Quantum Mechanics
Motion in a potential: Free particle

Gaussian wave packetIn classical mechanics the position and the momentum of a particle are known exactly. In quantum theory, the probability distribution to observe a particle at a certain position has a nonzero width. This implies that repetition of the experiment with the particle (under the same conditions etc.) will yield the average position with a statistical uncertainty, characterized by the variance of the distribution. According to the Heisenberg uncertainty relation, the product of the variance of the position and momentum cannot be made less than h**2 /4. The wave function for which this product is equal to h**2 /4 is called the minimum uncertainty wave packet. Such a wave packet takes the form of a Gaussian. In free space (i.e. in the absence of external forces) a Gaussian wave packet moves very much like a classical particle. As time departs from zero in both the past and future directions, the wave packet spreads, but remains Gaussian.

The following animations illustrate the spreading of the wave packet. The initial wave packet is

Wave packet

where r = (x,y), r_0 denotes the center of the wave packet, hk_0=(2*pi*h/\lamda,0) is the mean momentum and lambdasets the length scale.

sigma = lamda/sqrt(2) Simulation results for sigma = lambda/sqrt(2)
sigma = sqrt(2)*lambda Simulation results for sigma = sqrt(2)*lambda
sigma = 2*sqrt(2)*lambda Simulation results for sigma = 2*sqrt(2)*lambda

The spread of the initial wave packet determines the uncertainty Delta x(t) on the position of the particle. The more a particle is localized in space at time t = 0the faster its probability distribution spreads out with time, a consequence of the uncertainty principle. The consequences of the uncertainty principle on the spreading of the wave packets is further illustrated by comparing animations for wave packets of different initial width. The next animation shows the superposition of the animations for sigma = lambda/sqrt(2) (in red) and sigma = sqrt(2)*lambda (in transparent blue). For visualization purposes, the maxima of the initial probabilities are chosen to be the same.

Superposition of animations for sigma = lambda/sqrt(2) and sigma = sqrt92)*lambda

Narrow wave packets spread more rapidly than wide packets.

Suppose you could stop time and also reverse the arrow of time. What would happen to the wave packet ? Upon reversing the arrow of time would it

  1. continue expanding?
  2. contract forever?
  3. contract to its initial width and expand again?
  4. remain the same?