In classical mechanics there is (in theory) no limitation on how accurately the position and the momentum of a particle can be measured. Heisenberg's uncertainty relation implies that in quantum mechanics this is no longer the case. The highest precision in measurement of position and momentum is obtained by taking the minimum uncertainty wave packet as the initial wave function. 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
where r = ( x , y ), r0 denotes the center of the wave packet, hk0 = ( 2ph / l , 0 ) is the mean momentum and l sets the length scale.
s = l / Ö2 s = Ö2 l s = 2Ö2 l
The spread of the initial wave packet determines the uncertainty D x ( t ) on the position of the particle. The more a particle is localized in space at time t = 0 the 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 page shows the superposition of the animations for s = l / Ö2 (in red) and s = Ö2 l (in transparent blue). For visualization purposes, the maxima of the initial probabilities are chosen to be the same.
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
FOOTNOTE 1 The following animations show that as time goes on the probability distribution spreads over the whole box. Observe that in the third animation, the probability distribution expands in the direction of propagation only.
(kx , ky)
= (0.86 , 0.5) (kx , ky)
= (0.86 , 0.5) (kx
, ky) = (1 , 0)
In the third animation the initial wave packet
Y ( x , y , 0 ) = f ( x , 0) j ( y , 0 )
was chosen such that
| j ( y , t ) |2 = | j ( y , 0 ) |2
Probability distributions that do not change with time, e.g. | j ( y , t ) |2 , play an important role in quantum mechanics. If the Hamiltonian does not depend explicitly on time the solution of the time-dependent Schrödinger equation is intimately related to the solutions of the time-independent Schrödinger equation
H F( r ) = E F( r )
Looking for solutions of the time-dependent Schrödinger equation of the form
requires F( r ) and E to be a solution of the time-independent Schrödinger equation
H F( r ) = E F( r )
Such solutions represent stationary states of the particle.
Imagine a quantum particle initially described by a Gaussian wave packet centered at the middle of a square box, with momentum zero. As time goes on will the wave packet
Answer (Keep in mind that in the absence of forces a classical particle with zero momentum does not move at all.)
The initial wave function was chosen to be
for the first animation and
for the other two animations. Here
and A is a constant that is fixed by the requirement that the total probability be normalized to one.
The harmonic oscillator is one of the most important examples in classical and quantum mechanics. In one dimension harmonic oscillations occur when a particle experiences a restoring force proportional to the displacement from a fixed position. The concept of harmonic oscillators has proven to be useful to describe the motion of for instance pendulums, vibrating strings, individual atoms in molecules and in crystals and, on a more abstract level, the electromagnetic field. Whenever the forces between particles are linear functions of the relative displacements the motion of the particles can always be analyzed in terms of harmonic oscillators. Consider a particle of mass m oscillating in one dimension. The restoring force acting on the particle is F = -kx, where k is called the force or spring constant. This force derives from the potential energy V = kx2 / 2 through F = -dV / dx = -kx. Examples of the motion of a quantum particle in a two-dimensional harmonic potential are shown below. Note that in one of the movies the wave packet changes with time in one direction only. The wave function for the other direction was chosen to be a stationary state.
(sx , sy) = (l , l) (sx , sy) = (l , 2l) (sx , sy) = (l , 3l)
In the first animation the motion of the quantum particle is similar to that of a classical particle. The particle oscillates in the harmonic potential. Meanwhile the wave packet expands and contracts periodically. In the second and third animation the particle has zero momentum. The periodic expansion and contraction of the probability distribution reflects the fact that the particle is not in a stationary state of the harmonic oscillator system.
The potential energy of a charged particle in an electric field along the x-direction is given by V( x ) = qUx , where q denotes the charge of the particle and U sets the strength of the electric field. The force in the x-direction experienced by the particle is Fx = -qU. From Newton's equation of motion
max = Fx ; may = maz = 0
it follows that a classical particle will accelerate in the x-direction only, up to the point where the particle hits the boundary of the system. For simplicity it will be assumed that this collision is perfectly elastic.
In two dimensions the motion of a classical particle experiencing the force Fx is identical to that of an ideal billiard ball placed on an ideal billiard table that is tilted in the x-direction only. The following animation shows that the motion in the x-direction of a quantum particle on a tilted billiard resembles that of the classical particle: The wave packet is accelerated and bounces back and forth. In the y-direction the wave packet is that of a free particle in a box.
The acceleration deforms the wave packet, as clearly illustrated in the next movie.
As the velocity in the x-direction increases the wave packet looses its symmetric shape. The ratio of the spread of the wave packet in x- and y-direction increases.
Classically, a charged particle in a time-independent homogeneous magnetic field executes a circular motion in the plane perpendicular to the direction of the field. The period of this motion is the inverse of the cyclotron frequency wc = qB / m, where B is the strength of the field. A charged quantum particle in a time-independent homogeneous magnetic field also executes this circular motion. In addition the probability distribution oscillates harmonically with time.
In the coordinate system attached to the moving center of the wave packet, the particle experiences a harmonic potential leading to the harmonic, time-dependent distortion of the wave packet.
Sequence of snapshots illustrating the oscillation of the wave packet.
If the circular motion of a classical charged particle in the plane perpendicular to the direction of a uniform magnetic field is interrupted by reflection from a boundary, the particle executes so-called skipping orbits. Try to imagine what happens to the quantum particle.
The charged quantum particle follows the skipping orbits of the classical particle as long as the whole wave packet is reflected by a single boundary. Once the wave packet has entered a corner it becomes difficult to identify the skipping orbits.