Browse & Filter

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The Green function and the propagator for time dependent Schr\:{o}dinger equation are defined. The time dependent Schr\"{o}dinger equation is soled to obtain the solution for propagator.

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The energy eigenvalues and eigenfunctions are obtained for a free particle in one dimension. Properties and delta function normalization are discussed. It is shown that the energy eigenvalue must be positive. The free particle solution in three dimension is briefly given.

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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.

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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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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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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}

A brief account of representations in a finite dimensional vector spaces is presented. The use of an ortho norrnal basis along with Dirac notation makes all frequently used formula very intuitive. The formulas for representing a vector by a column vector and an operator by matrices are given.  The results  for change of o.n. bases are summarized.

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That assumption that the superposition principle be preserved under time evolution leads to unitary nature of the  them evolution operator. The state vector satisfies  differential equation, the Schrodinger equation, with Hamiltonian as the generator of time evolution.