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Main points of time evolution in Schrodinger picture are summarized.

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The tension $\tau$ in an elastic rubber band is given by
$$ \tau\,=\,aT\left(\frac{L}{L_0(T)}-\,\left(\frac{L_0(T)}{L}\right)^2\right),$$
where $a$ is a constant, $L_0(T)$is the unstretched length at zero tension, and is a function of temperature only.

(a) Write the first law using the work done when it is elongated and gets a supply of heat. ( Be careful of signs!)

(b) Use the first law to write $dF$, where $F$ is the free energy of the rubber band.

(c) Solve for the free energy $F$ and show that
$$ F(T,L)\,-\,F(T,L_0(T))\,=\,aT\left(\frac{L^2}{2L_0(T)}\,+\,\frac{L_0(T)^2}{L^2}\,-\,\frac{3L_0(T)}{2}\right)$$
and the entropy $S$
$$ S(T,L)\,-\,S(T,L_0(T))\,=\,a\left(\frac{3L_0}{2}\,-\,\frac{L_0^2}{L}\,-\,\frac{L^2}{2L_0}\right)\,-\,aT\left(\frac{3}{2}\,-\,\frac{2L_0}{L}\,+\,\frac{L^2}{2L_0^2}\right)\frac{dL_0(T)}{dT} $$

(d)Find the heat $Q$ transferred to the elastic band when it is stretched from $L_0$ to $L$ isothermally.

(e) Show that
$$ \left(\frac{\partial T}{\partial L}\right)_S \,=\,\frac{aTL_0^2}{c_LL^2}\left(-1\,+\,\left(\frac{L}{L_0}\right)^3\,+\,\frac{Ta}{L_0}\frac{dL_0}{dT}\left(2\,+\,\left(\frac{L}{L_0}\right)^3\right)\right)$$

where
$$c_L\,=\,\left(\frac{DQ}{\partial T}\right)_L.$$

The electric potential due to charge spread uniformly on a finite line segment is computed.The electric potential due to charge spread uniformly on a finite line segment is computed.

The basic notions of probability theory, simple events, sample space and ensemble, are introduced. The probability of compound events, independent events and joint and conditional probability are defined. Examples are given to illustrate the basic concepts.

The time evolution of a general quantum system is reviewed in an abstract setting. The eigen states of energy are seen to have all properties that make them qualify for being called stationary states.The stationary states have the property that all observable quantities remain constant in time.

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The vector potential is introduced using the Maxwell's equation \(\nabla \times \vec{B}=0\) and the equation \( \nabla \times \vec{B} = \mu_0 \vec{j}\) is derived. The expression for the magnetic field is obtained as volume integral, the Biot Savart law, is derived. The expressions for the magnetic field for the surface current and the line current are given.