A major difference between quantum mechanics and classical mechanics is that it is impossible to precisely know both position and momentum of a quantum particle. This is expressed through Heisenberg's famous uncertainty relation Δq Δp ≥ ℏ, with Δq and Δp the uncertainty of position and momentum, respectively. Planck's constant ℏ ≈ 10-34Js is one of the most fundamental constants of nature and sets a limit on how accurately we can measure q and p in any experiment.
Because of Heisenberg's uncertainty relation a phase space description of mechanics can be very helpful to understand quantum phenomena. In a phase space description one expresses all quantities of interest as a function of the particles initial position q0 and momentum p0. For instance, if no net force acts on a particle its position evolves like
q(t) = q0 + m-1 p0 t
Hence, when plotted as a function of q0 and p0 (which span the phase space) it corresponds to a tilted plane. The following figure shows this for the special case of infinite mass m (so that q(t) = q0).
For a harmonic oscillator we have so that the phase space description of the position is a rotating tilted plane:
The Kerr effect
In classical optics, the Kerr effect corresponds to a refractive index n that depends on the light intensity I: the more intense the light, the larger becomes n.
It turns out that for certain experiments one can describe the Kerr effect similarly to a disturbed harmonic oscillator. It is easiest to describe this using the complex amplitude
a = q+ i p of the oscillator. The energy of a simple harmonic oscillator
can then be written as E = ω |a|2. In optics, |a|2
is proportional to the light intensity. For the Kerr effect, the energy of the oscillator is
E = ω |a|2+ ω2 |a|4 (1)
In classical optics, the phase space description of q (i.e., the real part of a) looks like a spiral because the larger the amplitude is, the faster will the oscillations be:
Phase space description of the quantum Kerr effect
In quantum physics, the energy of light becomes quantized: light can only appear in wavepackets that have an energy E = ℏω. These wavepackets are called photons and can be described using the quantum harmonic oscillator that is discussed in PHYS 343, for instance. The quantum Kerr effect can be described similarly with the Hamiltonian given by Eq. (1).
Tom Osborn and Peter Marzlin have found an exact solution for the corresponding phase space version of Heisenberg's equation of motion (the so-called Moyal equation). Surprisingly, this solution has time-periodic singularities, i.e., the phase space description of the position diverges periodically as shown in the following figure.
This figure shows the ratio between Θ01, which is the phase space description of the quantum amplitude a of the oscillator, and the corresponding classical amplitude acl. When this ratio is one quantum and classical amplitudes agree. ξ is a parameter that interpolates between classical and quantum mechanics: ξ=1 corresponds to quantum optics and ξ=0 to classical optics.
A divergent result for an observable is always worrisome. However, Tom and Peter showed that this singularity will never appear in any observable quantity. It can be considered as an artifact of the Moyal phase space description of quantum mechanics.