## Section 8.1: Classical Free Particles and Wave Packets

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A classical free particle, like the ball shown in the animation (position given in meters and time given in seconds), obeys the kinematic equation: *x* = *x*_{0 }+ v_{0}*t*. Classical particles obey Newton's second law, Σ * F* =

*m*

**a**, and when there is no net force, there is no acceleration.

A classical wave, such as that of an idealized plucked string,^{1} obeys a classical wave equation,

[(1/v^{2}) ∂^{2}/∂*t*^{2} − ∂^{2}/∂*x*^{2}] *y*(*x*,*t*) = 0, (8.1)

where v is the wave velocity and *y*(*x*,*t*) is the wave function. Restart. In general, the solution to this equation is in the form *y*(*x*,*t*) = *f*(*x *± v*t*) = *g*(*kx* ± ω*t*). For harmonic waves, the solutions to the wave equation are

*y*(*x*,*t*) = *A* cos(*kx* ± ω*t*) + B sin(*kx* ± ω*t*) ,

where the upper/lower sign describes left-moving/right-moving solutions, respectively. In order to get a localized wave packet, we must add together many such traveling wave solutions. When we do so, we get a wave packet. One such wave packet is a Gaussian wave packet which is shown in the bottom panel of the animation and is described by the wave function

*y*(*x*, *t*) = A exp[−(*x *−* x*_{0 } − v_{0}*t*)^{2}/α^{2}] .

Here, *A* is the amplitude, *x*_{0 }+ v_{0}*t* is the position of the peak of the packet at time *t*, and v_{0} is the packet's velocity, the group velocity. In addition, α describes the width of the packet such that when *x *−* x*_{0 } + v_{0}*t* = ±α, the wave packet drops to 1/e of its maximum value. Note that this wave packet maintains its shape throughout its motion. This is because all of the underlying waves that make up this packet have exactly the same velocity since we have assumed harmonic waves with no dispersion.

^{1}This wave is dispersionless. Waves with dispersion are covered in Section 5.11.

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