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[NOTES/QM-12004] Free Particle in Three Dimensions

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The energy eigen functions of free particle are given. These are found to be eigen functions  momentum also. The energy eigen functions have infinite degeneracy. There an eigen function  corresponding to each momentum direction.

[NOTES/QM-11006] Optics Mechanics Analogy Road to Wave Mechanics

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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..

[NOTES/QM-11003] Schrodinger Equation for a Charged Particle

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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
\[ \psi(\vec{r},t) \to \psi^\prime (\vec{r},t) =  
e^{-i(q/c)\Lambda(\vec{r},t)}\psi(\vec{r},t). \]

[NOTES/QM-11007] Time Variation of Average Values

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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}