/kg • K. (c) Assume that all energy quanta emitted by the generator are absorbed by the pasta dish. How long must we wait until the dish in (b) is ready?(a) For what temperature is the peak of blackbody radiation spectrum at 400 nm? (b) If the temperature of a blackbody is 800 K, at what wavelength does it radiate the most energy?The tungsten elements of incandescent light bulbs operate at 3200 K. At what frequency does the filament radiate maximum energy?Interstellar space is filled with radiation of wavelength 970µn. This radiation is considered to be a remnant of the “big bang.” What is the corresponding blackbody temperature of this radiation?The radiant energy from the sun reaches its maximum at a wavelength of about 500.0 nm. What is the approximate temperature of the sun’s surface?A photon has energy 20 keV. What are its frequency and wavelength?The wavelengths of visible light range from approximately 400 to 750 nm. What is the corresponding range of photon energies for visible light?What is the longest wavelength of radiation that can eject a photoelectron from silver? Is it in the visible range?What is the longest wavelength of radiation that can eject a hotoelectron from potassium, given the work function of potassium 2.24 eV? Is it in the visible range?Estimate the binding energy of electrons in magnesium, given that the wavelength of 337 nm is the longest wavelength that a photon may have to eject a photoelectron from magnesium photoelectrode.The work function for potassium is 2.26 eV. What is the cutoff frequency when this metal is used as photoelectrode? What is the stopping potential when for the emitted electrons when this photo electrode is exposed to radiation of frequency 1200 THz?Estimate the work function of aluminum, given that the wavelength of 304 nm is the longest wavelength that a photon may have to eject a photoelectron from aluminum photoelectrode.What is the maximum kinetic energy of photoelectrons ejected from sodium by the incident radiation of wavelength 450 nm?A 120-nm UV radiation illuminates a gold-plated electrode. What is the maximum kinetic energy of the ejected photoelectrons?A 400-nm violet light ejects photoelectrons with a maximum kinetic energy of 0.860 eV from sodium photoelectrode. What is the work function of sodium?A 600-nm light falls on a photoelectric surface and electrons with the maximum kinetic energy of 0.17 eV are emitted. Determine (a) the work function and (b) the cutoff frequency of the surface. (c) What is the stopping potential when the surface is illuminated with light of wavelength 400 nm?The cutoff wavelength for the emission of photoelectrons from a particular surface is 500 nm. Find the maximum kinetic energy of the ejected photoelectrons when the surface is illuminated with light of wavelength 600 nm.Find the wavelength of radiation that can eject 2.00-eV electrons from calcium electrode. The work function for calcium is 2.71 eV. In what range is this radiation?Find the wavelength of radiation that can eject 0.10-eV electrons from potassium electrode. The work function for potassium is 2.24 eV. In what range is this radiation?Find the maximum velocity of photoelectrons ejected by an 80-nm radiation, if the work function of photoelectrode is 4.73 eV.What is the momentum of a 589-nm yellow photon?What is the momentum of a 4-cm microwave photon?In a beam of white light (wavelengths from 400 to 750 nm), what range of momentum can the photons have?What is the energy of a photon whose momentum is 3.01024 kg• m/s ?What is the wavelength of (a) a 12-keV X-ray photon; (b) a 2.O-MeV y -ray photon?Find the momentum and energy of a 1.0-Å photon.Find the wavelength and energy of a photon with momentum 5.001029 kg•m/s.A -ray photon has a momentum of 8.001021 kg • m/s. Find its wavelength and energy.(a) Calculate the momentum of a 2.5-pm photon. (b) Find the velocity of an electron with the same momentum. (c) What is the kinetic energy of the electron, and how does it compare to that of the photon?Show that p=h and Ef=hf are consistent with the relativistic formula E2=p2c2+m02c2 .Show that the energy E in eV of a photon is given by E=1.241106 m/A. where A is its wavelength in meters.For collisions with free electrons, compare the Compton shift of a photon scattered as an angle of 30° to that of a photon scattered at 45CX-rays of wavelength 12.3 pm are scattered from a block of carbon. What are the wavelengths of photons scattered at (a) 30°; (b) 90°; and, (c) 180° ?Calculate the wavelength of the first line in the Lyman series and show that this line lies in the ultraviolet part of the spectrum.Calculate the wavelength of the fifth line in the Lyman series and show that this line lies in the ultraviolet part of the spectrum.Calculate the energy changes corresponding to the transitions of the hydrogen atom: (a) from n=3 to n=4 ; (b) from n=2 to n=1 ; and (c) from n=3 to n= .Determine the wavelength of the third Balmer line (transition from n=5 to n=2 ).What is the frequency of the photon absorbed when the hydrogen atom makes the transition from the ground state to the n=4 state?When a hydrogen atom is in its ground state, what are the shortest and longest wavelengths of the photons it can absorb without being ionized?When a hydrogen atom is in its third excided state, what are the shortest and longest wavelengths of the photons it can emit?What is the longest wavelength that light can have if it is to be capable of ionizing the hydrogen atom in its ground state?For an electron in a hydrogen atom in the n=2 state, compute: (a) the angular momentum; (b) the kinetic energy; (c) the potential energy; and (d) the total energy.Find the ionization energy of a hydrogen atom in the fourth energy state.It has been measured that it required 0.850 eV to remove an electron from the hydrogen atom. In what state was the atom before the ionization happened?What is the radius of a hydrogen atom when the electron is in the first excited state?Find the shortest wavelength in the Balmer series. In what part of the spectrum does this line lie?Show that the entire Paschen series lies in the infrared part of the spectrum.Do the Balmer series and the Lyman series overlap? Why? Why not? (Hint: calculate the shortest Balmer line and the longest Lyman line.)(a) Which line in the Balmer series is the first one in the UV part of the spectrum? (b) How many Balmer lines lie in the visible part of the spectrum? (c) How many Balmer lines lie in the UV?A 4.653-urn emission line of atomic hydrogen corresponds to transition between the states nf=5 and ni. Find ni.At what velocity will an electron have a wavelength of 1.00 m?What is the de Brogue wavelength of an electron travelling at a speed of 5.0106 m/s ?What is the de Brogue wavelength of an electron that is accelerated from rest through a potential difference of 20 keV?What is the de Brogue wavelength of a proton whose kinetic energy is 2.0 MeV? 10.0 MeV?What is the de Brogue wavelength of a 10-kg football player running at a speed of 8.0 m/s?(a) What is the energy of an electron whose de Brogue wavelength is that of a photon of yellow light with wavelength 590 nm? (b) What is the de Brogue wavelength of an electron whose energy is that of the photon of yellow light?The de Brogue wavelength of a neutron is 0.01 nm. What is the speed and energy of this neutron?What is the wavelength of an electron that is moving at a 3% of the speed of light?At what velocity does a proton have a 6.0-fm wavelength (about the size of a nucleus)? Give your answer in units of c.What is the velocity of a 0.400-kg billiard ball if its wavelength is 7.50 fm?Find the wavelength of a proton that is moving at 1.00% of the speed of light (when =0.01).An AM radio transmitter radiates 500 kW at a frequency of 760 kHz. How many photons per second does the emitter emit?Find the Lorentz factor and de Brogue’s wavelength for a 50-GeV electron in a particle accelerator.Find the Lorentz factor and de Brogue’s wavelength for a 1.O-TeV proton in a particle accelerator.What is the kinetic energy of a 0.01-nm electron in a TEM?If electron is to be diffracted significantly by a crystal, its wavelength must be about equal to the spacing, d, of crystalline planes. Assuming d=0.250 nm, estimate the potential difference through which an electron must be accelerated from rest if it is to be diffracted by these planes.X-rays form ionizing radiation that is dangerous to living tissue and undetectable to the human eye. Suppose that a student researcher working in an X-ray diffraction laboratory is accidentally exposed to a fatal dose of radiation. Calculate the temperature increase of the researcher under the following conditions: the energy of X-ray photons is 200 keV and the researcher absorbs 41013 photons per each kilogram of body weight during the exposure. Assume that the specific heat of the student’s body is O.83kcal/kg K.Solar wind (radiation) that is incident on the top of Earth’s atmosphere has an average intensity of 1.3kW/ m2. Suppose that you are building a solar sail that is to propel a small toy spaceship with a mass of 0.1 kg in the space between the International Space Station and the moon. The sail is made from a very light material, which perfectly reflects the incident radiation. To assess whether such a project is feasible, answer the following questions, assuming that radiation photons are incident only in normal direction to the sail reflecting surface. (a) What is the radiation pressure (force per m2) of the radiation falling on the mirror-like sail? (b) Given the radiation pressure computed in (a), what will be the acceleration of the spaceship when the sail has of an area of 10.0 m2 ? (c) Given the acceleration estimate in (b), how fast will the spaceship be moving after 24 hours when it starts from rest?Treat the human body as a blackbody and determine the percentage increase in the total power of its radiation when its temperature increases from 98.6 °F to 103 ° F.Show that Wien’s displacement law results from Planck’s radiation law. (Him: substitute x=hckT and write Planck’s law in the form I(x,T)=Ax5(ex1) , where A=2( kT)5(h4c3). Now, for fixed T, find the position of the maximum in I(x,T) by solving for x in the equation dI(x,T)dx=0.Show that Stefan’s law results from Planck’s radiation law. Hin: To compute the total power of blackbody radiation emitted across the entire spectrum of wavelengths at a given temperature, integrate Planck’s law over the entire spectrum P(T)=0I(,T)d. Use the substitution x=hckT and the tabulated value of the integral 0dx x 3( e x 1)=415Determine the power intensity of radiation per unit wavelength emitted at a wavelength of 500.0 nm by a blackbody at a temperature of 10,000 K.The HC1 molecule oscillates at a frequency of 87.0 THz. What is the difference (in eV) between its adjacent energy levels?A quantum mechanical oscillator vibrates at a frequency of 250.0 THz. What is the minimum energy of radiation it can emit?In about 5 billion years, the sun will evolve to a red giant. Assume that its surface temperature will decrease to about half its present value of 6000 K, while its present radius of 7.0108 m will increase to 1.51011 m (which is the current Earth-sun distance). Calculate the ratio of the total power emitted by the sun in its red giant stage to its present power.A sodium lamp emits 2.0 W of radiant energy, most of which has a wavelength of about 589 nm. Estimate the number of photons emitted per second by the lamp.Photoelectrons are ejected from a photo electrode and are detected at a distance of 2.50 cm away from the photoelectrical. The work function of the photo electrode is 2.71 eV and the incident radiation has a wavelength of 420 nm. How long does it take a photoelectron to travel to the detector?If the work function of a metal is 3.2 eV, what is the maximum wavelength that a photon can have to eject a photoelectron from this metal surface?The work function of a photoelectric surface is 2.00 eV. What is the maximum speed of the photoelectrons emitted from this surface when a 450-nm light falls on it?A 400-nm laser beam is projected onto a calcium electrode. The power of the laser beam is 2.00 mW and the work function of calcium is 2.31 eV. (a) How many photoelectrons per second are ejected? (b) What net power is carried away by photoelectrons?(a) Calculate the number of photoelectrons per second that are ejected from a 1.00-mm2 area of sodium metal by a 500-nm radiation with intensity I .30kW/m2 (the intensity of sunlight above Earth’s atmosphere). (b) Given the work function of the metal as 2.28 eV, what power is carried away by these photoelectrons?A laser with a power output of 2.00 mW at a 400-nm wavelength is used to project a beam of light onto a calcium photoelectrode. (a) How many photoelectrons leave the calcium surface per second? (b) What power is carried away by ejected photoelectrons, given that the work function of calcium is 2.31 eV? (c) Calculate the photocurrent. (d) If the photoelectrode suddenly becomes electrically insulated and the setup of two electrodes in the circuit suddenly starts to act like a 2.00-pF capacitor, how long will current flow before the capacitor voltage stops it?The work function for barium is 2.48 eV. Find the maximum kinetic energy of the ejected photoelectrons when the barium surface is illuminated with: (a) radiation emitted by a 100-kW radio station broadcasting at 800 kHz; (b) a 633-nm laser light emitted from a powerful HeN e laser; and (c) a 434-nm blue light emitted by a small hydrogen gas discharge rube.(a) Calculate the wavelength of a photon that has the same momentum as a proton moving with 1% of the speed of light in a vacuum. (b) What is the energy of this photon in MeV? (c) What is the kinetic energy of the proton in MeV?(a) Find the momentum of a 100-keV X-ray photon. (b) Find the velocity of a neutron with the same momentum. (c) What is the neutron’s kinetic energy in eV?The momentum of light, as it is for particles, is exactly reversed when a photon is reflected straight back from a mirror, assuming negligible recoil of the mirror. The change in momentum is twice the photon’s incident momentum, as it is for the particles. Suppose that a beam of light has an intensity 1.0kW/m2 and falls on a -2.0-m2 area of a minor and reflects from it. (a) Calculate the energy reflected in 1.00 s. (b) What is the momentum imparted to the mirror? (c) Use Newton’s second law to find the force on the mirror. (d) Does the assumption of no-recoil for the mirror seem reasonable?A photon of energy 5.0 keV collides with a stationary electron and is scattered at an angle of 60°. What is the energy acquired by the electron in the collision?A 0.75-nm photon is scattered by a stationary electron. The speed of the electron’s recoil is 1.5106 m/s. (a) Find the wavelength shift of the photon. (b) Find the scattering angle of the photon.Find the maximum change in X-ray wavelength that can occur due to Compton scattering. Does this change depend on the wavelength of the incident beam?A photon of wavelength 700 nm is incident on a hydrogen atom. When this photon is absorbed, the atom becomes ionized. What is the lowest possible orbit that the electron could have occupied before being ionized?What is the maximum kinetic energy of an electron such that a collision between the electron and a stationary hydrogen atom in its ground state is definitely elastic?Singly ionized atomic helium He +1 is a hydrogen-like ion. (a) What is its ground-state radius? (b) Calculate the energies of its four lowest energy states. (c) Repeat the calculations for the Li2+ - ion.A triply ionized atom of betyllium Be3+ is a hydrogen-like ion. When Be3+ is in one of its excited states, its radius in this nth state is exactly the same as the radius of the first Bohr orbit of hydrogen. Find n and compute the ionization energy for this state of Be3+ .In extreme-temperature environments, such as those existing in a solar corona, atoms may be ionized by undergoing collisions with other atoms. One example of such ionization in the solar corona is the presence of C5+ ions, detected in the Fraunhofer spectrum. (a) By what factor do the energies of the C5+ ion scale compare to the energy spectrum of a hydrogen atom? (b) What is the wavelength of the first line in the Paschen series of C5+ ? (c) In what part of the spectrum are these lines located?(a) Calculate the ionization energy for He+. (b) What is the minimum frequency of a photon capable of ionizing He+?Experiments are performed with ultracold neutrons having velocities as small as 1.00 m/s. Find the wavelength of such an ultracold neutron and its kinetic energy.Find the velocity and kinetic energy of a 6.0-fm neutron. (Rest mass energy of neutron is E0=940 MeV.)The spacing between crystalline planes in the NaC1 crystal is 0.281 nm, as determined by X-ray diffraction with X-rays of wavelength 0.170 nm. What is the energy of neutrons in the neutron beam that produces diffraction peaks at the same locations as the peaks obtained with the X-rays?What is the wavelength of an electron accelerated from rest in a 30.0-kV potential difference?Calculate the velocity of a 1.0-m electron and a potential difference used to accelerate it from rest to this velocity.In a supercollider at CERN, protons are accelerated to velocities of 0.25c. What are their wavelengths at this speed? What are their kinetic energies? If a beam of protons were to gain its kinetic energy in only one pass through a potential difference, how high would this potential difference have to be? (Rest mass energy of a proton is E0=938 MeV).Find the de Brogue wavelength of an electron accelerated from rest in an X-ray rube in the potential difference of 100 keV. (Rest mass energy of an electron is E0=511 keV.)The cutoff wavelength for the emission of photoelectrons from a particular surface is 500 nm. Find the maximum kinetic energy of the ejected photoelectrons when the surface is illuminated with light of wavelength 450 nm.Compare the wavelength shift of a photon scattered by a free electron to that of a photon scattered at the same angle by a free proton.The spectrometer used to measure the wavelengths of the scattered X-rays in the Compton experiment is accurate to 5.0104 nm. What is the minimum scattering angle for which the X-rays interacting with the free electrons can be distinguished from those interacting with the atoms?Consider a hydrogen-like ion where an electron is orbiting a nucleus that has charge q=+Ze. Derive the formulas for the energy En of the electron in nth orbit and the orbital radius rn.Assume that a hydrogen atom exists in the n=2 excited state for 108 s before decaying to the ground state. How many times does the electron orbit the proton nucleus during this time? How long does it take Earth to orbit the sun this many times?An atom can be formed when a negative muon is captured by a proton. The muon has the same charge as the electron and a mass 207 times that of the electron. Calculate the frequency of the photon emitted when this atom makes the transition from n=2 to the n=1 state. Assume that the muon is orbiting a stationary proton.Check Your Understanding If a=3+4i , what is the product a* a?Check Your Understanding Suppose that a particle with energy E is moving along the x-axis and is in the region O and L. One possible wave function is (x,t)={AeiEt/hsinxL, when 0xL otherwise Determine the normalization constant.Check Your Understanding For the particle in the example, find the probability Of locating it 0 and L/4.Check Your Understanding A sodium atom nukes a transition from the first excited state the wound state, emitting a 589.0-nm photon with energy 2.105 eV. If the lifetime of this excited state is 16108s, what is the uncertainty in energy of this state? What is width of the corresponding line?Check Your Understanding A particle With mass m is moving along the x-axis in a given by the potential energy function U(x)=0.5m2x2. Compute the (x,t)*U(x)(x,t). Express your answer in terrns of the time-independent wave function, (x).Check Your Understanding Which of the following wave functions is valid wave-function solution for Schr0dinger’s equation?Check your Understanding (a) Consider an infinite square well with wall boundaries x=0 and x=L. What is the of finding a quantum panicle in its state somewhere between x=0 and x=L/4? (b) Repeat question (a) for a classical panicle.Check Your Understanding The vibrational frequency of the hydrogen iodide HI diatomic molecule is 6.691013Hz. (a) What is the force constant of the molecular bond between the hydrogen and the Iodine atoms? (b) What is the energy of the emitted photon when this molecule makes a transition between adjacent vibration energy levels?Check Your Understanding Find the expectation value of the position for a particle in the ground state of a harmonic oscillator using symmetry.Check Your Understanding A proton with kinetic energy 1.00 eV is incident on a square potential barrier with height 10.00 eV. If the proton is to have the same transmission probability as an electron of the same energy, what must the width of the barrier be relative to the barrier width encountered by an electron?What is the physical unit of a wave function, (x,t)? What is the physical unit of the square of this wave function?Can the magnitude of a wave function (*(x,t)(x,t)) be a negative number? Explain.What kind of physical quamtity does a wave function of an electron represent?What is the physical meaning of a wave function of a particle?What is the meaning of the expression "expectation value?" Explain.If the formalism of quantum mechanics is 'more exact' than that of classical mechanics, why don't we use quantum mechanics to describe the motion of a leaping frog? Explain.Can the de Broglie wavelength of a particle be known precisely? Can the position of a particle be known precisely?Can we measure the energy of a free localized particle with complete precision?Can we measure both the position and momentum of a particle with complete precision?What is the difference between a wave function (x,y,z) and a wave function (x,y,z,t) for the same particle?If a quantum particle is in a stationary state, does it mean that it does not move?Explain the difference between time-dependent and independent SchrÖdinger's equations.Suppose a wave function is discontinuous at some point. Can this function represent a quantum state of some physical particle? Why? Why not?Using the quantum particle in a box model, describe how the possible energies of the particle are related to the size of the box.Is it possible that when we measure the energy of a quantum particle in a box, the measurement may return a smaller value than the ground state energy? What is the highest value of the energy that we can measure for this particle?For a quantum panicle in a box, the first excited state (2) has zero value at the midpoint position in the box, so that the probability density of finding a particle at this point is exactly zero. Explain what is wrong with the following reasoning: "If the probability of finding a quantum particle at the midpoint is zero, the particle is never at this point, right? How does it come then that the particle can cross this point on its way from the left side to the right side of the box?Is it possible to measure energy of 0.75h for a quantum harmonic oscillator? Why? Why not? Explain.Explain the connection between Planck's hypothesis of energy quanta and the energies of the quantum harmonic oscillator.If a classical harmonic oscillator can at rest, why can the quantum harmonic oscillator never be at rest? Does this violate Bohr 's correspondence principle?Use an example of a quantum particle in a box or a quantum oscillator to explain the physical meaning of Bohr's correspondence principle.Can we simultaneously measure position and energy of a quantum oscillator? Why? Why not?When an electron and a proton of the same kinetic energy encounter a barrier of the same height and width, which one of them will tunnel through the barrier more easily? Why?What decreases the tunneling probability most: doubling the barrier width or halving the kinetic energy of the incident particle?Explain the difference between a box-potential and a of a quantum dot.Can a quantum particle 'escape' from an infinite potential well like that in a box? Why? Why not?A tunnel diode and a resonant-tunneling diode both utilize the same physics principle of quantum tunneling. In what important way are they different?Compute |(x,t)|2 for the function (x,t)=(x)sint, where is a real constant.Given the complex-valued function f(x,y)=(xiy)/(x+iy), calculate |f(x,y)|2.Which one of the following functions, and why, qualifies to be a wave function of a particle that can move along the entire real axis? (x)=Aex2; (x)=Aex; (x)=Atanx; (x)=A(sinx)/x; (x)=Ae|x|A particle with mass m moving along the x-axis and its quantum state is represented by the following wave function: (x,t)={0,x0Ax e ax e iEt/h,x0, where a a=2.01010m1. (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval 0xL. (c) Find the expectation value of position. (d) Find the expectation value of kinetic energy.A wave function of a particle with mass m is given by (x)={Acosax, 2ax+ 2a;0, otherwise where a =1.001010/m. (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval 0x0.51010m. (c) Find the particle's average position. (d) Find its average momentum. Find its average kinetic energy 0.51010mx+0.510-10m.A velocity measurement of an a-particle has been performed with a precision of 0.02 mm/s. What is the minimum uncertainty in its position?A gas of helium atoms at 273 K is in a cubical container with 25.0 cm on a side. (a) What is the minimum uncertainty in momentum components of helium atoms? (b) What is the minimum uncertainty in velocity components? (c) Find the ratio of the uncertainties in (b) to the mean speed of an atom in each direction.If the uncertainty in the y -component of a proton's position is 2.0 pm, find the minimum uncertainty in the simultaneous measurement of the proton's y -component of velocity. What is the minimum uncertainty in the simultaneous measurement of the proton's x -component of velocity?Some unstable elementary particle has a rest energy of 80.41 GeV and an uncertainty in rest energy of 2.06 GeV. Estimate the lifetime of this particle.An atom in a metastable state has a lifetime of 5.2 ms. Find the minimum uncertainty in the measurement of energy of the excited state.Measurements indicate that an atom remains in an excited state for an average time of 50.0 ns before making a transition to the ground state with the simultaneous emission of a 2.1-eV photon. (a) Estimate the uncertainty in the frequency of the photon. (b) What fraction of the photon's average frequency is this?Suppose an electron is confined to a region of length 0.1 nm (of the order of the size of a hydrogen atom) and its kinetic energy is equal to the ground state energy of the hydrogen atom in Bohr's model (13.6 eV). (a) What is the minimum uncertainty of its momentum? What fraction of its momentum is it? (b) What would the uncertainty in kinetic energy of this electron be if its momentum were equal to your answer in part (a)? What fraction of its kinetic energy is it?Combine Equation 7.17 and Equation 7.18 to show k2=2c2. 2Ex2=1c22Et2 E(x,t)=Asin(kxt),Show that (x,t)=Aei(kwt) is a valid solution to Schrödinger's time-dependent equation.Show that (x,t)=Asin(kxt) and (x,t)=Acos(kxt) do not obey Schrödinger's time-dependent equation.Show that when 1(x,t) and 2(x,t) are solutions to the time-dependent Schrödinger equation and A, B are numbers then a function (x,t) that is a superposition of these functions is also a solution: (x,t)=A1(x,t)+B1(x,t).A particle with mass m is described by the following wave function: (x)=Acoskx+Bsinkx, where A, B, and k are constants. Assuming that the particle is free, show that this function is the solution of the stationary SchrÖdinger equation for this particle and find the energy that the particle has in this state.Find the expectation value of the kinetic energy for the particle in the state, (x,t)=Aei(kxt). What conclusion can you draw from your solution?Find the expectation value of the square of the momentum squared for the particle in the state, (x,t)=Aei(kxt). What conclusion can you draw from your solution?A free proton has a wave function given by (x,t)=Aei(5.021011x8.001015t) . The coefficient of x is inverse meters (m1) and the coefficient on t is inverse seconds (s1). Find its momentum and energy.Assume that an electron in an atom can be treated as if it were confined to a box of width 2.0 A . What is the ground state energy of the electron? Compare your result to the ground state kinetic energy of the hydrogen atom in the Bohr's model of the hydrogen atom.Assume that a proton in a nucleus can be treated as if it were confined to a one-dimensional of width 10.0 fm. (a) What are the energies of the proton when it is in the states corresponding to n=1,n=2, and n=3? (b) What are the energies of the photons emitted when the proton makes the transitions from the first and second excited states to the ground state?An electron confined to a box has the ground state energy of 2.5 eV. What is the width of the box?What is the ground state energy (in eV) of a proton confined to a one-dimensional box the size of the uranium nucleus that has a radius of approximately 15.0 fm?What is the ground state energy (in eV) of an a -particle confined to a one-dimensional box the size of the uranium nucleus that has a radius of approximately 15.0 fm?To excite an election in a one-dimensional box from its first excited state to its third excited state requires 20.0 eV. What is the width of the box?An electron confined to a box of width 0.15 nm by infinite potential energy barriers emits a photon when it makes a transition from the first excited state to the ground state. Find the wavelength of the emitted photon.If the energy of the first excited state of the electron in the box is 25.0 eV, what is the width of the box?Suppose an electron confined to a emits photons. The longest wavelength that is registered is 500.0 nm. What is the width of the box?Hydrogen H2 molecules are kept at 300.0 K in a cubical container with a side length of 20.0 cm. Assume that you can treat the molecules as though they were moving in a one-dimensional box. (a) Find the ground state energy of the hydrogen molecule in the container. (b) Assume that the molecule has a thermal energy given by kBT/2 and find the corresponding quantum number n of the quantum state that would correspond to this thermal energy.An electron is confined to a box of width 0.25 nm. (a) Draw an energy-level diagram representing the first five states of the electron. (b) Calculate the wavelengths of the emitted photons when the electron makes transitions between the fourth and the second excited states, between the second excited state and the ground state, and between the third and the second excited states.An electron in a box is in the ground state with energy 2.0 eV. (a) Find the width of the box. (b) How much energy is needed to excite the electron to its first excited state? (c) If the electron makes a transition from an excited state to the ground state with the simultaneous emission of 30.0-eV photon, find the quantum number of the excited state?Show that the two lowest energy states of the simple harmonic oscillator, 0(x) and 1(x) from Equation 7.57, satisfy Equation 7.55. n(x)=Nne2x2/2Hn(x),n=0,1,2,3,.... h2md2(x)dx2+12m2x2(x)=E(x).If the ground state energy of a simple harmonic oscillator is 1.25 eV, what is the frequency of its motion?When a quantum harmonic oscillator makes a transition from the (n+1) state to the n state and emits a 450-nm photon, what is its frequency?Vibrations of the hydrogen molecule H2 can be modeled as a simple harmonic oscillator with the spring constant k=1.13103N/m and mass m=1.671027kg. (a) What is the vibrational frequency of this molecule? (b) What are the energy and the wavelength of the emitted photon when the molecule makes transition between its third and second excited states?A particle with mass 0.030 kg oscillates back-and- forth on a spring with frequency 4.0 Hz. At the equilibrium position, it has a speed of 0.60 m/s. If the particle is in a state of definite energy, find its energy quantum number.Find the expectation value x2 of the square of the position for a quantum harmonic oscillator in the ground state. Note: +dxx2ea x 2=(2a 3/2)1.Determine the expectation value of the potential energy for a quantum harmonic oscillator in the ground state. Use this to calculate the expectation value of the kinetic energy.Verify that given by Equation 7.57 is a solution of Schrödinger's equation for the quantum harmonic oscillator. n(x)=Nne2x2/2Hn(x),n=0,1,2,3,...Estimate the ground state energy of the quantum harmonic oscillator by Heisenberg's uncertainty principle. Start by assuming that the product of the uncertainties x and p is at its minimum. Write p in terms of x and assume that for the ground state xx and pp, then write the ground state energy in terms of x. Finally, find the value of x that minimizes the energy and find the minimum of the energy.A mass of 0.250 kg oscillates on a spring with the force constant 110 N/m. Calculate the ground energy level and the separation between the adjacent energy levels. Express the results in joules and in electron-volts. Are quantum effects important?Show that the wave function in (a) Equation 7.68 satisfies Equation 7.61, and (b) Equation 7.69 satisfies Equation 7.63.A 6.0-eV electron impacts on a barrier with height 11.0 eV. Find the probability of the electron to tunnel through the barrier if the barrier width is (a) 0.80 nm and (b) 0.40 nm.A 5.0-eV electron impacts on a barrier of with 0.60 nm. Find the probability of the electron to tunnel through the barrier if the barrier height is (a) 7.0 eV; (b) 9.0 eV; and (c) 13.0 eV.A 12.0-eV electron encounters a barrier of height 15.0 eV. If the probability of the electron tunneling through the barrier is 2.5 %, find its width.A quantum particle with initial kinetic energy 32.0 ev encounters a square barrier with height 41.0 ev and width 0.25 nm. Find probability that the particle tunnels through this barrier if the particle is (a) an electron and, (b) a proton.A simple model of a radioactive nuclear decay assumes that a-particles are trapped inside a well of nuclear potential that walls are the barriers of a finite width 2.0 fm and height 30.0 MeV. Find the tunneling probability across the potential barrier of the wall for a-particles having kinetic energy (a) 29.0 MeV and (b) 20.0 MeV. The mass of the a -particle is m=6.641027kg.A muon, a quantum particle with a mass approximately 2(X) times that of an electron, is incident on a potential barrier of height 10.0 eV. The kinetic energy of the impacting muon is 5.5 ev and only about 0.10% of the squared amplitude of its incoming wave function filters through the barrier. What is the barrier's width?A grain of sand with mass 1.0 mg and kinetic energy 1.0 J is incident on a potential energy barrier with height 1.00000 J and width 2500 nm. How many grains of sand have to fall on this barrier before, on the average, one passes through?Show that if the uncertainty in the position of a particle is on the order of its de Broglie's wavelength, then the uncertainty in its momentum is on the order of the value of its momentum.The mass of a -meson is measured to be 770MeV/c2 with an uncertainty of 100MeV/c2. Estimate the lifetime of this meson.A particle of mass m is confined to a box of width L. If the particle is in the first excited state, what are the probabilities of finding the particle in a region of width0.020 L around the given point x: (a) x=0.25L; (b) x=040L; (c) 0.75L and (d) x=0.90L.A particle in a box [0; L] is in the third excited state. What are its most probable positions?A 0.20-kg billiard ball bounces back and forth without losing its energy between the cushions of a 1.5 m long table. (a) If the ball is in its ground state, how many years does it need to get from one cushion to the other? You may compare this time interval to the age of the universe. (b) How much energy is required to make the ball go from its ground state to its first excited state? Compare it with the kinetic energy of the ball moving at 2.0 m/s.Find the expectation value of the position squared when the particle in the box is in its third excited state and the length of the box is L.Consider an infinite square well with wall boundaries x=0 and x=L. Show that the function (x)=Asinkx is the solution to the stationary Schrödinger equation for the particle in a box only if k=2mE/h. Explain why this is an acceptable wave function only if k is an integer multiple of /L.Consider an infinite square well with wall boundaries x=0 and x=L. Explain why the function (x)=Acoskx is not a solution to the stationary Schrödinger equation for the particle in a box.Atoms in a crystal lattice vibrate in simple harmonic motion. Assuming a lattice atom has a mass of 9.41026kg, what is the force constant of the lattice if a lattice atom makes a transition from the ground state to first excited state when it absorbs a 525m photon?A diatomic molecule behaves like a quantum harmonic oscillator with the force constant 12.0 N/m and mass 5.601026kg. (a) What is the wavelength of the emitted photon when the molecule makes the transition from the third excited state to the second excited state? (b) Find the ground state energy of vibrations for this diatomic molecule.An electron with kinetic energy 2.0 MeV encounters a potential energy barrier of height 16.0 MeV and width 2.00 nm. What is the probability that the electron emerges on the other side of the barrier?A beam of mono-energetic protons with energy 2.0 MeV falls on a potential energy barrier of height 20.0 MeV and of width 1.5 fm. What percentage of the beam is transmitted through the barrier?An electron in a long, organic molecule used in a dye laser behaves approximately like a quantum particle in a box with width 4.18 nm. Find the emitted photon when the electron makes a transition from the first excited state to the ground state and from the second excited state to the first excited state.In STM, an elevation of the tip above the surface being scanned can be determined with a great precision, because the tunneling-electron current between surface atoms and the atoms of the tip is extremely sensitive to the variation of the separation gap between them from point to point along the surface. Assuming that the tunneling-electron current is in direct proportion to the tunneling probability and that the tunneling probability is to a gotxi approximation expressed by the exponential function e2L with =10.0/nm , determine the ratio of the tunneling current when the tip is 0.500 nm above the surface to the current when the tip is 0.515 nm above the surface.If STM is to detect surface features with local heights of about 0.0200 nm, what percent change in tunneling-electron current must the STM electronics be able to detect? Assume that the tunneling-electron current has characteristics given in the preceding problem.Use Heisenberg's uncertainty principle to estimate the ground state energy of a particle oscillating on an spring with angular frequency, =k/m, where k is the spring constant and m is the mass.Suppose an infinite square well extends from L/2 to +L/2 . Solve the time-independent Schrödinger's equation to find the allowed energies and stationary states of a particle with mass m that is confined to this well. Then show that these solutions can be obtained by making the coordinate transformation x=xL/2 for the solutions obtained for the well extending between 0 and L.A particle of mass m confined to a box of width L is in its first excited state 2(x). (a) Find its average position (which is the expectation value of the position). (b) Where is the particle most likely to found?Check Your Understanding Can the magnitude of Lzever be equal to L?Check Your Understanding If the Stem-Gerlach experiment yielded four distinct bands instead of two, what might be concluded about the spin quantum number of the charged particle?Check Your Understanding X-rays are produced by bombarding a metal target with high-energy electrons. If the target is replaced by another with two times the atomic number, what happens to the frequency of X-rays?Identify the physical significance of each of the quantum numbers of the hydrogen atom.Describe the ground state of hydrogen in terms of wave function, probability density, and atomic orbitals.Distinguish between Bohr’s and Schiödinger’s model of the hydrogen atom. In particular, compare the energy and orbital angular momentum of the ground states.Explain why spectral lines of the hydrogen atom are split by an external magnetic field. What determines the number and spacing of these lines?A hydrogen atom is placed in a magnetic field. Which of the following quantities are affected? (a) total energy; (h) angular momentum; (c) z-component of angular momentum; (d) polar angle.On what factors does the orbital magnetic dipole moment of an electron depend?Explain how a hydrogen atom in the ground state (l = 0) can interact magnetically with an external magnetic field.Compare orbital angular momentum with spin angular momentum of an electron in the hydrogen atom.List all the possible values of s and msfor an electron. Are there particles for which these values are different?Are the angular momentum vectors L necessarily S aligned?