Problem/question
For page specific messages
For page author info
Find rotation matrix for a rotation by an angle \(\alpha\) about the axis \((1,2,1)\) where \(\cos\alpha=\frac{3}{5}, \sin\alpha =\frac{4}{5}\).
Consider a particle of mass \(\mu\) moving in a potential \[ V(r) = \frac{1}{2}\mu\omega^2 r^2 +\frac{\lambda^2}{2\mu r^2}. \] (a) Find condition on energy \(E\) and angular momentum \(L\) for circular orbits to exist.
(b) Does there exist a circular orbit for \(L=0\)?
(c) Assume \(L=0, E= \frac{25}{2}\mu\omega^2a^2\), \(\lambda=12\mu\omega^2\) Use initial conditions \[ r(t)\big|_{t=0} = 4a;\quad \dot{r}(t)\big|_{t=0} = 0 \text{ and } \dot{\theta(t)}\big|_{t=0}=0 \] solve the equations of motion and obtain \(r\), \(\theta\) as function of time. Describe the motion that takes place under conditions specified here.
KApoor
Comet Halley approaches the Sun to within 0.570 AU and its period is 75.6 years. How far from the Sun will this comet travel before it starts to return journey. Assume mass of Halley's comet is much smaller than the mass of the Sun.
[Given : 1 AU $= 1.5 \times 10^{11}$ m, $G= 6.63\times10^{-11}$ N.m\(^2\) /kg\(^2\), Mass of Sun=\( 1.99\times10^{30}\) kg ].
Obtain \(c_p\) for the ideal Bose gas for \(T > T_c\) . Show that ( for \(T > Tc\)) \[\frac{c_p}{c_v} = \frac{ 5 g_{5/2}(z) g_{1/2}(z)} {3 (g_{3/2}(z))^2}\]
HSMani
Compute the Bose- Einstein condensation temperature of a Bose-Einstein gas if \begin{equation} g(E) = \begin{cases} \alpha E^n, & \text{if } E>0 \\ 0 , & \text{if } E < 0 \end{cases} \end{equation} Assume the gas is in a cubical box of volume V.
HSMani
Find the density matrix of an ensemble of spin half prepared with 60 percent have $\sigma_z\,=\,+1$ and 40 percent have $\sigma_x\,=\,1$. Use the eigenstates of $\sigma_z$ as the basis. If the system is subjected to a magnetic field ( from time $t\,=\,0$) find the density matrix as a function of time. Assume the Hamiltonian is given by $H\,=-\,\mu B_0\sigma_z $ where $\mu\,B_0$ are a constants.
HSMani
Consider a cylindrical system of ideal gas with a movable piston as shown . All the wall.\, including the piston are thermally insulating. A weight W is placed on the piston. Treating this system as a closed one, Show that $$ \frac{W}{A}\,=\,\frac{2E}{3V}$$ where A is the area of cross section of the cylinder, E the energy and V the volume of gas.
HSMani
A box of volume, \(V = L^3\) , contains an ideal gas of \(N\) identical atoms, each of which has spin, \(s = 1/2\), and magnetic moment, \(\mu\). A magnetic field, \(B\) is applied to the system. (a) Compute the partition function for this system. (b) Compute the internal energy and the heat capacity. (c) What is the magnetization?
Show that in the canonical ensemble formalism, the entropy $S$ of the system is related to the partition function $Q$ as given below. $$S=k_B\left[\ln Q+T\left(\frac{\partial\ln Q}{\partial T}\right)_V\right].$$}
KPN
The canonical partition function of a system of $N$ hypothetical particles each of mass $m$, confined to a volume $V$ at temperature $T$ is given by, $$Q(T,V,N) = V^N\left(\frac{2\pi k_B T}{m}\right)^{5N/2}.$$ Determine the equation of state of the hypothetical system. Also find $C_V$ - heat capacity at constant volume. Identify the hypothetical system. How many degrees of freedom does each particle of the hypothetical system have ?
KPN
Pages
||Message]
