Chapter 39

What is the ground-state energy of (a) an electron and (b) a proton if each is trapped in a one-dimensional infinite potential well that is \(200 \mathrm{pm}\) wide?

The ground-state energy of an electron trapped in a onedimensional infinite potential well is \(2.6 \mathrm{eV}\). What will this quantity be if the width of the potential well is doubled?

An electron, trapped in a one-dimensional infinite potential well \(250 \mathrm{pm}\) wide, is in its ground state. How much energy must it absorb if it is to jump up to the state with \(n=4 ?\)

What must be the width of a one-dimensional infinite potential well if an electron trapped in it in the \(n=3\) state is to have an energy of \(4.7 \mathrm{eV} ?\)

a proton is confined to a one-dimensional infinite potential well \(100 \mathrm{pm}\) wide. What is its ground-state energy?

Consider an atomic nucleus to be equivalent to a onedimensional infinite potential well with \(L=1.4 \times 10^{-14} \mathrm{~m},\) a typical nuclear diameter. What would be the ground-state energy of an electron if it were trapped in such a potential well? (Note: Nuclei do not contain electrons.)

Suppose that an electron trapped in a one-dimensional infinite well of width \(250 \mathrm{pm}\) is excited from its first excited state to its third excited state. (a) What energy must be transferred to the electron for this quantum jump? The electron then de-excites back to its ground state by emitting light. In the various possible ways it can do this, what are the (b) shortest, (c) second shortest, (d) longest, and (e) second longest wavelengths that can be emitted? (f) Show the various possible ways on an energy-level diagram. If light of wavelength \(29.4 \mathrm{nm}\) happens to be emitted, what are the \((\mathrm{g})\) longest and (h) shortest wavelength that can be emitted afterwards?

An electron is trapped in a one-dimensional infinite potential well. For what (a) higher quantum number and (b) lower quantum number is the corresponding energy difference equal to the energy difference \(\Delta E_{43}\) between the levels \(n=4\) and \(n=3 ?\) (c) Show that no pair of adjacent levels has an energy difference equal to \(2 \Delta E_{43}\).

An electron is trapped in a one-dimensional infinite potential well. For what (a) higher quantum number and (b) lower quantum number is the corresponding energy difference equal to the energy of the \(n=5\) level? (c) Show that no pair of adjacent levels has an energy difference equal to the energy of the \(n=6\) level.

An electron is trapped in a one-dimensional infinite well of width \(250 \mathrm{pm}\) and is in its ground state. What are the (a) longest, (b) second longest, and (c) third longest wavelengths of light that can excite the electron from the ground state via a single photon absorption?

A one-dimensional infinite well of length \(200 \mathrm{pm}\) con- tains an electron in its third excited state. We position an electrondetector probe of width \(2.00 \mathrm{pm}\) so that it is centered on a point of maximum probability density. (a) What is the probability of detection by the probe? (b) If we insert the probe as described 1000 times, how many times should we expect the electron to materialize on the end of the probe (and thus be detected)?

An electron is in a certain energy state in a one-dimensional, infinite potential well from \(x=0\) to \(x=L=200 \mathrm{pm} .\) The electron's probability density is zero at \(x=0.300 L,\) and \(x=0.400 L ;\) it is not zero at intermediate values of \(x\). The electron then jumps to the next lower energy level by emitting light. What is the change in the electron's energy?

An electron is trapped in a one-dimensional infinite potential well that is \(100 \mathrm{pm}\) wide; the electron is in its ground state. What is the probability that you can detect the electron in an interval of width \(\Delta x=5.0 \mathrm{pm}\) centered at \(x=(\mathrm{a}) 25 \mathrm{pm},\) (b) \(50 \mathrm{pm}\) and (c) \(90 \mathrm{pm} ?\)

A rectangular corral of widths \(L_{x}=L\) and \(L_{y}=2 L\) holds an electron. What multiple of \(h^{2} / 8 m L^{2}\) where \(m\) is the electron mass, gives (a) the energy of the electron's ground state, (b) the energy of its first excited state, (c) the energy of its lowest degenerate states, and (d) the difference between the energies of its second and third excited states?

An electron (mass \(m\) ) is contained in a rectelectron. What multiple of \(h^{2} / 8 m L^{2}\), where \(m\) is the electron mass, is (a) the energy of the electron's ground state, (b) the energy of its second excited state, and (c) the difference between the energies of its second and third excited states? How many degenerate states have the energy of (d) the first excited state and (e) the fifth excited state?

A cubical box of widths \(L_{x}=L_{y}=L_{z}=L\) contains an electron. What multiple of \(h^{2} / 8 m L^{2}\), where \(m\) is the electron mass, is (a) the energy of the electron's ground state, (b) the energy of its second excited state, and (c) the difference between the energies of its second and third excited states? How many degenerate states have the energy of (d) the first excited state and (e) the fifth excited state?

An electron is in the ground state in a two-dimensional, square, infinite potential well with edge lengths \(L .\) We will probe for it in a square of area \(400 \mathrm{pm}^{2}\) that is centered at \(x=L / 8\) and \(y=L / 8 .\) The probability of detection turns out to be \(4.5 \times 10^{-8}\) What is edge length \(L ?\)

What is the ratio of the shortest wavelength of the Balmer series to the shortest wavelength of the Lyman series?

An atom (not a hydrogen atom) absorbs a photon whose associated wavelength is \(375 \mathrm{nm}\) and then immediately emits a photon whose associated wavelength is \(580 \mathrm{nm}\). How much net energy is absorbed by the atom in this process?

What are the (a) energy, (b) magnitude of the momentum, and (c) wavelength of the photon emitted when a hydrogen atom undergoes a transition from a state with \(n=3\) to a state with \(n=1 ?\)

Calculate the radial probability density \(P(r)\) for the hydrogen atom in its ground state at (a) \(r=0,\) (b) \(r=a,\) and (c) \(r=2 a,\) where \(a\) is the Bohr radius.

A neutron with a kinetic energy of \(6.0 \mathrm{eV}\) collides with a stationary hydrogen atom in its ground state. Explain why the collision must be elastic - that is, why kinetic energy must be conserved. (Hint: Show that the hydrogen atom cannot be excited as a result of the collision.)

An atom (not a hydrogen atom) absorbs a photon whose associated frequency is \(6.2 \times 10^{14} \mathrm{~Hz}\). By what amount does the energy of the atom increase?

What are the (a) wavelength range and (b) frequency range of the Lyman series? What are the (c) wavelength range and (d) frequency range of the Balmer series?

A hydrogen atom, initially at rest in the \(n=4\) quantum state, undergoes a transition to the ground state, emitting a photon in the process. What is the speed of the recoiling hydrogen atom?

In the ground state of the hydrogen atom, the electron has a total energy of \(-13.6 \mathrm{eV}\). What are (a) its kinetic energy and (b) its potential energy if the electron is one Bohr radius from the central nucleus?

A hydrogen atom in a state having a binding energy (the energy required to remove an electron) of \(0.85 \mathrm{eV}\) makes a transition to a state with an excitation energy (the difference between the energy of the state and that of the ground state) of \(10.2 \mathrm{eV}\). (a) What is the energy of the photon emitted as a result of the transition? What are the (b) higher quantum number and (c) lower quantum number of the transition producing this emission?

Calculate the probability that the electron in the hydrogen atom, in its ground state, will be found between spherical shells whose radii are \(a\) and \(2 a,\) where \(a\) is the Bohr radius.

Light of wavelength \(121.6 \mathrm{nm}\) is emitted by a hydrogen atom. What are the (a) higher quantum number and (b) lower quantum number of the transition producing this emission? (c) What is the name of the series that includes the transition?

Light of wavelength \(102.6 \mathrm{nm}\) is emitted by a hydrogen atom. What are the (a) higher quantum number and (b) lower quantum number of the transition producing this emission? (c) What is the name of the series that includes the transition?

Let \(\Delta E_{\text {adj }}\) be the energy difference between two adjacent energy levels for an electron trapped in a one-dimensional infinite potential well. Let \(E\) be the energy of either of the two levels. (a) Show that the ratio \(\Delta E_{a d j} / E\) approaches the value \(2 / n\) at large values of the quantum number \(n .\) As \(n \rightarrow \infty,\) does (b) \(\Delta E_{\text {adj }},\) (c) \(E\), or (d) \(\Delta E_{\text {adj }} / E\) approach zero? (e) What do these results mean in terms of the correspondence principle?

An electron is trapped in a one-dimensional infinite potential well. Show that the energy difference \(\Delta E\) between its quantum levels \(n\) and \(n+2\) is \(\left(h^{2} / 2 m L^{2}\right)(n+1)\)

An electron is confined to a narrow evacuated tube of length \(3.0 \mathrm{~m} ;\) the tube functions as a one-dimensional infinite potential well. (a) What is the energy difference between the electron's ground state and its first excited state? (b) At what quantum number \(n\) would the energy difference between adjacent energy levels be \(1.0 \mathrm{eV}-\) which is measurable, unlike the result of (a)? At that quantum number, (c) what multiple of the electron's rest energy would give the electron's total energy and (d) would the electron be relativistic?

(a) For a given value of the principal quantum number \(n\) for a hydrogen atom, how many values of the orbital quantum number \(\ell\) are possible? (b) For a given value of \(\ell,\) how many values of the orbital magnetic quantum number \(m_{\ell}\) are possible? (c) For a given value of \(n\), how many values of \(m_{\ell}\) are possible?

In atoms there is a finite, though very small, probability that, at some instant, an orbital electron will actually be found inside the nucleus. In fact, some unstable nuclei use this occasional appearance of the electron to decay by electron capture. Assuming that the proton itself is a sphere of radius \(1.1 \times 10^{-15} \mathrm{~m}\) and that the wave function of the hydrogen atom's electron holds all the way to the proton's center, use the ground-state wave function to calculate the probability that the hydrogen atom's electron is inside its nucleus.

From the energy-level diagram for hydrogen, explain the observation that the frequency of the second Lyman-series line is the sum of the frequencies of the first Lyman-series line and the first Balmer-series line. This is an example of the empirically discovered Ritz combination principle. Use the diagram to find some other valid combinations.

A hydrogen atom can be considered as having a central point-like proton of positive charge \(e\) and an electron of negative charge \(-e\) that is distributed about the proton according to the volume charge density \(\rho=A \exp \left(-2 r / a_{0}\right) .\) Here \(A\) is a constant, \(a_{0}=0.53 \times 10^{-10} \mathrm{~m},\) and \(r\) is the distance from the center of the atom. (a) Using the fact that the hydrogen is electrically neutral, find \(A\). Then find the (b) magnitude and (c) direction of the atom's electric field at \(a_{0}\)

An old model of a hydrogen atom has the charge \(+e\) of the proton uniformly distributed over a sphere of radius \(a_{0},\) with the electron of charge \(-e\) and mass \(m\) at its center. (a) What would then be the force on the electron if it were displaced from the center by a distance \(r \leq a_{0} ?\) (b) What would be the angular frequency of oscillation of the electron about the center of the atom once the electron was released?

In a simple model of a hydrogen atom, the single electron orbits the single proton (the nucleus) in a circular path. Calculate (a) the electric potential set up by the proton at the orbital radius of \(52.9 \mathrm{pm},\) (b) the electric potential energy of the atom, and (c) the kinetic energy of the electron. (d) How much energy is required to ionize the atom (that is, to remove the electron to an infinite distance with no kinetic energy)? Give the energies in electron-volts.

Consider a conduction electron in a cubical crystal of a conducting material. Such an electron is free to move throughout the volume of the crystal but cannot escape to the outside. It is trapped in a three-dimensional infinite well. The electron can move in three dimensions, so that its total energy is given by $$E=\frac{h^{2}}{8 L^{2} m}\left(n_{1}^{2}+n_{2}^{2}+n_{3}^{2}\right)$$ in which \(n_{1}, n_{2},\) and \(n_{3}\) are positive integer values. Calculate the energies of the lowest five distinct states for a conduction electron moving in a cubical crystal of edge length \(L=0.25 \mu \mathrm{m}\).