# The Correspondence Principle

The Correspondence Principle provides a condition for new theories in quantum physics. The principle states that quantum theories, though generally focused on science at atomic scales, should predict classical behaviour in the limit of large quantum numbers. In other words, new quantum theories should agree with established classical mechanics when considering large systems.

First put forward by Niels Bohr in 1923, widely acknowledged as one of the founders of quantum theory, this seemingly simple rule was instrumental to the development of quantum mechanics. His initial statement is interpreted in several ways by different scientists; we focus on the intensity interpretation.

Classical physics and quantum physics model particles using entirely different models. Classically, we consider a particle to be a singular point in space. In quantum physics, particles do not have a definite position before measurement; they are represented by wave functions. The square of the wave function gives the probability of finding the particle at a given point in space.

##### Infinite Square Well

The infinite square well is a convenient model used to approximate a particle experiencing a very large potential. The particle is restricted to the well; it can move freely anywhere within the well but cannot leave. Classically, we would expect the particle to be equally likely to be found anywhere in the well.

The period of the wave function decreases as the energy of the particle increases. For large energies, the period becomes much smaller than the size of the well, making it insignificant. The wavefunction is constant within the well, meaning the particle is equally likely to be found at any point, in agreement with our classical prediction. The probability of finding the particle at different points along an infinite square well of length L. At low energies (left) the particle is more likely to be found in the centre of the well. For higher energies (right) the particle is roughly equally likely to be found anywhere in the well.
##### Simple Harmonic Oscillator

Consider a simple harmonic oscillator, composed of a mass on a spring which moves in 1D, oscillating back and forth. The mass moves fastest as it passes the equilibrium point and slows down as it changes direction. The mass therefore spends most of its time furthest away from equilibrium; if you were to picture the mass at a random time, there would be a higher probability of capturing it away from the centre.

An oxygen molecule contains a pair of atoms held together by a chemical bond that can be modelled as a spring. This is an example of the quantum harmonic oscillator. When the system is in the ground state, the wavefunction is a Gaussian and the particle is most likely to be found at equilibrium.

As the energy of the system increases, the probability of finding the mass away from equilibrium becomes increasingly large. At very large energies the probability of finding the particle at equilibrium is extremely small and very large at the edges. At large energies, the classical case is reproduced. The probability of finding a particle at different points in its oscillation. As the energy is increased, the probability of finding the particle at maximum distance from equilibrium becomes very large, in agreement with the classical, macroscopic oscillator.
##### Ehrenfest Theorem

Statistical fluctuation of physical values observed at a microscopic level causes quantum fuzziness. As a result, in quantum physics, we must work with the average value of physical variables, known as the expectation value.

Ehrenfest theorem, named after the Austrian-Dutch physicist Paul Ehrenfest, reproduces the classical result, F=ma, Newton’s second law. It derives this equation using expectation values of the position and momentum associated with a quantum system. It is important to note that this is an instance of the correspondence principle and not a proof of the statement.

##### Continuity

Many properties of quantum systems, including their energy and angular momentum, can only take a set of discrete values. These values are integer multiples of a minimum value, each separated by small amounts. For example, the allowed energies of the simple harmonic oscillator, discussed previously, are separated by multiples of Planck’s constant.

For systems containing a small number of particles at low energy, this separation is significant and the discrete nature of the energy levels is apparent. However, for macroscopic objects in the real world, the number of particles is so large that the separation of possible energies is not clear, giving the illusion of a continuum. This is why, in everyday applications, we are able to model quantities such as time, energy and position as continuous values.