Chapter 38

(II) The neutrons in a parallel beam, each having kinetic energy \(0.030 \mathrm{eV}\), are directed through two slits \(0.60 \mathrm{~mm}\) apart. How far apart will the interference peaks be on a screen \(1.0 \mathrm{~m}\) away? [Hint: First find the wavelength of the neutron.

(1I) The neutrons in a parallel beam, each having kinetic energy \(0.030 \mathrm{eV},\) are directed through two slits 0.60 \(\mathrm{mm}\) apart. How far apart will the interference peaks be on a apart. How far apart will the interference peaks be on a screen 1.0 \(\mathrm{m}\) away? [Hint: First find the wavelength of the neutron.

(II) Pellets of mass \(3.0 \mathrm{~g}\) are fired in parallel paths with speeds of \(150 \mathrm{~m} / \mathrm{s}\) through a hole \(3.0 \mathrm{~mm}\) in diameter. How far from the hole must you be to detect a 1.0 -cm-diameter spread in the beam of pellets?

(I) An electron remains in an excited state of an atom for typically \(10^{-8}\) s. What is the minimum uncertainty in the energy of the state (in eV)?

(1) An electron remains in an excited state of an atom for typically \(10^{-8} \mathrm{s}\) . What is the minimum uncertainty in the energy of the state (in \(\mathrm{eVV}\) )

(I) If an electron's position can be measured to a precision of \(2.6 \times 10^{-8} \mathrm{~m}\), how precisely can its speed be known?

(I) The lifetime of a typical excited state in an atom is about 10 ns. Suppose an atom falls from one such excited state and emits a photon of wavelength about \(500 \mathrm{nm}\). Find the fractional energy uncertainty \(\Delta E / E\) and wavelength uncertainty \(\Delta \lambda / \lambda\) of this photon.

(1) The lifetime of a typical excited state in an atom is about 10 ns. Suppose an atom falls from one such excited state and emits a photon of wavelength about 500 \(\mathrm{nm}\) . Find the fractional energy uncertainty \(\Delta E / E\) and wavelength uncertainty \(\Delta \lambda / \lambda\) of this photon.

(II) A 12-g bullet leaves a rifle horizontally at a speed of \(180 \mathrm{~m} / \mathrm{s}\). ( \(a\) ) What is the wavelength of this bullet? \((b)\) If the position of the bullet is known to a precision of \(0.65 \mathrm{~cm}\) (radius of the barrel), what is the minimum uncertainty in its vertical momentum?

(II) A 12 -g bullet leaves a ritle horizontally at a speed of 180 \(\mathrm{m} / \mathrm{s} .(a)\) What is the wavelength of this bullet? \((b)\) If the position of the bullet is known to a precision of 0.65 \(\mathrm{cm}\) (radius of the barrel), what is the minimum uncertainty in its vertical momentum?

(II) What is the uncertainty in the mass of a muon \(\left(m=105.7 \mathrm{MeV} / c^{2}\right),\) specified in eV/c\(^{2}\) , given its lifetime of 2.20\(\mu \mathrm{s} ?\)

(1I) A free neutron \(\left(m=1.67 \times 10^{-27} \mathrm{kg}\right)\) has a mean life of 900 s. What is the uncertainty in its mass (in kg)?

(II) Use the uncertainty principle to show that if an electron were present in the nucleus \(\left(r \approx 10^{-15} \mathrm{m}\right),\) its kinetic energy (use relativity) would be hundreds of MeV. (Since such electron energies are not observed, we conclude that electrons are not present in the nucleus) [Hint: Assume a particle can have energy as large as its uncertainty.]

(II) An electron in the \(n=2\) state of hydrogen remains there on average about \(10^{-8}\) s before jumping to the \(n=1\) state. (a) Estimate the uncertainty in the energy of the \(n=2\) state. (b) What fraction of the transition energy is this? (c) What is the wavelength, and width (in \(\mathrm{nm}\) ), of this line in the spectrum of hydrogen?

(11) How accurately can the position of a 3.50 -keV electron be measured assuming its energy is known to 1.00\(\% ?\)

(III) In a double-slit experiment on electrons (or photons), suppose that we use indicators to determine which slit each electron went through (Section \(38-2\) ). These indicators must tell us the \(y\) coordinate to within \(d / 2,\) where \(d\) is the distance between slits. Use the uncertainty principle to show that the interference pattern will be destroyed. [Note: First show that the angle \(\theta\) between maxima and minima of the interference pattern is given by \(\frac{1}{2} \lambda / d\),

(II) Show that the superposition principle holds for the timedependent Schrödinger equation. That is, show that if \(\Psi_{1}(x, t)\) and \(\Psi_{2}(x, t)\) are solutions, then \(A \Psi_{1}(x, t)+B \Psi_{2}(x, t)\) is also a solution where \(A\) and \(B\) are arbitrary constants.

(I) A free electron has a wave function \(\psi(x)=\) \(A \sin \left(2.0 \times 10^{10} x\right),\) where \(x\) is given in meters. Determine the electron's \((a)\) wavelength, \((b)\) momentum, \((c)\) speed, and (d) kinetic energy.

(I) Write the wave function for \((a)\) a free electron and (b) a free proton, each having a constant velocity \(v=3.0 \times 10^{5} \mathrm{~m} / \mathrm{s}\)

(II) An \(n=4\) to \(n=1\) transition for an electron trapped in a rigid box produces a 340 -nm photon. What is the width of the box?

(II) For a particle in a box with rigid walls, determine whether our results for the ground state are consistent with the uncertainty principle by calculating the product \(\Delta p \Delta x\). Take \(\Delta x \approx \ell\), since the particle is somewhere within the box. For \(\Delta p,\) note that although \(p\) is known \((=\hbar k),\) the direction of \(\overrightarrow{\mathbf{p}}\) is not known, so the \(x\) component could vary from \(-p\) to \(+p ;\) hence take \(\Delta p \approx 2 p\)

(II) Determine the lowest four energy levels and wave functions for an electron trapped in an infinitely deep potential well of width 2.0 \(\mathrm{nm} .\)

(II) Consider an atomic nucleus to be a rigid box of width \(2.0 \times 10^{-14} \mathrm{~m} .\) What would be the ground-state energy for (a) an electron, (b) a neutron, and ( \(c\) ) a proton in this nucleus?

(II) A proton in a nucleus can be roughly modeled as a particle in a box of nuclear dimensions. Calculate the energy released when a proton confined in a nucleus of width \(1.0 \times 10^{-14} \mathrm{~m}\) makes a transition from the first excited state to the ground state.

(II) Consider a single oxygen molecule confined in a onedimensional rigid box of width \(4.0 \mathrm{~mm}\). \((a)\) Treating this as a particle in a rigid box, determine the ground-state energy. ( \(b\) ) If the molecule has an energy equal to the onedimensional average thermal energy \(\frac{1}{2} k T\) at \(T=300 \mathrm{~K},\) what is the quantum number \(n ?(c)\) What is the energy difference between the \(n\) th state and the next higher state?

(III) If an infinitely deep well of width \(\ell\) is redefined to be located from \(x=-\frac{1}{2} \ell\) to \(x=\frac{1}{2} \ell\) (as opposed to \(x=0\) to \(x=\ell\) ), speculate how this will change the wave function for a particle in this well. Investigate your speculation(s) by determining the wave functions and energy levels for this newly defined well. [Hint: \(\operatorname{Try} \psi=A \sin (k x+\phi)\).

(II) An electron with \(180 \mathrm{eV}\) of kinetic energy in free space passes over a finite potential well \(56 \mathrm{eV}\) deep that stretches from \(x=0\) to \(x=0.50 \mathrm{nm}\). What is the electron's wavelength (a) in free space, \((b)\) when over the well? ( \(c\) ) Draw a diagram showing the potential energy and total energy as a function of \(x\), and on the diagram sketch a possible wave function.

(1I) An electron with 180 \(\mathrm{eV}\) of kinetic energy in free space passes over a finite potential well 56 \(\mathrm{eV}\) deep that stretches from \(x=0\) to \(x=0.50 \mathrm{nm}\) . What is the electron's wavelength (a) in free space, ( \(b\) ) when over the well? (c) Draw a diagram showing the potential energy and total energy as a function of \(x,\) and on the diagram sketch a possible wave function.

(II) An electron is trapped in a 0.16 -nm-wide finite square well of height \(U_{0}=2.0 \mathrm{keV}\). Estimate at what distance outside the walls of the well the ground state wave function drops to \(1.0 \%\) of its value at the walls.

(II) A potential barrier has a height \(U_{0}=14 \mathrm{eV}\) and thickness \(\ell=0.85 \mathrm{nm}\) . If the transmission coefficient for an incident electron is \(0.00050,\) what is the electron's energy?

(II) An electron approaches a potential barrier \(18 \mathrm{eV}\) high and \(0.55 \mathrm{nm}\) wide. If the electron has a \(1.0 \%\) probability of tunneling through the barrier, what is the electron's energy?

(II) A proton and a helium nucleus approach a 25-MeV potential energy barrier. If each has a kinetic energy of \(5.0 \mathrm{MeV}\) what is the probability of each to tunnel through the barrier, assuming it is rectangular and \(3.6 \mathrm{fm}\) thick?

(1I) A proton and a helium nucleus approach a \(25-\) MeV potential energy barrier. If each has a kinetic energy of 5.0 \(\mathrm{MeV}\) . what is the probability of each to tunnel through the barrier, assuming it is rectangular and 3.6 \(\mathrm{fm}\) thick?

(II) An electron with an energy of \(8.0 \mathrm{eV}\) is incident on a potential barrier which is \(9.2 \mathrm{eV}\) high and \(0.25 \mathrm{nm}\) wide. (a) What is the probability that the electron will pass through the barrier? (b) What is the probability that the electron will be reflected?

The \(Z^{0}\) boson, discovered in \(1985,\) is the mediator of the weak nuclear force, and it typically decays very quickly. Its average rest energy is 91.19 \(\mathrm{GeV}\) , but its short lifetime shows up as an intrinsic width of 2.5 \(\mathrm{GeV}\) . What is the lifetime of this particle?

Estimate the lowest possible energy of a neutron contained in a typical nucleus of radius \(1.2 \times 10^{-15} \mathrm{~m}\). [Hint: A particle can have an energy at least as large as its uncertainty.

A neutron is trapped in an infinitely deep potential well 2.5 \(\mathrm{fm}\) in width. Determine \((a)\) the four lowest possible energy states and \((b)\) their wave functions. (c) What is the wavelength and energy of a photon emitted when the neutron makes a transition hetween the two lowest states? In what region of the EM spectrum does this photon lie? \([\) Note: This is a rough model of an atomic nucleus.]

An electron and a proton, each initially at rest, are accelerated across the same voltage. Assuming that the uncertainty in their position is given by their de Broglie wavelength. find the ratio of the uncertainty in their momentum.

Simple Harmonic Oscillator. Suppose that a particle of mass \(m\) is trapped not in a square well, but in one whose potential energy is that of a simple harmonic oscillator: \(U(x)=\frac{1}{2} C x^{2} .\) That is, if the particle is displaced from \(x=0\) a restoring force \(F=-C x\) acts on it, where \(C\) is constant. (a) Sketch this potential energy. (b) Show that \(\psi=A e^{-B x^{2}}\) is a solution to the Schrödinger equation and that the energy of this state is \(E=\frac{1}{2} \hbar \omega,\) where \(\omega=\sqrt{C / m}\) (as classically, Eq. \(14-5\) ) and \(B=m \omega / 2 \hbar\). [Note: This is the ground state, and this energy \(\frac{1}{2} \hbar \omega\) is the zero-point energy for a harmonic oscillator. The energies of higher states are \(E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega,\) where \(n\) is an integer.

Estimate the kinetic energy and speed of an alpha particle \(\left(q=+2 e, M=4 M_{\text {proton }}\right)\) trapped in a nucleus \(1.5 \times 10^{-14} \mathrm{~m}\) wide. Assume an infinitely deep square well potential.

By how much does the tunneling current through the tip of an STM change if the tip rises \(0.020 \mathrm{nm}\) from some initial height above a sodium surface with a work function \(W_{0}=2.28 \mathrm{eV} ?\) [Hint: Let the work function (see Section \(37-2\) ) equal the energy needed to raise the electron to the top of the barrier.

By how much does the tunneling current through the tip of an STM change if the tip rises 0.020 nm from some initial height above a sodium surface with a work function \(W_{0}=2.28 \mathrm{cV}\) ? [Hint: Let the work function equal the energy needed to raise the electron to the top of the barrier.

Consider a particle that can exist anywhere in space with a wave function given by \(\psi(x)=b^{-\frac{1}{2}}|x / b|^{\frac{1}{2}} e^{-(x / b)^{2} / 2},\) where \(b=1.0 \mathrm{nm} .\) (a) Check that the wave function is normalized. (b) What is the most probable position for the particle in the region \(x>0 ?\) (c) What is the probability of finding the particle between \(x=0 \mathrm{nm}\) and \(x=0.50 \mathrm{nm} ?\)

(III) Consider a particle of mass \(m\) and energy \(E\) traveling to the right where it encounters a narrow potential barrier of height \(U_{0}\) and width \(\ell\) as shown in Fig. \(38-21 .\) It can be shown that: (i) for \(EU_{0},\) the transmission probability is $$ T=\left[1+\frac{\sin ^{2}\left(G^{\prime} \ell\right)}{4\left(E / U_{0}\right)\left(E / U_{0}-1\right)}\right]^{-1} $$ where $$ G^{\prime}=\sqrt{\frac{2 m\left(E-U_{0}\right)}{\hbar^{2}}} $$ and \(R=1-T .\) Consider that the particle is an electron and it is incident on a rectangular barrier of height \(U_{0}=10 \mathrm{eV}\) and width \(\ell=1.0 \times 10^{-10} \mathrm{~m} .\) ( \(a\) ) Calculate \(T\) and \(R\) for the electron from \(E / U_{0}=0\) to \(10,\) in steps of \(0.1 .\) Make a single graph showing the two curves of \(T\) and \(R\) as a function of \(E / U_{0}\). (b) From the graph determine the energies \(\left(E / U_{0}\right)\) at which the electron will have transmission probabilities of \(10 \%\) $$ 20 \%, 50 \%, \text { and } 80 \% $$