Modern Classical Mechanics 1st Edition T. M. Helliwell, V. V. Sahakian - Solutions

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(a) Using the thermodynamic relation\[C_{P}-C_{\mathrm{V}}=T(\partial P / \partial T)_{\mathrm{V}}(\partial V / \partial T)_{P}=-T(\partial P / \partial T)_{\mathrm{V}}^{2} /(\partial P / \partial V)_{T}\]and the equation of state (10.3.9), we get\[\frac{C_{P}-C_{\mathrm{V}}}{N k}=-\frac{T(\partial
Since, by definition,\[\alpha=\mathrm{v}^{-1}(\partial \mathrm{v} / \partial T)_{P} \text { and } B^{-1} \equiv \kappa_{T}=-\mathrm{v}^{-1}(\partial \mathrm{v} / \partial P)_{T}\]we must have:\[\begin{equation*}[\partial(\alpha \mathrm{v}) / \partial P]_{T}=-\left[\partial\left(\mathrm{v} B^{-1}
To the desired approximation,\[\begin{equation*}\frac{P}{k T} \equiv \frac{1}{V} \ln \mathscr{Q}=\frac{1}{\lambda^{3}}\left(z-a_{2} z^{2}\right), \quad n=\frac{N}{V}=\frac{1}{\lambda^{3}}\left(z-2 a_{2} z^{2}\right) \tag{1a,b}\end{equation*}\]where \(a_{2}\) is the second virial coefficient of the
We consider a volume element \(d x_{1} d y_{1} d z_{1}\) around the point \(P\left(x_{1}, 0,0 \right)\) in solid 1 and a volume element \(d x_{2} d y_{2} d z_{2}\) around the point \(Q\left(x_{2}, y_{2}, z_{2} \right)\) in solid 2. The force of attraction between these elements will
Referring to equation (10.5.31) for the phase shifts \(\eta_{l}(k)\) of a hard-sphere gas, show that for \(k D \ll 1\)\[\eta_{l}(k) \simeq-\frac{(k D)^{2 l+1}}{(2 l+1)\{1 \cdot 3 \cdots(2 l-1)\}^{2}}\]
The symmetrized wave functions for a pair of non-interacting bosons/fermions are given by\[\Psi_{\alpha}\left(\mathbf{r}_{1}, \mathbf{r}_{2}\right)=\frac{1}{\sqrt{2} V}\left(e^{i \mathbf{k}_{1} \cdot \mathbf{r}_{1}} e^{i \mathbf{k}_{2} \cdot \mathbf{r}_{2}} \pm e^{i \mathbf{k}_{1} \cdot
A particle with spin \(J\) can be in any one of the \((2 J+1)\) spin states characterized by the spin functions \(\chi_{m}(m=-J, \ldots, J)\). For a pair of such particles, we will have \((2 J+1)^{2}\) spin states characterized by the symmetrized spin functions\[\chi_{m_{1}}(1) \chi_{m_{2}}(2) \pm
To derive the desired results, we make the following observations:(i) Since a pair of particles with spin \(J\) has \((2 J+1)^{2}\) possible spin states while a pair of spinless particles has only one, we have to divide the expression for \(b_{2}\) pertaining to the former by \((2 J+1)^{2}\) so
Expand the definition of the pair density \(n_{2}\left(\boldsymbol{r}, \boldsymbol{r}^{\prime} \right)\) in powers of the fugacity \(z\) using the grand canonical partition function and the Mayer functions \(f_{i j}=\exp \left(-\beta u\left(r_{i j} \right)
Let \(P(r)\) be the cumulative probability that no particles are closer than \(r\) to a given particle. Breaking up the interval between zero and \(r\) into small intervals starting ar \(r_{k}=k \Delta r\) with width \(\Delta r\) gives\[P(r)=\prod_{k=0}^{r / \Delta r}\left(1-4 \pi n g\left(r_{k}
(a) In this problem we are concerned with the integral\[I=\int_{0}^{\infty} \frac{\partial u}{d r} e^{-\beta u} r^{3} d r\]Integrating by parts, we get\[I=-\left.\frac{1}{\beta}\left[e^{-\beta u}+c \right] r^{3} \right|_{0} ^{\infty}+\frac{1}{\beta} \int_{0}^{\infty}\left(e^{-\beta u}+c \right) 3
Use\[\begin{aligned}{\left[\kappa_{T}(n, T) \right]^{-1} } & =n\left(\frac{\partial p}{\partial n} \right)_{T} \\P(n, T) & =n^{2}\left(\frac{\partial f}{\partial n} \right)_{T}\end{aligned}\]where \(f=A / N\) is the Helmholtz free energy per particle. Then\[\begin{aligned}P(n, T) & =p\left(n_{0}, T
The most general Gaussian distribution of variables \(\left\{u_{1}, \cdots, u_{N} \right\}\) is of the form\[P\left(u_{1}, \cdots, u_{N} \right) \sim \exp \left(-\frac{1}{2} \boldsymbol{u}^{T} A \boldsymbol{u} \right)\]where \(A\) is a symmetric positive definite matrix. The matrix has only
(a) We replace the sum over pp appearing in eqn. (11.3.14) by an integral, viz.∫∞0{ε(p)−p22m−4πaℏ2NmV+(4πaℏ2NmV)2mp2}V⋅4πp2dph3∫0∞{ε(p)−p22m−4πaℏ2NmV+(4πaℏ2NmV)2mp2}V⋅4πp2dph3Substituting p=(8πaℏ2N/V)1/2xp=(8πaℏ2N/V)1/2x, we
Assume that in the virial expansionPvkT=1−∞∑j=1jj+1βj(λ3v)jwhere βj are the irreducible cluster integrals of the system, only terms with j=1 and j=2 are appreciable in the critical region. Determine the relationship between β1 and β2 at the critical point, and show that kTc/Pcvc=3.
The given equation of state isP=kTv−be−a/kTv(1)(1)P=kTv−be−a/kTvIt follows thatMisplaced &Misplaced &At the critical point, both these derivatives vanish - with the result thatExtra \left or missing \rightExtra \left or missing \rightwhence vc=2bvc=2b and kTc=a/4bkTc=a/4b. Equation (1) then
The given equation of state (for one mole) of the gas is\[\begin{equation*}P=R T /(\mathrm{v}-b)-a / \mathrm{v}^{n} \quad(n>1) . \tag{1}\end{equation*}\]Equating \((\partial P / \partial \mathrm{v})_{T}\) and \(\left(\partial^{2} P / \partial \mathrm{v}^{2}\right)_{T}\) to zero, we
The partition function of the system may be written as \(\sum_{L} \exp f(L)\), where\[f(L)=\ln N !-\ln (N p) !-\ln (N q) !+\beta N\left(\frac{1}{2} q J L^{2}+\mu B L\right)\]Using the Stirling approximation (B.29), we get\[f(L) \approx-N p \ln p-N q \ln q+\beta N\left(\frac{1}{2} q J L^{2}+\mu B
The relevant results of the preceding problem are\[\begin{align*}& \frac{A}{N}=k T\left\{\frac{1+L^{*}}{2} \ln \frac{1+L^{*}}{2}+\frac{1-L^{*}}{2} \ln \frac{1-L^{*}}{2}\right\}-\frac{1}{2} q J L^{* 2}-\mu B L^{*} \tag{1}\\& \bar{N}_{+}=\frac{1}{2} N\left(1+L^{*}\right), \bar{N}_{-}=\frac{1}{2}
The Hamiltonian of this model may be written as\[H=-\frac{1}{2} c \sum_{i eq j} \sigma_{i} \sigma_{j}-\mu B \sum_{i} \sigma_{i}\]The double sum here is equal to \(\sum_{i} \sigma_{i} \sum_{j} \sigma_{j}-\sum_{i} \sigma_{i}^{2}=(N L)^{2}-N\) which, for \(N \gg 1\), is essentially equal to \(N^{2}
Let us concentrate on one particular spin, \(\mathrm{s}_{0}\), in the lattice and look at the part of the energy \(E\) that involves this spin, viz. \(-2 J \sum_{j=1}^{q} \mathbf{s}_{0} \cdot \mathbf{s}_{j}-g \mu_{B} \mathbf{s}_{0} \cdot \mathbf{H}\); for notation, see Secs. 3.9 and 12.3. In the
Let us concentrate on one particular spin, \(\mathrm{s}_{0}\), in the lattice and look at the part of the energy \(E\) that involves this spin, viz. \(-2 J \sum_{j=1}^{q} \mathbf{s}_{0} \cdot \mathbf{s}_{j}-g \mu_{B} \mathbf{s}_{0} \cdot \mathbf{H}\); for notation, see Secs. 3.9 and 12.3. In the
We shall consider only the Heisenberg model; the study of the Ising model is somewhat simpler. Following the procedure of Problem 12.7, we find that the "effective field" \(\mathbf{H}_{a}\) experienced by any given spin \(\mathbf{s}\) on the sublattice \(a\) would
We shall consider only the Heisenberg model; the study of the Ising model is somewhat simpler. Following the procedure of Problem 12.7, we find that the "effective field" \(\mathbf{H}_{a}\) experienced by any given spin \(\mathbf{s}\) on the sublattice \(a\) would
To determine the equilibrium distribution \(f(\sigma)\), we minimize the free en\(\operatorname{ergy}(E-T S)\) of the system under the obvious constraint \(\sum_{\sigma} f(\sigma)=1\). For this, we vary the function \(f(\sigma)\) to \(f(\sigma)+\delta f(\sigma)\) and require that the resulting
The configurational energy of the lattice is given by\[\begin{aligned}E= & \frac{1}{2} q N\left[\varepsilon_{11} \cdot x_{A}(1+X) \cdot x_{A}(1-X)+\varepsilon_{12}\left\{x_{A}(1+X)\left(x_{B}+x_{A} X\right)\right.\right. \\& \left.\left.+x_{A}(1-X)\left(x_{B}-x_{A}
Consider a two-component solution of \(N_{A}\) atoms of type \(A\) and \(N_{B}\) atoms of type \(B\), which are supposed to be randomly distributed over \(N\left(=N_{A}+N_{B}\right)\) sites of a single lattice. Denoting the energies of the nearest-neighbor pairs \(A A, B B\), and \(A B\) by
(a) Setting \(\bar{N}_{++}+\bar{N}_{--}+\bar{N}_{+-}=\frac{1}{2} q N\), we find that, in equilibrium, \(\gamma=1 /\left(1+s \bar{L}^{2}\right)\). So, in general, it may be written as \(1 /\left(1+s L^{2}\right)\).(b) As in Problem 12.4, we write \(Q(B, T)=\sum_{L} \exp f(L)\),
Show that in the Bethe approximation the entropy of the Ising lattice at \(T=T_{C}\) is given by the expression\[\frac{S_{c}}{N k}=\ln 2+\frac{q}{2} \ln \left(1-\frac{1}{q}\right)-\frac{q(q-2)}{4(q-1)} \ln \left(1-\frac{2}{q}\right) .\]Compare this result with the one following from the
Examine the critical behavior of the low-field susceptibility, \(\chi_{0}\), of an Ising model in the Bethe approximation of Section 12.6, and compare your results with equations (12.5.22) of the Bragg-Williams approximation.Equation (12.5.22) Xo(T) (N/k)(T-Tc) (TTC, B 0), (22a) Xo(T)
A function \(f(x)\) is said to be concave over an interval \((a, b)\) if it satisfies the property\[f\left\{\lambda x_{1}+(1-\lambda) x_{2}\right\} \geq \lambda f\left(x_{1}\right)+(1-\lambda) f\left(x_{2}\right),\]where \(x_{1}\) and \(x_{2}\) are two arbitrary points in the interval \((a, b)\)
In view of the thermodynamic relationshipCV=TV(∂2P/∂T2)V−TN(∂2μ/∂T2)VCV=TV(∂2P/∂T2)V−TN(∂2μ/∂T2)Vfor a fluid, μμ being the chemical potential of the system, Yang and Yang (1964) pointed out that, if CVCV is singular at T=TcT=Tc, then either (∂2P/∂T2)V(∂2P/∂T2)V or
Determine the numerical values of the coefficients \(r_{1}\) and \(s_{0}\) of equation (12.9.5) in (i) the Bragg-Williams approximation of Section 12.5 and (ii) the Bethe approximation of Section 12.6. Using these values of \(r_{1}\) and \(s_{0}\), verify that equations (12.9.4), (12.9.9),
Consider a system with a modified expression for the Landau free energy, namely\[\psi_{h}(t, m)=-h m+q(t)+r(t) m^{2}+s(t) m^{4}+u(t) m^{6},\]with \(u(t)\) a fixed positive constant. Minimize \(\psi\) with respect to the variable \(m\) and examine the spontaneous magnetization \(m_{0}\) as a
In the preceding problem, put \(s=0\) and approach the tricritical point along the \(r\)-axis, setting \(r \approx r_{1} t\). Show that the critical exponents pertaining to the tricritical point in this model are\[\alpha=\frac{1}{2}, \beta=\frac{1}{4}, \gamma=1, \text { and } \delta=5\]
Consider a fluid near its critical point, with isotherms as sketched in Figure 12.3. Assume that the singular part of the Gibbs free energy of the fluid is of the form\[G^{(s)}(T, P) \sim|t|^{2-\alpha} g\left(\pi /|t|^{\Delta}\right)\]where \(\pi=\left(P-P_{c}\right) / P_{c}, t=\left(T-T_{c}\right)
Consider a model equation of state which, near the critical point, can be written as\[h \approx a m\left(t+b m^{2}\right)^{\Theta} \quad(1
Assuming that the correlation function \(g\left(\boldsymbol{r}_{i}, \boldsymbol{r}_{j}\right)\) is a function only of the distance \(r=\left|\boldsymbol{r}_{j}-\boldsymbol{r}_{i}\right|\), show that \(g(r)\) for \(r eq 0\) satisfies the differential equation\[\frac{d^{2} g}{d r^{2}}+\frac{d-1}{r}
Consider the correlation function \(g(r ; t, h)\) of Section 12.11 with \(h>0 .{ }^{22}\) Assume that this function has the following behavior:\[g(r) \sim e^{-r / \xi(t, h)} \times \text { some power of } r \quad(t \gtrsim 0) \text {, }\]such that \(\xi(0, h) \sim h^{-v^{c}}\). Show that \(v^{c}=v
Liquid He4He4 undergoes a superfluid transition at T≃2.17 KT≃2.17 K. The order parameter in this case is a complex number ΨΨ, which is related to the Bose condensate density ρ0ρ0 asρ0∼|Ψ|2∼|t|2β(t≲0)ρ0∼|Ψ|2∼|t|2β(t≲0)The superfluid density ρsρs, on the other hand,
The surface tension, \(\sigma\), of a liquid approaches zero as \(T \rightarrow T_{c}\) from below. Define an exponent \(\mu\) by writing\[\sigma \sim|t|^{\mu} \quad(t \lesssim 0) .\]Identifying \(\sigma\) with the "free energy associated with a unit area of the liquid-vapor interface," argue that
This problem is similar to Problem 7.8 of the Bose gas and can be done the same way. In the limit \(z \rightarrow \infty\), which corresponds to \(T \rightarrow 0 K\),\[w^{2} \approx 2 k T \ln z / 3 m\]which tends to the limiting value \(2 \varepsilon_{F} / 3 m\). Thus\[w_{0}=\left(2
This problem is similar to Problem 7.8 of the Bose gas and can be done the same way. In the limit \(z \rightarrow \infty\), which corresponds to \(T \rightarrow 0 K\),\[w^{2} \approx 2 k T \ln z / 3 m\]which tends to the limiting value \(2 \varepsilon_{F} / 3 m\). Thus\[w_{0}=\left(2
Obtain numerical estimates of the Fermi energy (in \(\mathrm{eV}\) ) and the Fermi temperature (in \(\mathrm{K}\) ) for the following systems:(a) conduction electrons in silver, lead, and aluminum;(b) nucleons in a heavy nucleus, such as \({ }_{80} \mathrm{Hg}^{200}\), and(c) \(\mathrm{He}^{3}\)
This problem is similar to Problem 7.14 of the Bose gas and can be done the same way.Parts (i) and (ii) are straightforward. For part (iii), we have to show that\[\begin{equation*}\frac{C_{P}}{C_{\mathrm{V}}}=1+\left(\frac{s}{n}\right)^{2} \frac{C_{\mathrm{V}}}{N k} \frac{f_{(n / s)-1}(z)}{f_{n /
For \(T \gg T_{F}\), we get\[\frac{C_{\mathrm{V}}}{N k} \simeq \frac{n}{s}, \frac{C_{P}-C_{\mathrm{V}}}{N k} \simeq 1, \quad \text { so that } \frac{C_{P}}{N k} \simeq\left(\frac{n}{s}+1 \right)\]For \(T \ll T_{F}\), we obtain [see formula (E.17)]\[\begin{aligned}\frac{C_{\mathrm{V}}}{N k} &
8.13. The Fermi energy of the gas is given by the obvious relation\[\begin{equation*}N=\int_{0}^{\varepsilon_{F}} a(\varepsilon) d \varepsilon \tag{1}\end{equation*}\]At the same time, the quantities \(N\) and \(U\), as functions of \(\mu\) and \(T\), are given by the standard
In the notation of Sec. 3.9, the potential energy of a magnetic dipole in the presence of a magnetic field \(\boldsymbol{B}=(0,0, B)\) is given by the expression \(-\left(g \mu_{B} m\right) B\), where \(m=-J, \ldots,+J\). The total energy \(\varepsilon\) of the dipole is then given by
We note that the symbol \(\mu_{0}(x N)\) denotes the chemical potential ( \(\left.\equiv k T \ln z \right)\) of an ideal gas of \(x N\) "spinless" \((g=1)\) fermions. The corresponding fugacity \(z\) is determined by the equation\[\begin{equation*}f_{3 / 2}(z)=x N \lambda^{3} / V
The ground-state energy of a relativistic gas of electrons is given by\[E_{0}=\frac{8 \pi V}{h^{3}} \int_{0}^{p_{F}} m c^{2}\left[\left\{1+(p / m c)^{2} \right\}^{1 / 2}-1 \right] p^{2} d p\]Making the substitution (8.5.9), we get\[\begin{equation*}E_{0}=\frac{8 \pi m^{4} c^{5} V}{h^{3}}
Utilizing the result obtained in Problem 8.13, we have for a Fermi gas at low temperatures\[\begin{equation*}\frac{C_{\mathrm{V}}}{N k}=\frac{\pi^{2}}{3} \frac{a\left(\varepsilon_{F}\right)}{N} k T \tag{1}\end{equation*}\]Now, the density of states for the relativistic gas is given by, see eqn.
The number of fermions in the trap is\[N(T, \mu)=\int \frac{d \varepsilon \varepsilon^{2}}{2(\hbar \omega)^{3}} \frac{1}{e^{\beta(\varepsilon-\mu)}-1}=\int_{0}^{\varepsilon_{F}} \frac{d \varepsilon \varepsilon^{2}}{2(\hbar \omega)^{3}}=\frac{\varepsilon_{F}^{3}}{6(\hbar \omega)^{3}} .\]Using \(k
Consider the equation of motion\[\frac{d^{2} u}{d \varphi^{2}}+u-\frac{1}{p}=3 \lambda u^{2}\]where \(p\) and \(\lambda\) are constants. Find the solution using perturbation theory to first order in the small parameter \(\lambda\). Assess whether your approximate solution is a good one by solving
Using numerical methods, solve the differential equation\[\frac{d q}{d t}=-\alpha q+\beta\]Compare your results with the exact solution as a function of the discrete time-step you use, and the order/method of the algorithm you adopt. Try in particular contrasting 4th order Runge-Kutta with another
Consider the logistic map discussed in the text. To gauge the density of bifurcations, one uses a measure of distance between fixed points as follows. Define \(d=x^{*}-(1 / 2)\) as the distance between the fixed point \(1 / 2\) and the nearest fixed point to it - labeled \(x^{*}\). That is, you
Consider a particle in a Newtonian potential \(V(r)=-k / r+\epsilon / r^{n}\) for some integer \(n\). Using the alternate variable \(u=1 / r\), (a) show that the radial equation of motion can be put into the form\[\frac{d^{2} u}{d \varphi^{2}}+u=\frac{1}{p}+n \kappa u^{n+1}\]where\[p \equiv
Consider the one dimensional harmonic oscillator with angular frequency \(\omega\) perturbed by the small non-linear potential \(\epsilon q^{4}\).(a) Find the solution using the perturbation technique introduced in the text to first order in the small perturbation;(b) Improve your solution from
The celebrated Lorentz attractor is described by the differential equations\[\frac{d x}{d t}=-\sigma x+\sigma y \quad, \quad \frac{d y}{d t}=-x z+\alpha x-y \quad, \quad \frac{d z}{d t}=x y-\beta z,\]and is used to described chaotic fluid dynamics involving heat flow. It is parameterized by
Consider the recursion relation\[x_{n+1}=x_{n}+y_{n} \quad, \quad y_{n+1}=a y_{n}-b \cos \left(x_{n}+y_{n}\right)\]where \(a\) and \(b\) are constants; this system is known as the standard map. Analyze the system as we did for the logistic map in the text. In particular, explore regions of the
Show that the standard map of the previous problem described by\[x_{n+1}=x_{n}+y_{n} \quad, \quad y_{n+1}=a y_{n}-b \cos \left(x_{n}+y_{n}\right),\]can be obtained by discretizing time in the Hamiltonian equations of motion of the following physical system: a planar pendulum in the absence of
Consider the variant of the standard map described by the recursion relation\[y_{n+1}=y_{n}+k \sin x_{n} \quad, x_{n+1}=x_{n}+y_{n+1}\]where \(k\) is a constant. (a) Study the distortion of the KAM tori as \(k\) is taken from \(k=0\) to \(k=0.6\). (b) Analyze the system when \(k=0.9716\). Compute
Consider the map given by\[x_{n+1}=x_{n} e^{\alpha\left(1-x_{n}\right)}\]used to study population growth limited by disease. Analyze the system as done for the logistic map in the text, identifying the onset of chaos and bifurcations, if any. Consider in particular values of \(\alpha=1.5,2,2.7\).
Consider the map given by\[x_{n+1}=\alpha \sin \left(\pi x_{n}\right)\]where \(0
Consider the two dimensional recursion\[x_{n+1}=y_{n}+1-\alpha x_{n}^{2} \quad, \quad y_{n+1}=\beta x_{n},\]introduced by Hénon to describe chaotic behavior in the trajectories of asteroids. Study the sequence for \(\alpha=1.4\) and \(\beta=0.3\) with the initial condition \(x_{0}=0.63135448\) and
Consider the logistic map analyzed in the text. Divide the range \((0,1)\) into \(N\) equal intervals and numerically compute the probabilities \(p_{k}\) that the recursion lands in the \(k\) th interval. Then compute the entropy\[S \equiv-\sum_{k=1}^{N} p_{k} \ln p_{k}\]Do this so as to build up
Consider a magnetic compass needle with moment of inertia \(I\) and magnetic dipole moment \(\boldsymbol{\mu}\), free to rotate in the \(x-y\) plane. Denote the polar angle by \(\theta\). A time dependent external magnetic field \(\mathbf{B}=B_{0} \cos \omega t \hat{\boldsymbol{x}}\) applies a
Consider the two dimensional map\[x_{n+1}=\alpha\left(x_{n}-\frac{1}{4}\left(x_{n}+y_{n}\right)^{2}\right) \quad, \quad y_{n+1}=\frac{1}{\alpha}\left(y_{n}+\frac{1}{4}\left(x_{n}+y_{n}\right)^{2}\right)\]that approximates the chaotic scattering behavior of a projectile off a region near the origin
A ball of mass \(m\) is dropped from rest above the surface of an airless moon in essentially uniform gravity \(g\). (a) If \(y\) is the vertical axis, show that the Hamilton-Jacobi equation for the ball is\[\frac{1}{2 m}\left(\frac{\partial S}{\partial y}\right)^{2}+m g y+\frac{\partial
Starting from rest at time \(t=0\) and at altitude \(h_{0}\), a block of mass \(M\) slides down a frictionless plane inclined at angle \(\alpha\) to the horizontal. There is a uniform gravitational field \(g\) directed vertically downward. (a) Write the Hamilton-Jacobi equation for the block. (b)
A spaceship drifts in gravity-free space. If its velocity is \(v_{0}\) in the positive \(x\) direction at time \(t=0\), find its subsequent motion using the Hamilton-Jacobi method.
A projectile is fired in a uniform gravitational field \(g\) with initial speed \(v_{0}\) and angle \(\theta_{0}\) relative to the horizontal. Note that the Hamiltonian is\[H=\frac{p_{x}^{2}}{2 m}+\frac{p_{y}^{2}}{2 m}+m g y\]Find the projectile's motion \(x(t)\) and \(y(t)\) using the
A block \(m\) can slide along a frictionless table top in the \(x, y\) plane, subject to the forces exerted by one spring that lies along the \(x\) axis and has force-constant \(k_{1}\), and another spring that lies along the \(y\) axis and has force-constant \(k_{2}\). Assume the motions of \(m\)
A thin, stiff metal ring of radius \(R\) is placed in a vertical plane, and made to spin with constant angular velocity \(\omega\) about a vertical axis that passes through the center of the ring. A bead of mass \(m\) is free to slide around the ring, with its position defined by its angle
The Hamiltonian for a particle of mass \(m\), with arbitrary initial position and velocity, and subject to an inverse-square attractive force, can be written\[H=\frac{p_{r}^{2}}{2 m}+\frac{p_{\theta}^{2}}{2 m r^{2}}-\frac{k}{r}\]where \(k\) is a positive constant. (a) Write the Hamilton-Jacobi
Using the action-angle variables approach, find the oscillation frequency of a one-dimensional simple harmonic oscillator of mass \(m\) and force-constant \(k\).
Using the action-angle variables approach, find the frequencies of oscillation of a planet orbiting the sun, for both the radial and angular motions. What is the consequence of the fact that these frequencies turn out to be the same? You might need to use contour integration to evaluate an integral.
A particle of mass \(m\) moves in one dimension subject to a force \(F\) of constant magnitude, but directed toward the left for positive \(x\) and to the right for negative \(x\). Thus the potential energy has the form \(U=k|x|\) for some constant \(k\). Using action-angle variables, find the
As an example of adiabatic invariance, consider the preceding problem in the case where the magnitude of the force \(F\) is slowly changed, i.e., slow relative to the oscillation period of the particle. Which (if any) of the following quantities remain constant in the adiabatic limit? (a) the
A particle of mass \(m\) can move along the positive \(x\) axis only, subject to a constant force to the left. That is, there is an impenetrable wall at \(x=0\) preventing it from reaching negative \(x\), and for positive \(x\) there is a constant force attracting the particle back towards the
A particle of mass \(m\) can move along the positive \(x\) axis only, subject to a Hooke's-law spring force \(F=-k x\). That is, there is an impenetrable wall at \(x=0\) preventing it from reaching negative \(x\), and for positive \(x\) there is a spring force attracting the particle back towards
A particle of mass \(m\) is confined to move inside a cubical box of side length \(L\), with potential energy zero. Find the allowed energies of the particle according to the "old quantum theory."
One end of a spring of rest-length zero and force-constant \(k\) is attached to a fixed point while the other end is attached to a ball of mass \(m\), which is otherwise free to move as it likes in three-dimensional space. Find the allowed energies of the system according to the "old quantum
From the classical point of view, the electron in a hydrogen atom moves under the influence of a central attractive force \(F=-e^{2} / r^{2}\) caused by the proton nucleus. According to "old quantum theory" the phase integrals over \(r\) and \(\theta\) are given by\[\oint p_{r} d r=n_{1} h \quad
If a quantum mechanical particle has definite energy \(E\) we can write its wave function in the form \(\Psi(\mathbf{r}, t)=\psi(\mathbf{r}) e^{-i E t / \hbar}\). (a) Substitute this into the full Schrödinger equation to show that the time independent Schrödinger equation for \(\psi(\mathbf{r})\)
A particle of mass \(m\) is trapped inside a three-dimensional box with sides of length \(L\). The potential energy of the particle is zero for \(0
The lowest-energy (i.e., "ground state") eigenfunction of the electron in a hydrogen atom is spherically symmetric. (a) Using the time-independent Schrödinger equation, find this eigenfunction in terms of the radius \(r\) of the electron from the nucleus as origin. (b) Find also the electron's
The ground-state wave function of a one-dimensional simple harmonic oscillator of mass \(m\) and force-constant \(k\) is the Gaussian function \(\psi(x)=A e^{-\alpha x^{2}}\) where \(A\) is a normalization constant (adjusted to make \(\int \psi^{*} \psi d x=1\) ) and \(\alpha\) is also a constant.
The angular momentum vector of a particle of mass m is written as \(\mathbf{L}=\mathbf{r} \times \mathbf{p}\). Find the Poisson brackets of any two components of the angular momentum vector in Cartesian coordinates. Do this using the explicit representation of the Poisson bracket as derivatives
(a) Repeat the previous problem but instead use only the four properties of the Poisson bracket and the facts that \(\left\{x, p^{x}\right\}=\left\{y, p^{y}\right\}=\left\{z, p^{z}\right\}=1\) while the other brackets of positions and momenta vanish.(b) Write the quantum version of this algebra
Using equation 11.140, find the generator of the transformation that can translate a function of the canonical momentum \(f(p)\) by \(p \rightarrow p+\epsilon\).
Using Eq. (11.140), show that if we use G= ϵLx as a generator of a transformation (where \(L_{x}\) is the \(x\)-component of the angular momentum), we end up rotating the components of the position vector \(\mathbf{r}\) by an infinitesimal angle \(\epsilon\) about the \(x\)-axis. Show this by
Inspired by the previous problem, find the generators that rotate the momentum vector \(\mathbf{p}\) about the \(x, y\), and \(z\) axes by infinitesimal angles.Data from previous problemUsing Eq. (11.140), show that if we use G= ϵLx as a generator of a transformation (where \(L_{x}\) is the
Compute the Poisson bracket of any components of position or momentum with any component of angular momentum. Use the Poisson bracket representation as derivatives with respect to canonical coordinates and momenta. You might find it useful to write \(L^{i}=\varepsilon^{i j k} x^{j} p^{k}\), and use
Repeat the previous problem but use only the four properties of the Poisson bracket and the particular Poisson brackets between the components of the position and momentum vectors. From this, deduce the corresponding commutation relations in quantum mechanics.Data from previous problemCompute the
Find the generator that performs a Galilean boost in the \(x\)-direction by an infinitesimal speed \(\epsilon\). Do this using equation 11.140, working backwards and considering expected effects of the transformation on position and momentum of a particle.
(a) Find the generator that performs an infinitesimal scale transformation, where \(\mathbf{r}^{\prime}=(1+\epsilon) \mathbf{r}\), and similarly for momentum. (b) Find the brackets of this generator with the components of angular momentum of a particle.
Show that\[\hat{q}\left(t_{0}+\Delta t\right)=e^{\frac{i}{\hbar} \hat{H} \Delta t} \hat{q}\left(t_{0}\right) e^{-\frac{i}{\hbar} \hat{H} \Delta t}\]where we define\[e^{-\frac{i}{\hbar} \hat{H} \Delta t}=\sum_{n=0}^{\infty} \frac{1}{n !}\left(\frac{-i
Verify the particular solutions given of the inhomogeneous first-order equations for the perihelion precession, as given in equation 10.34.Data from equation 10.34 GMe E (u2+u)-GMmu -u. 2m mc
The metric of flat, Minkowski spacetime in Cartesian coordinates is ds2 = \(-c^{2} d t^{2}+d x^{2}+d y^{2}+d z^{2}\). Show that the geodesics of particles in this spacetime correspond to motion in straight lines at constant speed.
The geodesic problem in the Schwarzschild geometry is to make stationary the functional \(S=\int I d \tau\), where\[I=\sqrt{(1-2 \mathcal{M} / r) c^{2} \dot{t}^{2}-(1-2 \mathcal{M} / r)^{-1} \dot{r}^{2}-r^{2} \dot{\varphi}^{2}}\]where \(\dot{t}=d t / d \tau\), etc. Use this integrand in the

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Modern Classical Mechanics 1st Edition T. M. Helliwell, V. V. Sahakian - Solutions

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