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Monday, 17 October 2016
Problem set 10
The function f(z)=u(x,y)+iv(x,y) is analytic at z=x+iy. The value of \nabla^2u at this point is:
0
undefined
\pi
e^{-\pi^2}
The function f(z)=u(x,y)+iv(x,y) is analytic at z=x+iy means u(x,y) and v(x,y) satisfy Cauchy-Reiman equations namely
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\therefore\nabla^2u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\nabla^2u=\frac{\partial }{\partial x}\left(\frac{\partial u}{\partial x}\right)+\frac{\partial }{\partial y}\left(\frac{\partial u}{\partial y}\right)\begin{align*}
\nabla^2u&=\frac{\partial }{\partial x}\left(\frac{\partial v}{\partial y}\right)-\frac{\partial }{\partial y}\left(\frac{\partial v}{\partial x}\right)\\
&=\frac{\partial^2 v}{\partial x\partial y}-\frac{\partial^2 v}{\partial y\partial x}=0
\end{align*}
A head of mass m slides on a smooth rod which is rotating about one end in a vertical plane with uniform angular velocity \omega. The Lagrangian of the system is :
Kinetic energy of head due to sliding on the rod =\frac{1}{2}m\dot r^2
Kinetic energy of head due to rotation=\frac{1}{2}mr^2\dot\theta^2
Potential energy of the head due to vertical height=mgr\sin\thetaL=\frac{1}{2}m\left(\dot r^2+r^2\dot\theta^2\right)-mgr\sin\theta
A partition function of two Bose particles each of which can occupy any of the two energy levels 0 and \epsilon is
1+e^{-2\epsilon/kT}+2e^{-\epsilon/kT}
1+e^{-2\epsilon/kT}+e^{-\epsilon/kT}
2+e^{-2\epsilon/kT}+e^{-\epsilon/kT}
e^{-2\epsilon/kT}+e^{-\epsilon/kT}
The partition function for Bose particles is given by Z=\sum\limits_Re^{-(n_1\epsilon_1+n_2\epsilon_2+\dots)/kT}, where n_1 is number of particles with energy \epsilon_1 and so on. Bose particles are indistinguishable particles and any number of particles can be in any state. Hence states of the system can be obtained as follow.
A one dimensional random walker takes steps to left or right with equal probability. The probability that the random walker starting from origin is back to origin after N even number of steps is
The probability p(m,N) that the particle will be at the position x = ml after N steps is
given by
{\scriptstyle p(m,N)=\frac{N!}{\left(\frac{N+m}{2}\right)!\left(\frac{N-m}{2}\right)!}p^{\frac{1}{2}\left(N+m\right)}q^{\frac{1}{2}\left(N-m\right)}}
Here p=q=\frac{1}{2}, the probability of taking steps to right and left, l is step length and x=ml=0. Hence m=0.
Hence
p(0,N)=\frac{N!}{\left(\frac{N}{2}\right)!\left(\frac{N}{2}\right)!}\left(\frac{1}{2}\right)^N
Five electrons (Fermions with spin 1/2\hbar) are kept in a one-dimensional infinite potential well with width a. (Ground state energy of single electron well =\frac{\hbar^2\pi^2}{2ma^2}). The first absorption line corresponds to energy:
\frac{\hbar^2\pi^2}{2ma^2}
\frac{5\hbar^2\pi^2}{2ma^2}
\frac{7\hbar^2\pi^2}{2ma^2}
\frac{11\hbar^2\pi^2}{2ma^2}
As electrons are fermions, there can be two electrons with same energy. Hence ground state configuration will be
two electrons in n=1 state, two electrons in n=2 state and one electron in n=3 state. The first absorption line corresponds to transition of electron from n=3 state to n=4 state. Energy of n=3 state is \frac{9\hbar^2\pi^2}{2ma^2} and energy of n=4 state is \frac{16\hbar^2\pi^2}{2ma^2}. Hence first absorption line corresponds to energy \frac{16\hbar^2\pi^2}{2ma^2}-\frac{9\hbar^2\pi^2}{2ma^2}=\frac{7\hbar^2\pi^2}{2ma^2}
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