Chapter 3: The Gaussian Wavepacket


A. The initial Gaussian has width parameter The Actual Formul The center of the Gaussian satisfies the classical equations of motion, however the width first, spreads and then contracts periodically in time, twice per period.
(Animation) A.
B. The initial width is the same as the ground state width (this is so-called "coherent state": The Actual Formul where The Actual Formul is the angular frequency of the oscillator) and hence the Gaussian moves without spreading.
(Animation) B.
C. The initial Gaussian lias width parameter The Actual Formul The center of the Gaussian satisfies the classical equations of motion. Now the width first contracts and then spreads, twice per period.
(Animation) C.
Gaussian wavepacket. in a harmonic potential. The envelope is the absolute value and the oscillatory curve in the interior is the imaginary part of the wavepacket. The zero-point, phase Formul has been removed for clarity. Note that the average position and momentum change according to the classical equations of motion, i.e. the average momentum vanishes at the classical turning points and is maximum at the potential minimum.