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Starting from the time dependent Schr\"{o}dinger equation, an equation of continuity \[{\partial\rho\over\partial t} + \vec{\nabla}.\vec{j}=0\] is derived. Physical interpretation of the continuity equation is given in analogy with charge conservation in electromagnetic theory. The equation of continuity represents conservation of probability in quantum mechanics.
$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d#2}} \newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}} \newcommand{\average}[2]{\langle#1|#2|#1\rangle}$ Making use of analogy between optics ad mechanics we motivate the introduction of the Schrodinger equation.Here Fermi's "Lectures on Quantum Mechanics" has been followed very closely..
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We discuss the solution of time dependent one particle Schrodinger equation and obtain an expression for the propagator giving the time development.
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Time reversal operation in quantum mechanics of one particle is discussed.
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Using the classical Hamiltonian and the correspondence rule \(\vec p \to -i\hbar \nabla\), the expression for the Hamiltonian operator for a charged particle is written giving the time dependent Schr\"{o}dinger equation. The Schrodinger equation retains its form under gauge transformations if the wave function is assumed to transform as\[ \psi(\vec{r},t) \to \psi^\prime (\vec{r},t) = e^{-i(q/c)\Lambda(\vec{r},t)}\psi(\vec{r},t). \]
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For conservative systems, we show how solution of time dependent Schrodinger equation can be found by separation of variables. Explicit expression for the wave function at arbitrary time \(t\) is obtained in terms of energy eigenfunctions and eigenvalues.
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Starting from the time dependent Schrodinger equation, it is proved that the average value a dynamical variable \(\hat F\)obeys the equation\begin{equation} {d\over dt}\, \langle \hat{F} \rangle = \,\langle{\partial\over \partial t} \hat{F} \rangle + {1\over i\hbar} \langle\, [\hat{F},\hat{H} ]\, \rangle.\end{equation}
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