Questions and Answers of Mechanics

Show that, for a sufficiently large momentum \(\hbar k\) (in fact, such that the slope \(d \varepsilon / d k\) of the energy spectrum is greater than the initial slope \(\hbar c\) ), a state of
Establish the asymptotic formula (11.7.30) for the function \(F\left(z_{0}ight)\).[Write the coefficient that appears in the sum (11.7.24) in the form\[\begin{aligned}& \frac{1}{\sqrt{ }(r s
The grand partition function of a gaseous system composed of mutually interacting, spin-half fermions has been evaluated by Lee and Yang (1957), with the result \({ }^{27}\)\[\begin{aligned}\ln
The energy spectrum \(\varepsilon(p)\) of a gas composed of mutually interacting, spin-half fermions is given by (Galitskii, 1958; Mohling, 1961)\[\begin{aligned}\frac{\varepsilon(p)}{p_{F}^{2} / 2
At \(T=0 K\), the chemical potential of a thermodynamic system is given by\[\mu=\left(\frac{\partial E}{\partial N}ight)_{v}=\frac{\partial(E / V)}{\partial(N / V)}\]It follows that, in the ground
Correction to the first printing of third edition: In line 3, the definition of the dimensionless wavefunction should read: \(\psi=a_{\text {osc }}^{3 / 2} \Psi / \sqrt{N}\). Using that substitution
The solution for the case \(V=0\) is \(\Psi=\sqrt{N / V}\) which gives \(\mu=N u_{0} / V\) and \(E=\left(2 \pi a \hbar^{2} N^{2}ight) /(m V)\).
For the case \(a ightarrow 0\) the dimensionless G-P equation is\[-\frac{1}{2} \tilde{abla}^{2} \psi+\frac{1}{2} s^{2} \psi+=\mu \psi\]which has solution \(\psi=\frac{1}{\pi^{3 / 4}} \exp
Use the dimensionless form from problem 11.17. Ignoring the kinetic energy term\[\psi=\frac{\sqrt{\tilde{\mu}-\frac{s^{2}}{2}}}{\sqrt{4 \pi N a / a_{0}}} .\]The normalization is\[1=\frac{4 \pi
Referring to Fig. 8.11 and noting that the slope of the tangent at the point \(x=\xi\) is \(-1 / 4\), the approximate distribution is given by\[f(x)=\left\{\begin{array}{cc}1 & 0 \leq x \leq(\xi-2)
By eqns. (8.1.4) and (8.1.5), the temperature \(T_{0}\) is given by\[\begin{equation*}T_{0}=\left(\frac{N}{g V f_{3 / 2}(1)}ight)^{2 / 3}\left(\frac{h^{2}}{2 \pi m k}ight) \tag{1}\end{equation*}\]At
This problem is similar to Problem 7.4 of the Bose gas and can be done the same way - only the functions \(g_{\mathrm{v}}(z)\) get replaced by \(f_{\mathrm{v}}(z)\).To obtain the low-temperature
(a) Show that the isothermal compressibility \(\kappa_{T}\) and the adiabatic compressibility \(\kappa_{S}\) of an ideal Fermi gas are given by\[\kappa_{T}=\frac{1}{n k T} \frac{f_{1 / 2}(z)}{f_{3 /
Evaluate \(\left(\partial^{2} P / \partial T^{2}ight)_{u},\left(\partial^{2} \mu / \partial T^{2}ight)_{u}\), and \(\left(\partial^{2} \mu / \partial T^{2}ight)_{P}\) of an ideal Fermi gas and check
By eqns. (8.1.4, 5 and 24 ), the Fermi energy \(\varepsilon_{F}\) is given by\[\varepsilon_{F}=\left\{\frac{3}{4 \pi} f_{3 / 2}(z)ight\}^{2 / 3} \frac{h^{2}}{2 m \lambda^{2}}=\left\{\frac{3 \pi^{1 /
For a Fermi gas confined to a two-dimensional region of area A,N=Aλ2f1(zF)=Aλ2ln(1+zF),EF=AkTλ2f2(zF)while the corresponding results for the Bose gas
Calculate the fraction of the conduction electrons in tungsten \(\left(\varepsilon_{F}=9.0 \mathrm{eV}ight)\) at \(3000 \mathrm{~K}\) whose kinetic energy \(\varepsilon\left(=\frac{1}{2} m
The total energy \(E\) of the electron cloud in an atom can be written as\[E=K+V_{n e}+V_{e e},\]where \(K\) is the kinetic energy of the electrons, \(V_{n e}\) the interaction energy between the
Study the density matrix and the partition function of a system of free particles, using the unsymmetrized wavefunction (5.4.3) instead of the symmetrized wavefunction (5.5.7). Show that, following
Show that in the first approximation the partition function of a system of \(N\) noninteracting, indistinguishable particles is given by\[Q_{N}(V, T)=\frac{1}{N ! \lambda^{3 N}} Z_{N}(V,
Determine the values of the degeneracy discriminant \(\left(n \lambda^{3}ight)\) for hydrogen, helium, and oxygen at NTP. Make an estimate of the respective temperature ranges where the magnitude of
Show that the quantum-mechanical partition function of a system of \(N\) interacting particles approaches the classical form\[Q_{N}(V, T)=\frac{1}{N ! h^{3 N}} \int e^{-\beta E(\boldsymbol{q},
Prove the following theorem due to Peierls. "If \(\hat{H}\) is the hermitian Hamiltonian operator of a given physical system and \(\left\{\phi_{n}ight\}\) an arbitrary orthonormal set of
Show that the entropy of an ideal gas in thermal equilibrium is given by the formula\[S=k \sum_{\varepsilon}\left[\left\langle n_{\varepsilon}+1ightangle \ln \left\langle
Derive, for all three statistics, the relevant expressions for the quantity \(\left\langle n_{\varepsilon}^{2}ightangle-\left\langle n_{\varepsilon}ightangle^{2}\) from the respective probabilities
Refer to Section 6.2 and show that, if the occupation number \(n_{\varepsilon}\) of an energy level \(\varepsilon\) is restricted to the values \(0,1, \ldots, l\), then the mean occupation number of
The potential energy of a system of charged particles, characterized by particle charge \(e\) and number density \(n(\boldsymbol{r})\), is given by\[U=\frac{e^{2}}{2} \iint \frac{n(\boldsymbol{r})
Show that the root-mean-square deviation in the molecular energy \(\varepsilon\), in a system obeying Maxwell-Boltzmann distribution, is \(\sqrt{ }(2 / 3)\) times the mean molecular energy
Show that, for any law of distribution of molecular speeds,\[\left\{\langle uangle\left\langle\frac{1}{u}ightangleight\} \geq 1\]Check that the value of this quantity for the Maxwellian distribution
Through a small window in a furnace, which contains a gas at a high temperature \(T\), the spectral lines emitted by the gas molecules are observed. Because of molecular motions, each spectral line
An ideal classical gas composed of \(N\) particles, each of mass \(m\), is enclosed in a vertical cylinder of height \(L\) placed in a uniform gravitational field (of acceleration \(g\) ) and is in
Centrifuge-based uranium enrichment: Natural uranium is composed of two isotopes: \({ }^{238} \mathrm{U}\) and \({ }^{235} \mathrm{U}\), with percentages of \(99.27 \%\) and \(0.72 \%\),
(a) Show that, if the temperature is uniform, the pressure of a classical gas in a uniform gravitational field decreases with height according to the barometric formula\[P(z)=P(0) \exp \{-m g z / k
(a) Show that the momentum distribution of particles in a relativistic Boltzmannian gas, with \(\varepsilon=c\left(p^{2}+m_{0}^{2} c^{2}ight)^{1 / 2}\), is given by\[f(\boldsymbol{p}) d
(a) Considering the loss of translational energy suffered by the molecules of a gas on reflection from a receding wall, derive, for a quasistatic adiabatic expansion of an ideal nonrelativistic gas,
(a) Determine the number of impacts made by gas molecules on a unit area of the wall in a unit time for which the angle of incidence lies between \(\theta\) and \(\theta+d \theta\).(b) Determine the
Consider the effusion of molecules of a Maxwellian gas through an opening of area \(a\) in the walls of a vessel of volume \(V\).(a) Show that, while the molecules inside the vessel have a mean
A polyethylene balloon at an altitude of \(30,000 \mathrm{~m}\) is filled with helium gas at a pressure of \(10^{-2} \mathrm{~atm}\) and a temperature of \(300 \mathrm{~K}\). The balloon has a
Consider two Boltzmannian gases \(A\) and \(B\), at pressures \(P_{A}\) and \(P_{B}\) and temperatures \(T_{A}\) and \(T_{B}\), respectively, contained in two regions of space that communicate
A small sphere, with initial temperature \(T\), is immersed in an ideal Boltzmannian gas at temperature \(T_{0}\). Assuming that the molecules incident on the sphere are first absorbed and then
Show that the mean value of the relative speed of two molecules in a Maxwellian gas is \(\sqrt{ } 2\) times the mean speed of a molecule with respect to the walls of the container.
What is the probability that two molecules picked at random from a Maxwellian gas will have a total energy between \(E\) and \(E+d E\) ? Verify that \(\langle Eangle=3 k T\).
The energy difference between the lowest electronic state \({ }^{1} S_{0}\) and the first excited state \({ }^{3} S_{1}\) of the helium atom is \(159,843 \mathrm{~cm}^{-1}\). Evaluate the relative
Derive an expression for the equilibrium constant \(K(T)\) for the reaction \(\mathrm{H}_{2}+\mathrm{D}_{2} \leftrightarrow 2 \mathrm{HD}\) at temperatures high enough to allow classical
With the help of the Euler-Maclaurin formula (6.5.19), derive high-temperature expansions for \(r_{\text {even }}\) and \(r_{\text {odd }}\), as defined by equations (6.5.29) and (6.5.30), and obtain
The potential energy between the atoms of a hydrogen molecule is given by the (semiempirical) Morse potential\[V(r)=V_{0}\left\{e^{-2\left(r-r_{0}ight) / a}-2 e^{-\left(r-r_{0}ight) / a}ight\}\]where
Show that the fractional change in the equilibrium value of the internuclear distance of a diatomic molecule, as a result of rotation, is given by\[\frac{\Delta r_{0}}{r_{0}}
The ground state of an oxygen atom is a triplet, with the following fine structure:\[\varepsilon_{J=2}=\varepsilon_{J=1}-158.5 \mathrm{~cm}^{-1}=\varepsilon_{J=0}-226.5 \mathrm{~cm}^{-1} .\]Calculate
Calculate the contribution of the first excited electronic state, namely \({ }^{1} \Delta\) with \(g_{e}=2\), of the \(\mathrm{O}_{2}\) molecule toward the Helmholtz free energy and the specific heat
The rotational kinetic energy of a rotator with three degrees of freedom can be written as\[\varepsilon_{\mathrm{rot}}=\frac{M_{\xi}^{2}}{2 I_{1}}+\frac{M_{\eta}^{2}}{2 I_{2}}+\frac{M_{\zeta}^{2}}{2
Determine the translational, rotational, and vibrational contributions toward the molar entropy and the molar specific heat of carbon dioxide at NTP. Assume the ideal-gas formulae and use the
Determine the molar specific heat of ammonia at a temperature of \(300 \mathrm{~K}\). Assume the ideal-gas formula and use the following data: the principal moments of inertia: \(I_{1}=4.44 \times
Derive the equilibrium concentration equation (6.6.6) from the equilibrium condition (6.6.3).Data From Equation (6.6.6) [X]x[Y]VY = K(T) = exp(-BA(0)), [A]VA [B]VB
Use the following values to determine the equilibrium constant for the reaction \(2 \mathrm{CO}+\mathrm{O}_{2} ightleftarrows 2 \mathrm{CO}_{2}\). At a combustion temperature of \(T=1500 \mathrm{~K}:
Derive an expression for the equilibrium constant \(K(T)\) for the reaction \(\mathrm{N}_{2}+\mathrm{O}_{2} ightleftarrows 2 \mathrm{NO}\) in terms of the ground state energy change \(\Delta
Analyze the combustion reaction\[\begin{equation*}\mathrm{CH}_{4}+2 \mathrm{O}_{2} ightleftarrows \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O}, \tag{6.6.8}\end{equation*}\]
Determine the equilibrium ionization fraction for the reaction\[\mathrm{Na} ightleftarrows \mathrm{Na}^{+}+e^{-}\]in a sodium vapor. Treat all three species as ideal classical monatomic gases. The
By considering the order of magnitude of the occupation numbers \(\left\langle n_{\varepsilon}ightangle\), show that it makes no difference to the final results of Section 7.1 if we combine a finite
Deduce the virial expansion (7.1.13) from equations (7.1.7) and (7.1.8), and verify the quoted values of the virial coefficients.
Combining equations (7.1.24) and (7.1.26), and making use of the first two terms of formula (D.9) in Appendix D, show that, as \(T\) approaches \(T_{c}\) from above, the parameter \(\alpha(=-\ln z)\)
Show that for an ideal Bose gas\[\frac{1}{z}\left(\frac{\partial z}{\partial T}ight)_{P}=-\frac{5}{2 T} \frac{g_{5 / 2}(z)}{g_{3 / 2}(z)}\]compare this result with equation (7.1.36). Hence show
(a) Show that the isothermal compressibility \(\kappa_{T}\) and the adiabatic compressibility \(\kappa_{S}\) of an ideal Bose gas are given by\[\kappa_{T}=\frac{1}{n k T} \frac{g_{1 / 2}(z)}{g_{3 /
Show that for an ideal Bose gas the temperature derivative of the specific heat \(C_{V}\) is given by\[\frac{1}{N k}\left(\frac{\partial C_{V}}{\partial T}ight)_{V}=
Evaluate the quantities \(\left(\partial^{2} P / \partial T^{2}ight)_{u},\left(\partial^{2} \mu / \partial T^{2}ight)_{u}\), and \(\left(\partial^{2} \mu / \partial T^{2}ight)_{P}\) for an ideal Bose
The velocity of sound in a fluid is given by the formula\[w=\sqrt{ }(\partial P / \partial ho)_{s}\]where \(ho\) is the mass density of the fluid. Show that for an ideal Bose gas\[w^{2}=\frac{5 k
Show that for an ideal Bose gas\[\langle uangle\left\langle\frac{1}{u}ightangle=\frac{4}{\pi} \frac{g_{1}(z) g_{2}(z)}{\left\{g_{3 / 2}(z)ight\}^{2}}\]\(u\) being the speed of a particle. Examine and
Consider an ideal Bose gas in a uniform gravitational field of acceleration \(g\). Show that the phenomenon of Bose-Einstein condensation in this gas sets in at a temperature \(T_{c}\) given
Consider an ideal Bose gas consisting of molecules with internal degrees of freedom. Assuming that, besides the ground state \(\varepsilon_{0}=0\), only the first excited state \(\varepsilon_{1}\) of
Consider an ideal Bose gas in the grand canonical ensemble and study fluctuations in the total number of particles \(N\) and the total energy \(E\). Discuss, in particular, the situation when the gas
Consider an ideal Bose gas confined to a region of area \(A\) in two dimensions. Express the number of particles in the excited states, \(N_{e}\), and the number of particles in the ground state,
Consider an \(n\)-dimensional Bose gas whose single-particle energy spectrum is given by \(\varepsilon \propto p^{s}\), where \(s\) is some positive number. Discuss the onset of Bose-Einstein
At time \(t=0\), the ground state wavefunction of a one-dimensional quantum harmonic oscillator with potential \(V(x)=\frac{1}{2} m \omega_{0}^{2} x^{2}\) is given by\[\psi(x, 0)=\frac{1}{\pi^{1 / 4}
At time \(t=0\), a collection of classical particles is in equilibrium at temperature \(T\) in a threedimensional harmonic oscillator potential \(V(\boldsymbol{r})=\frac{1}{2} m
As shown in Section 7.1, \(n \lambda^{3}\) is a measure of the quantum nature of the system. Use equations (7.2.11) and (7.2.15) to determine \(n \lambda^{3}\) at the center of the harmonic trap at
Show that the integral of the semiclassical spatial density in equation (7.2.15) gives the correct counting of the atoms that are not condensed into the ground state.
Construct a theory for \(N\) bosons in an isotropic two-dimensional trap. This corresponds to a trap in which the energy level spacing due to excitations in the \(z\) direction is much larger than
The (canonical) partition function of the blackbody radiation may be written as\[Q(V, T)=\prod_{\omega} Q_{1}(\omega, T)\]so that\[\ln Q(V, T)=\sum_{\omega} \ln Q_{1}(\omega, T) \approx
Show that the mean energy per photon in a blackbody radiation cavity is very nearly \(2.7 k T\).
Considering the volume dependence of the frequencies \(\omega\) of the vibrational modes of the radiation field, establish relation (7.3.17) between the pressure \(P\) and the energy density \(U /
The sun may be regarded as a black body at a temperature of \(5800 \mathrm{~K}\). Its diameter is about \(1.4 \times 10^{9} \mathrm{~m}\) while its distance from the earth is about \(1.5 \times
Calculate the photon number density, entropy density, and energy density of the \(2.725 \mathrm{~K}\) cosmic microwave background.
Figure 7.20 is a plot of \(C_{V}(T)\) against \(T\) for a solid, the limiting value \(C_{V}(\infty)\) being the classical result \(3 N k\). Show that the shaded area in the figure,
Show that the zero-point energy of a Debye solid composed of \(N\) atoms is equal to \(\frac{9}{8} N k \Theta_{D}\).
Show that, for \(T \ll \Theta_{D}\), the quantity ( \(C_{P}-C_{V}\) ) of a Debye solid varies as \(T^{7}\) and hence the ratio \(\left(C_{P} / C_{V}ight) \simeq 1\).
Determine the temperature \(T\), in terms of the Debye temperature \(\Theta_{D}\), at which one-half of the oscillators in a Debye solid are expected to be in the excited states.
Determine the value of the parameter \(\Theta_{D}\) for liquid \(\mathrm{He}^{4}\) from the empirical result (7.4.28).
(a) Compare the "mean thermal wavelength" \(\lambda_{T}\) of neutrons at a typical room temperature with the "minimum wavelength" \(\lambda_{\min }\) of phonons in a typical crystal.(b) Show that the
Proceeding under conditions (7.4.16) rather than (7.4.13), show that\[C_{V}(T)=N k\left\{D\left(x_{0, L}ight)+2 D\left(x_{0, T}ight)ight\}\]where \(x_{0, L}=\left(\hbar \omega_{D, L} / k Tight)\) and
A mechanical system consisting of \(n\) identical masses (each of mass \(m\) ) connected in a straight line by identical springs (of stiffness \(K\) ) has natural vibrational frequencies given
Assuming the dispersion relation \(\omega=A k^{\mathcal{s}}\), where \(\omega\) is the angular frequency and \(k\) the wave number of a vibrational mode existing in a solid, show that the respective
Assuming the excitations to be phonons \((\omega=A k)\), show that their contribution toward the specific heat of an \(n\)-dimensional Debye system is proportional to \(T^{n}\).
The (minimum) potential energy of a solid, when all its atoms are "at rest" at their equilibrium positions, may be denoted by the symbol \(\Phi_{0}(V)\), where \(V\) is the volume of the solid.
Apply the general formula (6.4.3) for the kinetic pressure of a gas, namely\[P=\frac{1}{3} n\langle p uangle\]to a gas of rotons and verify that the result so obtained agrees with the Boltzmannian
Show that the free energy \(A\) and the inertial density \(ho\) of a roton gas in mass motion are given by\[A(v)=A(0) \frac{\sinh x}{x}\]and\[ho(v)=ho(0) \frac{3(x \cosh x-\sinh x)}{x^{3}},\]where
Integrating (7.6.17) by parts, show that the effective mass of an excitation, whose energymomentum relationship is denoted by \(\varepsilon(p)\), is given by\[m_{\mathrm{eff}}=\left\langle\frac{1}{3
The relativity of simultaneity. Two clocks are placed at rest on the \(x^{\prime}\) axis of the primed frame, clock A at \(x^{\prime}=0\) and clock B at \(x^{\prime}=L_{0}\). They are therefore a
A primed frame moves at \(V=(3 / 5) c\) relative to an unprimed frame. Just as their origins pass, clocks at the origins of both frames read zero, and a flashbulb explodes at that point. Later, the
Synchronized clocks A and B are at rest in our frame of reference, a distance five light-minutes apart. Clock \(\mathrm{C}\) passes \(\mathrm{A}\) at speed (12/13) \(c\) bound for \(\mathrm{B}\),
Two spaceships are approaching one another. According to observers in our frame,(a) the left-hand ship moves to the right at \((4 / 5) c\) and the right-hand ship moves to the left at \((3 / 5) c\).
Astronaut A boards a spaceship leaving earth for the star Alpha Centauri, 4 light-years from earth, while her friend B stays at home. The ship travels at speed \(4 / 5 c\), and upon arrival

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