Free Particle and the Schroedinger Equation (22)

The solutions to the Schrodinger equation with potential everywhere zero, the free particle solutions, are introduced and briefly discussed, including the lack of quantization and problems with normalization.  (This lecture is part of a series for a course based on Griffiths' Introduction to Quantum Mechanics.

Free Particle Wave Packets and Stationary States (23)

The construction of wave packets from infinite traveling wave stationary states is described, including a qualitative derivation of Fourier transforms and a discussion of wave crest velocity (phase velocity) vs wave packet velocity (group velocity).

Free Particle Wave Packets Example (24)

An example is done of the construction of a wave packet from given initial conditions using the results of Fourier analysis.  In this case, the initial wavefunction is a triangle function, the representation in terms of k is a sinc^2, and while the time evolution doesn't have a nice closed form solution, numerical results (as animated) behave as expected.

The Dirac Delta Function (25)

A description of the Dirac delta function as a distribution, its use in integrals, shifted delta functions, delta functions of other functions, derivatives of delta functions, and delta functions in Fourier analysis.

Note: there is a mistake in the example integral done on slide 6 around 17:15: the exponentials in the answer should be evaluated at the values of x picked out by the delta functions, i.e. -1 and 1, not x and -x.  The answer should be 1 + e/2 + 1/(2*e).  Apologies for the confusion this probably caused, and thanks to Manu Kamin and Unni Barchamua for pointing it out.