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1. Optics Mechanics Analogy --- Road to Wave Mechanics 2. Time Dependent Schrodinger Equation in Coordinate Representation 2.1 Probability Conservation We shall discuss some aspects of the Schrodinger equation using the coordinate representation for a particle in a potential \(V(\vec{r})\) 2.2 Schrodinger Equation for a Charged Particle 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\[\psi(\vec{r},t) \to \psi^\prime (\vec{r},t) = e^{-i(q/c)\Lambda(\vec{r},t)}\psi(\vec{r},t).\] Time reversal operation in quantum mechanics of one particle is discussed. 3 Solution of time dependent Schr\"{o}dinger equation 3.1 Dependent Schrodinger Equation :Solution for Wave function at time \(t\) We show how solution of time dependent Schrodinger equation can be found. Explicit expression for the wave function at arbitrary time \(t\) is obtained in terms of energy eigen functions and eigen values. 3,2 Time Variation of Average Values Starting from the time dependent Schrodinger equation, it is proved that the average value a dynamical variable \(\hat F\)obeys the equation\[\begin{equation} {d\over dt}\, \langle \hat{F} \rangle = \,\langle{\partial\over \partial t} \hat{F} \rangle + {1\over i\hbar} \langle\, [\hat{F},\hat{H} ]\, \rangle.\end{equation}\] We discuss the solution of time dependent one particle Schrodinger equation and obtain an expression for the propagator giving the time development. |
