Correspondence principle

The correspondence principle is a physical principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers.

The rules of quantum mechanics are highly successful in describing microscopic objects, such as atoms and elementary particles. On the other hand, we know from experiment that a variety of macroscopic systems (springs, capacitors, llamas, and so forth) can be accurately described by classical theories such as classical mechanics and classical electrodynamics. However, it is not unreasonable to believe that the ultimate laws of physics must be independent of the size of the physical objects being described. This is the motivation for Bohr's correspondence principle, which states that classical physics must emerge as an approximation to quantum physics as systems become "larger".

The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or the classical limit. Bohr provided a rough prescription for the correspondence limit: it occurs when the quantum numbers describing the system are large, meaning either some quantum numbers of the system are excited to a very large value, or the system is described by a large set of quantum numbers, or both.

The correspondence principle is one of the tools available to physicists for selecting quantum theories corresponding to reality. The principles of quantum mechanics are fairly broad - for example, they state that the states of a physical system occupy a Hilbert space, but do not state what type of Hilbert space. The correspondence principle limits the choices to those that reproduce classical mechanics in the correspondence limit. For this reason, Bohm has argued that classical physics does not emerge from quantum physics in the same way that classical mechanics emerges as an approximation of special relativity at small velocities; rather, classical physics exists independently of quantum theory and cannot be derived from it.

An example: the Quantum Harmonic Oscillator

We provide a demonstration of how large quantum numbers can give rise to classical behavior. Consider the one-dimensional quantum harmonic oscillator. Quantum mechanics tells us that the (kinetic) energy of the oscillator, E, has a set of discrete values:

E  =  (n + 1/2) ℏ ω,      n = 0, 1, 2, 3, ....,

where ω is the angular frequency of the oscillator. However, in a classical harmonic oscillator such as a lead ball attached to the end of a spring, we do not perceive any discreteness. Instead, the energy of such a macroscopic system appears to vary sinusoidally over a continuum of values.

We can verify that our idea of "macroscopic" systems fall within the correspondence limit. The average kinetic energy of the classical harmonic oscillator is equal to the average potential energy, which is:

<math> E_{avg} = \frac{m \omega ^2 [x^2]}{2} </math>

where [x²] denotes the average value of the squared displacement. Thus, the quantum number has the value

<math> n = \frac{E}{\kappa \cdot \omega} - \frac{1}{2} = \frac{\frac{m \omega [x^2]}{\kappa -1}}{2}</math>

If we apply the appropriately "human-scale" values m = 1kg, ω = 1Hz, and [x²] = 1m, then n ≈ 4.74×10³³. This is a very large number, so the system is indeed in the correspondence limit.

It is simple to see why we perceive a continuum of energy in the correspondence limit. With ω = 1Hz, the difference between each energy level is ℏω ≈ 1.05×10^-34J, well below what we can detect.