Questions and Answers of Essential University Physics

Using the table below, make a plot of atomic volume versus atomic number, for the elements from Z = 30 to Z = 59 listed in the table. Comment on the structure of your graph in relation to the
You work for a company that makes red helium?neon lasers widely used in physics experiments. Figure 36.19 shows an energy-level diagram for this laser. An electric current excites helium to a
The ratio of the magnetic moment, in units of the Bohr magneton μB, to the angular momentum, in units of ℏ, is called the g-factor.(a) Show that the classical orbital g-factor for an atomic
An ensemble of one-electron square-well systems of width 1.17 nm all have their electrons in highly excited states. They undergo all possible transitions in dropping toward the ground state, obeying
A selection rule for the infinite square well allows only those transitions in which n changes by an odd number. Suppose an infinite square well of width 0.200 nm contains an electron in the n = 4
Excimer lasers for vision correction generally use a combination of argon and fluorine to form a molecular complex that can exist only in an excited state. Stimulated de-excitation produces 6.42-eV
(a) Verify Equation 36.8 by considering a single-electron atom with nuclear charge Ze instead of e.(b) Calculate the ionization energies for single-electron versions of helium, oxygen, lead, and
Form the radial probability density P2(r) associated with the ψ2s state of Equation 36.7, and find the electron’s most probable radial position. 1 -r/2ao (36.7) 25 2 ao 4V2ma
Show that the maximum number of electrons in an atom’s nth shell is 2n2.
A hydrogen atom is in the 2s state. Find the probability that its electron will be found(a) beyond one Bohr radius(b) beyond 10 Bohr radii.
A hydrogen atom is in an F state.(a) Find the possible values for its total angular momentum.(b) For the state with the greatest angular momentum, find the number of possible values for the component
What’s the most orbital angular momentum that could be added to an atomic electron initially in the 6d state without changing its principal quantum number? What would be the new state?
Repeat Problem 58 for the case of an infinite square well containing N particles, rather than for a harmonic oscillator. Express your answer in terms of the ground-state energy E1.Data from Problem
An infinite square well containing a number of particles is in its lowest possible energy state, which is 19 times the ground-state energy E1 for the same well when it contains a single particle.(a)
A harmonic oscillator potential with natural frequency ω contains N electrons and is in its state of lowest energy. Find expressions for the total energy for(a) N even (b) N odd.
A harmonic oscillator potential with natural frequency ω contains N electrons and is in its state of lowest energy. Find expressions for the energy of the highest-energy electron for(a) N
A harmonic oscillator potential with natural frequency v contains a number of electrons and is in its state of lowest energy. If that energy is 6.5ℏω(a) how many electrons are in the potential
Singly ionized oxygen (so-called O-II) is a prevalent species in the tenuous gas between stars, and O-II emits a doublet spectral line at 372.60 nm and 372.88 nm. Astrophysicists analyze this line to
You’ve acquired a laser for your dental practice. It produces 400-mJ pulses at 2.94 μm wavelength. A patient wonders about the number of photons in each pulse, and where they lie in the EM
Find the probability that the electron in the hydrogen ground state will be found in the radial-distance range r = a0 ±0.1a0.
For hydrogen, fine-structure splitting of the 2p state is only about 50 μeV. What percentage is this difference of the photon energy emitted in the 2p → 1s transition? Your answer shows why it’s
A solid-state laser made from lead–tin selenide has a lasing transition at a wavelength of 30 μm. If its power output is 2.0mW, how many lasing transitions occur each second?
An electron in a highly excited state of hydrogen (n1 >> 1) drops into the state n = n2. Find the lowest value of n2 for which the emitted photon will be in the infrared.
Determine the electronic configuration of copper.
You work for a nanotechnology company developing a new quantum device that operates essentially as a one-dimensional infinite square well of width 2.5 nm. You’re asked to specify the maximum number
A harmonic oscillator potential of natural frequency v contains eight electrons and is in its lowest-energy state.(a) What is its energy?(b) What would the lowest energy be if the electrons were
Suppose you put five electrons into an infinite square well of width L. Find an expression for the minimum energy of this system, consistent with the exclusion principle.
A hydrogen atom is in the 4F5/2 state. Find(a) its energy in units of the ground-state energy,(b) its orbital angular momentum in units of ℏ, (c) the magnitude of its total angular momentum
Differentiate the radial probability density for the hydrogen ground state, and set the result to zero to show that the electron is most likely to be found at one Bohr radius.
Substitute Equation 36.3 into Equation 36.4 and carry out the differentiation to show that you get the first unnumbered equation following Equation 36.4. 4 = Ae rlao (36.3) h? d r2 dr ke? = Ep (36.4)
An electron in hydrogen is in the 5f state. What possible values, in units of ℏ, could a measurement of the orbital angular momentum component on a given axis yield?
A hydrogen atom has energy E = -0.850 eV. Find the maximum possible values for(a) its orbital angular momentum (b) the component of that angular momentum on a given axis.
A hydrogen atom is in an l = 2 state. What are the possible angles its orbital angular momentum vector can make with a given axis?
The maximum possible angular momentum for a hydrogen atom in a certain state is 30√11ℏ.. Find(a) the principal quantum number(b) the energy.
Assuming the Moon’s orbital angular momentum is quantized, estimate its orbital quantum number λ.
Find(a) the energy(b) the magnitude of the orbital angular momentum for an electron in the 5d state of hydrogen.
Determine the principal and orbital quantum numbers for a hydrogen atom whose electron has energy 20.850 eV and orbital angular momentum L = √6ℏ.
The 4p → 3s transition in sodium produces a double spectral line at 330.237 nm and 330.298 nm. What’s the energy splitting of the 4p level?
The 4f → 3p transition in sodium produces a spectral line at 567.0 nm. Find the energy difference between these two levels.
Show that the wavelength l in nm of a photon with energy E in eV is λ = 1240/E.
Write the full electronic structure of bromine.
Write the full electronic structure of scandium.
Use shell notation to characterize rubidium’s outermost electron.
A quantum harmonic oscillator with frequency ω contains 21 electrons. What’s the energy of the highest-energy electron?
An infinite square well contains nine electrons. Find the energy of the highest-energy electron in terms of the ground-state energy E1.
What are the possible j values for a hydrogen atom in the 3D state?
Some very short-lived particles known as delta resonances have spin 3 2. Find(a) the magnitude of their spin angular momentum (b) the number of possible spin states.
Theories of quantum gravity predict a spin-2 particle called the graviton. What would be the magnitude of the graviton’s spin angular momentum?
Verify the value of the Bohr magneton ?B in Equation 36.14 eh M. = ±; 2m (36.14)
Give a symbolic description for the state of the electron in a hydrogen atom with total energy -1.51 eV and orbital angular momentum √6ℏ.
A hydrogen atom is in the 6f state. Find(a) its energy(b) the magnitude of its orbital angular momentum.
What’s the orbital quantum number for an electron whose orbital angular momentum has magnitude L = √30ℏ?
The orbital angular momentum of the electron in a hydrogen atom has magnitude 2.585x10-34 Js. Find its minimum possible energy.
Which of the following is not a possible value for the magnitude of the orbital angular momentum in hydrogen:(a) √12ℏ;(b) √20ℏ;(c) √30ℏ;(d) √40ℏ;(e) √56ℏ?
Find the maximum possible magnitude for the orbital angular momentum of an electron in the n = 7 state of hydrogen.
A group of hydrogen atoms is in the same excited state, and photons with at least 1.5-eV energy are required to ionize these atoms. What’s the quantum number n for the initial excited state?
Using physical constants accurate to four significant figures (see inside front cover), verify the numerical values of the Bohr radius a0 and the hydrogen ground-state energy E1.
Quantum dots, or qdots, are nanoscale crystals of semiconductor material that trap electrons in a potential well closely resembling the three-dimensional square well discussed in Section 35.4.
Quantum dots, or qdots, are nanoscale crystals of semiconductor material that trap electrons in a potential well closely resembling the three-dimensional square well discussed in Section 35.4.
Quantum dots, or qdots, are nanoscale crystals of semiconductor material that trap electrons in a potential well closely resembling the three-dimensional square well discussed in Section 35.4.
Quantum dots, or qdots, are nanoscale crystals of semiconductor material that trap electrons in a potential well closely resembling the three-dimensional square well discussed in Section 35.4.
The wave functions of Problem 58, as well as their derivatives, need to be continuous at x = L if these functions are to represent the quantum state of a particle in the finite square well.(a) Show
The next three problems solve the Schrödinger equation for finite square wells like that shown in Fig. 35.14. It’s convenient to work in dimensionless forms of the particle energy E and well depth
The table below lists the wavelengths emitted as electrons in identical square-well potentials drop from various states n to the ground state. Determine a quantity that, when you plot l against it,
You’re trying to convince a friend that nuclear energy represents a much more concentrated energy source than fossil fuels, whose combustion involves rearranging atomic electrons. For a rough
(a) Using the potential energy U = 1/2mω2x2, develop the Schrödinger equation for the harmonic oscillator.(b) Show by substitution that ψ0(x) = A0e-a^2x^2/2 satisfies your equation, where α2
A particle is in the nth quantum state of an infinite square well.(a) Show that the probability of finding it in the left-hand quarter of the well is (b) Show that for odd n, the probability
A large number of electrons are confined to infinite square wells 1.2 nm wide. They’re undergoing transitions among all possible states. How many visible lines (400 nm to 700 nm) are in the
A 9-W laser beam shines on an ensemble of 1024 electrons, each in the ground state of a one-dimensional infinite square well 0.72 nm wide. The photon energy is just high enough to raise an electron
The generalization of the Schrödinger equation to three dimensions isFor a particle confined to the cubical region 0 ≤ x ≤ L, 0 ≤ y ≤ L, 0 ≤ z ≤ L, show by direct substitution that the
(a) Use Equation 35.8 to draw an energy-level diagram for the first six energy levels of a particle in a cubical box, in terms of h2 /8mL2 ,?(b) give the degeneracy of each. E 8ml? (m? + n} + a?)
A particle of mass m is in a region where its total energy E is less than its potential energy U. Show that the Schr?dinger equation has nonzero solutions of the form Ae?.?Such solutions describe the
Find the probability that a particle in an infinite square well is located in the central one-fourth of the well for the quantum states n = (a) 1, (b) 2, (c) 5, and (d) 20. (e) What’s the classical
A particle detector has a resolution 15% of the width of an infinite square well. What’s the probability that the detector will find a particle in the ground state of the square well if the
In your physical chemistry course, you model hydrogen chloride as a hydrogen atom on a spring; the other end of the spring is attached to a rigid wall (the massive chlorine atom). In order to
Is quantization significant for macromolecules confined to biological cells? To find out, consider a protein of mass 250,000 u confined to a 10μm-diameter cell. Treating this as a particle in a
What’s the probability of finding a particle in the central 80% of an infinite square well, assuming it’s in the ground state?
A laser emits 1.96-eV photons. If this emission is due to electron transitions from the n = 2 to n = 1 states of an infinite square well, what’s the well width?
A particle is in the ground state of an infinite square well. What’s the probability of finding the particle in the left-hand third of the well?
An infinite square well extends from -L/2 to L/2.(a) Find expressions for the normalized wave functions for a particle of mass m in this well, giving separate expressions for even and odd quantum
Sketch the probability density for the n = 2 state of an infinite square well extending from x = 0 to x = L, and determine where the particle is most likely to be found.
Electrons in an ensemble of 0.834-nm-wide square wells are all initially in the n = 4 state.(a) How many different wavelengths of spectral lines could be emitted as the electrons cascade to the
The ground-state energy for an electron in infinite square well A is equal to the energy of the first excited state for an electron in well B. How do the wells’ widths compare?
An electron is in a narrow molecule 4.4 nm long, a situation that approximates a one-dimensional infinite square well. If the electron is in its ground state, what is the maximum wavelength of
Show explicitly that the difference between adjacent energy levels in an infinite square well becomes arbitrarily small compared with the energy of the upper level, in the limit of large quantum
An electron drops from the n = 7 to the n = 6 level of an infinite square well 1.5 nm wide. Find(a) the energy(b) the wavelength of the photon emitted.
An electron is trapped in an infinite square well 25 nm wide. Find the wavelengths of the photons emitted in these transitions: (a) n = 2 to n = 1;(b) n = 20 to n = 19;(c) n = 100 to n = 1.
Suppose ψ1 and ψ2 are solutions of the Schrödinger equation for the same energy E. Show that the linear combination aψ1 + bψ2 is also a solution, where a and b are arbitrary constants.
Find an expression for the normalization constant A for the wave function given by ψ = 0 for |x| > b and ψ = A (b2-x2) for -b ≤x ≤b
An electron is confined to a cubical box. For what box width will a transition from the first excited state to the ground state result in emission of a 950-nm infrared photon?
A very crude model for an atomic nucleus is a cubical box 1 fm on a side. What would be the energy of a gamma ray emitted if a proton in such a nucleus made a transition from its first excited state
If all sides of a cubical box are doubled, what happens to the ground-state energy of a particle in that box?
Your roommate is taking Newtonian physics, while you’ve moved on to quantum mechanics. He claims that QM can’t be right, because he didn’t see any evidence of quantized energy levels in a
The ground-state energy of a harmonic oscillator is 4.0 eV. Find the energy separation between adjacent quantum states.
A harmonic oscillator emits a 1.1-eV photon as it undergoes a transition between adjacent states. Find its classical oscillation frequency f.
Find the ground-state energy for a particle in a harmonic oscillator potential whose classical angular frequency ω is 1.0x1017 s-1.
A quantum harmonic oscillator has ground-state energy 0.14 eV. What would be the system’s classical oscillation frequency f?
An alpha particle (mass 4 u) is trapped in a uranium nucleus with diameter 15 fm. Treating the system as a one-dimensional square well, what would be the minimum energy for the alpha particle?
A 3-g snail crawls at 0.5 mm/s between two rocks 15 cm apart. Treating this system as an infinite square well, determine the approximate quantum number. Does the correspondence principle permit the
A particle is confined to a 1.0-nm-wide infinite square well. If the energy difference between the ground state and the first excited state is 1.13 eV, is the particle an electron or a proton?

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