Correspondence Principle
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This article discusses quantum theory. For other uses, see Correspondence principle (disambiguation).

In physics, the correspondence principle is a quantitative tool in the old quantum theory, explicitly formulated by Niels Bohr in 1923. It says that the behavior of quantum mechanics systems reproduce classical mechanics in the limit of large quantum numbers.

Quantum Mechanics The rules of quantum mechanics are highly successful in describing microscopic objects, atoms and particle physics. But macroscopic systems like spring (device)s and capacitors are accurately described by classical theories like classical mechanics and electromagnetism. The laws of physics should be independent of the size of the physical objects being described, so there must be some limit in which quantum mechanics reduces to classical mechanics. Bohr's principle demands that classical physics and quantum physics give the same answer when the systems become large.

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 some quantum numbers of the system are excited to a very large value. A more elaborated analysis of quantum-classical correspondence (QCC) in wavepacket spreading leads to the distinction between robust "restricted QCC" and fragile "detailed QCC". See Ref. and references therein. "Restricted QCC" refers to the first two moments of the probability distribution and is true even when the wave packets diffract, while "detailed QCC" requires smooth potentials which vary over scales much larger than the wavelength, which is what Bohr considered.

The post-1925 new quantum theory came in two different formulations. In matrix mechanics, the correspondence principle was built in and was used to construct the theory. In the Schrödinger equation classical behavior is not clear because the waves spread out as they move. Once the Schrödinger equation was given a probabilistic interpretation, Ehrenfest's theorem showed that Newton's laws hold on average--- The quantum statistical expectation value of the position and momentum obey Newton's laws.

The correspondence principle is one of the tools available to physicists for selecting quantum theories corresponding to reality. The mathematical formulation of quantum mechanics are broad - they say that the states of a physical system form a Hilbert space, but they do not say which operators correspond to physical quantities or measurements. The correspondence principle limits the choices to those that reproduce classical mechanics in the correspondence limit.

Because quantum mechanics only reproduces classical mechanics in a statistical interpretation, and because the statistical interpretation only gives the probabilities of different classical outcomes, Niels Bohr 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 velocity. He argued that classical physics exists independently of quantum theory and cannot be derived from it. His position is that it is inappropriate to understand the experiences of observers using purely quantum mechanical notions such as wavefunctions because the different states of experience of an observer are defined classically, and do not have a quantum mechanical analog.

The many worlds interpretation of quantum mechanics is an attempt to understand the experience of observers using only quantum mechanical notions. Neils Bohr was an early opponent of such interpretations.

Other Scientific Theories The term "correspondence principle" is used in a more general sense to mean the reduction of a new scientific theory#science to an earlier scientific theory in appropriate circumstances. This requires that the new theory explain all the phenomena under circumstances for which the preceding theory was known to be valid, the "correspondence limit".

For example, Einstein's special relativity satisfies the correspondence principle, because it reduces to classical mechanics in the limit of velocities small compared to the speed of light (example below). General relativity reduces to gravity in the limit of weak gravitational fields. Kepler's theory of heliocentric orbits reproduces Ptolmei's geocentric theory when the Earth is made stationary. Statistical mechanics reproduces thermodynamics when the number of particles is large. In biology, chromosome inheritance theory reproduces Mendel's laws of inheritance, in the domain that the inherited factors are protein coding genes.

In order for there to be a correspondence, the earlier theory has to have a domain of validity--- it must work under some conditions. Not all theories have a domain of validity. For example, there is no limit where Newton's mechanics reduces to Aristotle's mechanics because Aristotle's mechanics, although academically viable for many centuries, does not have any domain of validity.

Examples Bohr Model If an electron in an atom is moving on an orbit with period T, the electromagnetic radiation will classically repeat itself every orbital period. If the coupling to the electromagnetic field is weak, so that the orbit doesn't decay very much in one cycle, the radiation will be emittedin a pattern which repeats every period, so that the fourier transform will have frequencies which are only multiples of 1/T. This is the classical radiation law: the frequencies emitted are integer multiples of 1/T.

In quantum mechanics, this emission must be of quanta of light. The frequency of the quanta emitted should be integer multiples of 1/T so that classical mechanics is an approximate description at large quantum numbers. This means that the energy level corresponding to a classical orbit of period 1/T must have nearby energy levels which differ in energy by h/T, and they should be equally spaced near that level:

\Delta E_n= { h\over T(E_n) }

Bohr worried whether the energy spacing 1/T should be best calculated with the period of the energy state E_n or E_{n+1} or some average. In hindsight, there is no need to quibble, since this theory is only the leading semiclassical approximation.

Bohr considered circular orbits. These orbits classically decay into smaller circular orbits, so they must also decay to smaller circles when they emit photons. The level spacing between circular orbits can be calculated with the correspondence formula. For a hydrogen atom, the classical orbits have a period T which is determined by Keplers laws to scale as r^{3/2}. The energy scales as 1/r, so the level spacing formula says that:

\Delta E \propto { 1 \over r^{3\over 2} } \propto E^{3 \over 2}

It is possible to determine the energy levels by recursively stepping down orbit by orbit, but there is a shortcut. The angular momentum L of the circular orbit scales as \scriptstyle \sqrt{r} . The energy in terms of the angular momentum is then

E \propto {1\over r} \propto {1 \over L^2}

Assuming that quantized values of L are equally spaced, the spacing between neighboring energies is

\Delta E \propto {1 \over (L+1)^2 } - {1 \over L^2} \approx {d \over dL} {1 \over L^2} = - {2 \over L^3} = -2 E^{3 \over 2}

Which is what we want. So that the solution is to have equally spaced angular momentum. If you keep track of the constants, the spacing is \scriptstyle \hbar, so the angular momentum should be an integer multiple of \scriptstyle \hbar

L = {nh \over 2\pi} = n \hbar

This is how Bohr arrived at his Bohr Model. Since only the level spacing is determined by the correspondence principle, you could always add a small fixed offset to the quantum number--- L could just as well have been \scriptstyle (n+.338)\hbar. Bohr used his physical intuition to decide which quantities were best to quantize. It is a testimony to his skill that he was able to get so much from what is only the leading order approximation.

Bohr's condition can be solved for the level energies in a general one dimensional potential. Define a quantity J(E) which is a function only of the energy, and has the property that:

{dJ \over dE} = T

This is the analog of the angular momentum in the case of the circular orbits. The orbits selected by the correspondence principle are the ones that obey J=nh for n integer, since

\Delta E = E_{n+1} - E_n = {dE \over dJ}(J_{n+1} - J_n) = {1 \over T} \Delta J

This quantity J is canonically conjugate to a variable \theta which, by the Hamiltonian mechanics changes with time as the gradient ofenergy with J. Since this is equal to the inverse period at all times, the variable \theta increases steadily from 0 to 1 over one period.

The angle variable comes back to itself after 1 unit of increase, so the geometry of phase space in J,\theta coordinates is that of a half-cylinder, capped off at J=0, which is the motionless orbit at the lowest value of the energy. These coordinates are just as canonical as x,p, but the orbits are now lines of constant J instead of nested ovoids in x-p space. The area enclosed by an orbit is Liouville's theorem under canonical transformations, so it is the same in x-p space as in J-\theta. But in the J,\theta coordinates this area is the area of a cylinder of unit circumference between 0 and J, or just J. So J is equal to the area enclosed by the orbit in x-p coordinates too:

J = \int_0^T p {d x \over dt} dt

The quantization rule is that the action-angle variables J is an integer multiple of h.

So Bohr's correspondence principle provided a way to find the semiclassical quantization rule for an arbitrary system. It was an argument for the quantum conditions mostly independent from the one developed by Wien and Einstein, which focused on adiabatic invariant. But both pointed to the same quantity.

The angle variable for a circular orbit is the angular position divided by 2\pi, since this variable increases uniformly in time and increases by one unit inone orbit. The conjugate momentum is the angular momentum times 2\pi, andthis is the quantity that is quantized in units of h:

\, 2 \pi L = n h

For multidimensional motions, the correspondence principle requires each of the independent action variables to be integers. This extension allowed Arnold Sommerfeld to formulate a more accurate model of the hydrogen atom. He could solve this model even with a relativistic electron motion, and found the correct fine structure to the spectral lines of hydrogen. This is very mysterious because the full quantum mechanical treatment of the hydrogen fine structure requires spin, which is not included in a semiclassical description. Still the semiclassical method gives the same formula as the Dirac equation, which is nowadays understood as a consequence of a hidden supersymmetry of the Dirac equation in a Coulomb field.

For chaotic systems, which do not have action angle variables, a description like this one will never work, a point that troubled Einstein. Once Louis DeBroglie reinterpreted the quantum condition as the condition that matter forms a standing wave, Einstein associated the phase of a quantum wave with the solution of the classical Hamilton-Jacobi equation. This was still only correct semiclassically. Erwin Schrödinger modified the Hamilton Jacobi equation to get a Schrödinger equation that determined both the amplitude and phase of the wave. This equation no longer required any classical concepts.

The Bohr correspondence principle was extended by Hendrik Kramers and Werner Heisenberg to attempt to account for emission intensities of different states. This requires understanding the details of the orbit, which means including all the fourier coefficients of the motion. The Fourier coefficients of the dipole moment, which in an atom is just the position of the electron relative to the nucleus, determines the classical emission intensities. These fourier coefficients must have a quantum analog which have the right correspondence limit. They introduced matrices to describe these fourier coefficients in a joint publication, and this line of reasoning led Heisenberg to develop matrix mechanics.

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 total (kinetic and potential) energy of the oscillator, E, has a set of discrete values:

E=(n+1/2)\hbar \omega, \ n=0, 1, 2, 3, \dots

where \omega\, is the angular frequency of the oscillator. However, in a 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 over a continuum of values.

We can verify that our idea of "macroscopic" systems fall within the correspondence limit. The energy of the classical harmonic oscillator with amplitude A\, is

E = \frac{m \omega ^2 A^2}{2}

Thus, the quantum number has the value

n = \frac{E}{\hbar \cdot \omega} - \frac{1}{2} = \frac{m \omega A^2}{2\hbar} -\frac{1}{2}

If we apply typical "human-scale" values m = 1kilogram, \omega\, = 1 radian/second, and A = 1metre, 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 said limit. With \omega\, = 1 rad/s, the difference between each energy level is \hbar \omega\approx 1.05\times 10^{-34}Joule, well below what we can detect.

Relativistic kinetic energy Here we show that the expression of kinetic energy from special relativity becomes arbitrarily close to the classical expression for speeds that are much slower than the speed of light.

Einstein's famous mass-energy equation

E = m c^2 \

represents the total energy of a body with relativistic mass

m = \frac{m_0} {\sqrt{1 - v^2/c^2--> \

where the velocity, v \ is the velocity of the body relative to the observer, m_0 \ is the rest mass (the observed mass of the body at zero velocity relative to the observer), and c \ is the speed of light.

When the velocity v \ is zero, the energy expressed above is not zero and represents the rest energy:

E_0 = m_0 c^2 \ .

When the body is in motion relative to the observer, the total energy exceeds the rest energy by an amount that is, by definition, the kinetic energy:

T = E - E_0 = m c^2 - m_0 c^2 = \frac{m_0 c^2} {\sqrt{1 - v^2/c^2--> \ - \ m_0 c^2 \

Using the approximation

( 1 + x )^n \approx 1 + nx \ ::for |x| \ll 1 \

we get when speeds are much slower than that of light or v \ll c \ {||-| T \ |=m_0 c^2 \left( \frac{1} {\sqrt{1 - v^2/c^2--> - 1 \right) \ |-||=m_0 c^2 \left( \left( 1 - v^2/c^2 \right) ^{-\frac{1}{2--> - 1 \right) \ |-||\approx m_0 c^2 \left( (1 - (-\begin{matrix} \frac{1}{2} \end{matrix} )v^2/c^2) - 1 \right) \ |-||=m_0 c^2 \left( \begin{matrix} \frac{1}{2} \end{matrix} v^2/c^2 \right) \ |-||= \begin{matrix} \frac{1}{2} \end{matrix} m_0 v^2 \ |}

which is the classical physics expression for kinetic energy.

References Weidner, Richard T., and Sells, Robert L. (1980) Elementary Modern Physics. ISBN 0-205-06559-7

A. Stotland and D. Cohen, J. Phys. A 39, 10703 (2006).

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