Principle of correspondence

In Physical, the principle of correspondence , proposed for the first time by Niels Bohr in 1923, is a principle establishing that the quantum behavior of a system can be reduced to a behavior of traditional physics, when the quantum numbers concerned in the system are very large, or when the quantity of action represented by the Constante of Planck can be neglected in front of the action implemented in the system.

Origin and required

The laws of quantum mechanics are extremely effective in the description of the microscopic objects, like the Atome S or the Particule S. On another side, the experiment reveals that many macroscopic systems - for example the Ressort S or the Condensateur S - can be completely described by classical theories, utilizing only Newtonian mechanics and the electromagnetism not relativist. Thus, since there is no particular reason so that laws of physics, presumedly universal, depend on the size on a system, Bohr proposed this principle, according to which: “traditional mechanics must be found, like approximation of quantum mechanics for objects larger” .

This formula is however ambiguous: when does one have to consider that a system is not subjected any more to the traditional laws? The quantum physics pose a limit of correspondence , or traditional limit . Bohr provided a coarse measurement of this limit: “when the quantum numbers describing the system are large” , which means either that the system is very energy, or that it consists of many quantum numbers, or both.

The principle of correspondence is one of the fundamental tools which make it possible to check the quantum theories which have a reality. Indeed, the formulation - very mathematical - quantum physics is very open: it is known for example that the states of a physical system occupy a Espace of Hilbert, but nothing of this space is known. The principle of correspondence thus limits the choices, when they are presented, to solutions which do not contradict traditional mechanics with large scales.

Mathematical expression of the principle of correspondence

One can pass from laws in traditional physics to the laws treating of the same subject in quantum physics in:

• Replacing the dynamic variables V of the system by observable has acting on the function of state of the system.

• Substitute $\backslash \; frac\; \{D\; \{V\; (T)\}\}\; \{dt\}$ by the switch of the associated operators $\backslash \; frac\; \{1\}\; \{I\; \backslash \; hbar\}$ has (T) being the Observable and H the Hamiltonian .

One can show that $\backslash \; frac\; \{1\}\; \{I\; \backslash \; hbar\}\; =\; \backslash \; left\; \backslash \; \{H,\; has\; \backslash \; right\; \backslash \}\; +\; O\; (\backslash \; hbar^2)$, $\backslash \; left\; \backslash \; \{H,\; has\; \backslash \; right\; \backslash \}$ being the Crochet of Poisson Hamiltonian and dynamic variable.

However, $\backslash \; frac\; \{D\; \{V\; (T)\}\}\; \{dt\}\; =\; \backslash \; left\; \backslash \; \{H,\; V\; \backslash \; right\; \backslash \}$ in traditional mechanics. The traditional expression well is found if $O\; (\backslash \; hbar^2)$ is negligible, which is the case in the applicability of the principle of correspondence.

What leads to the Théorème of Ehrenfest: the measurements made in traditional physics equal to the averages of observable are associated with the variables used.

Examples

Quantum harmonic oscillator

It is shown here in what, in this example, of the significant quantum numbers allow to find traditional mechanics.

One considers a quantum harmonic oscillator with 1 dimension. According to quantum mechanics, its énegie total (kinetics and potential), noted E , is one of the discrete values:

$E=\; (n+1/2)\; \backslash \; hbar\; \backslash \; Omega,\; \backslash \; n=0,\; 1,2,3,\; \backslash \; dots$

where $\backslash \; Omega\; \backslash ,$ is the angular frequency of the oscillator. However, in a traditional harmonic oscillator - for example a lead ball attached to a spring - there is a continuous motion. Moreover, the energy of such a system seems continuous.

It is here that the principle of correspondence intervenes, which makes it possible to pass to a “macroscopic” case. By noting $A\; \backslash ,$ the amplitude of the traditional oscillator:

$E\; =\; \backslash \; frac\; \{m\; \backslash \; Omega\; ^2\; A^2\}\; \{2\}$

In extreme cases of correspondence, the two approaches are equivalent: quantum mechanics gives:

$N\; =\; \backslash \; frac\; \{E\}\; \{\backslash \; hbar\; \backslash \; cdot\; \backslash \; Omega\}\; -\; \backslash \; frac\; \{1\}\; \{2\}\; =\; \backslash \; frac\; \{m\; \backslash \; Omega\; A^2\}\; \{2\; \backslash \; hbar\}\; -\; \backslash \; frac\; \{1\}\; \{2\}$

Let us take values on our scale: m = 1 kg, $\backslash \; Omega\; \backslash ,$ = 1 rad/s and has = 1 Mr. One finds N ≈ 4.74× 10³³. It is a system indeed very broad, the system is well within the limit of correspondence.

One includes/understands then why one with the impression of a continuous motion: with $\backslash \; Omega\; \backslash ,$ = 1 rad/s, the difference between two successive energy levels is $\backslash \; hbar\; \backslash \; Omega\; \backslash \; approx\; 1.05\; \backslash \; times\; 10^\; \{-\; 34\}$ J, well in on this side the EC what it is possible easily to detect.

Equation of Schrödinger

The equation of the energy of a body in traditional physics is: $\backslash \; E\; =\; \backslash \; frac\; \{1\}\; \{2m\}.\; (\backslash \; vec\; p~)\; ^2\; +\; V\; (\backslash \; vec\; \{R\},\; T)$

The correspondence between the traditional values and the operators is: $E\; \backslash \; longleftrightarrow\; I\; \backslash \; hbar\; \backslash \; frac\; \{\backslash \; share\}\; \{\backslash \; share\; T\}$ and $\backslash \; vec\; p\; \backslash \; longleftrightarrow\; -\; I\; \backslash \; hbar\; \backslash \; frac\; \{\backslash \; share\}\; \{\backslash \; share\; \backslash \; vec\; X\}\; =\; -\; I\; \backslash \; hbar\; \backslash \; vec\; \backslash \; nabla$

While replacing in traditional energy, one obtains: $\backslash \; I\; \backslash ,\; \backslash \; hbar\; \backslash \; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; T\}\; \backslash \; =\; -\; \backslash \; frac\; \{\backslash \; hbar^2\}\; \{2m\}\; \backslash ,\; \backslash \; nabla\; ^2\; \backslash \; +\; \backslash \; V\; (\backslash \; vec\; \{R\},\; T)\; \backslash ,\; \backslash$

This equality is an equality of operators acting on the whole of the functions. By applying it to a function $\backslash \; psi\; (\backslash \; vec\; \{R\},\; T)$, one obtains: $\backslash \; I\; \backslash ,\; \backslash \; hbar\; \backslash \; \backslash \; frac\; \{\backslash \; partial\; \backslash \; psi\; (\backslash \; vec\; \{R\},\; T)\}\; \{\backslash \; partial\; T\}\; \backslash \; =\; -\; \backslash \; frac\; \{\backslash \; hbar^2\}\; \{2m\}\; \backslash ,\; \backslash \; nabla\; ^2\; \backslash \; psi\; (\backslash \; vec\; \{R\},\; T)\; \backslash \; +\; \backslash \; V\; (\backslash \; vec\; \{R\},\; T)\; \backslash \; psi\; (\backslash \; vec\; \{R\},\; T)\; \backslash ,\; \backslash$

What is anything else only the equation of Schrödinger.

This manner of obtaining it has only one virtue of illustration of the use of the principle and is by no means a demonstration of the équation.
It is to be noticed that this technique applied to the equation of energy in restricted Relativité makes it possible to find the equation of Klein-Gordon.

Other meanings

Since 1915, Einstein used a principle of substitution to pass from the equations of the restricted Relativité to that of the General relativity, and to help themselves to determine the equations of the field of gravitation.

In addition, the term of “principle of correspondence” is also included in a philosophical direction more , according to which a new scientific theory including a preceding theory must be able to explain all that the latter explained.

For example, the theory of the restricted Relativité satisfies this principle: at lower speeds in front of that of the light, one finds traditional mechanics. In the same way, the General relativity gives traditional mechanics for weak gravitational fields.

In General relativity

Einstein understood that in restricted relativity derivation along an axis of coordinate is replaced, in general relativity, by the derivation covariante which consists in deriving according to the substitu with the concept of right-hand side in a curved space: the way followed by the inertial movement, which one calls a Géodésique.
It is an application of the Principe of equivalence and Principe of relativity: the equations of restricted relativity are seen like equations of tensors in an inertial reference frame (moving following geodetic, or in freefall in the field of gravitation), and to find the equations in the other reference frames, it is enough to replace each tensor used by its general expression in general relativity, and each derivative according to a line of the reference frame of inertia, therefore this line is geodetic, by the expression of derivation according to this geodetic.

To formally find the equations of general relativity starting from the equations of the restricted Relativity, it is necessary:

• to make sure that the equation of restricted relativity is well an equality of tensors. For example quadri-speed is a tensor, but the quadri-coordinates east do not form one.

• Y to replace metric the $\backslash \; eta\; ^\; \{ij\}$ by metric the $\backslash \; g^\; \{ij\}$.

• Y to replace the differentials $\backslash \; part\_i$ by the differentials covariantes $\backslash \; D\_i$ or $\backslash \; nabla\; \_i$, according to the selected notation.

• Y to replace, in the integrals, elementary volume $\backslash \; d^4x$ by $\backslash \; sqrt.d^4x$.

One then obtains the equation of the general relativity which relates to the same subject as that treated in relativity restreinte.

Of course, one cannot thus obtain the equations of the field of Gravitation which cannot be treated in restricted relativity. However, it was an invaluable guide:

In restricted relativity, the tensor of energy $\backslash \; T^\; \{ij\}$ checks $\backslash \; part\_i\; T^\; \{ij\}\; =\; 0$, which gives, in general relativity, $\backslash \; nabla\_i\; T^\; \{ij\}\; =\; 0$. Extremely of the idea that the gravitation is a deformation of the geometry of space which had with the energy of the body present, Einstein thus sought a geometrical tensor $\backslash \; G^\; \{ij\}$ checking the same equality, to be able to write $\backslash \; \backslash \; chi.\; T^\; \{ij\}\; =\; G^\; \{ij\}$, where $\backslash \; chi$ is a constant homogenizing dimensions. His/her friend Marcel Grossmann indicated to him, about 1912, that one could take $\backslash \; G^\; \{ij\}\; =\; R^\; \{ij\}\; -\; \backslash \; frac\; \{1\}\; \{2\}\; g^\; \{ij\}\; R$, which left Einstein incrédule. In 1915, finally, Einstein admitted that it was the sought geometrical tensor there, and thereafter, Élie Cartan showed that it was only checking the equation $\backslash \; nabla\_i\; G^\; \{ij\}\; =\; 0$, except for an additive constant (which will give the cosmological Constante), and containing only the derivative first and seconds of the coefficients $g\_\; \{ij\}$ of metric of space (condition meaning that position, speed and acceleration are enough to describe the evolution of any physical system, which is an assumption inherited Newton).

But the precise history of discovered general relativity is rich scientific and human details, and is prone to controversies.

Example: relativistic kinetic energy

One shows here how the relativistic expression of the kinetic energy, within the framework of restricted relativity, is roughly equal to the traditional kinetic energy when speed is much lower than that of the light.

One leaves the famous equivalence of Einstein:

For energy when the body is at rest compared to the observer $E\_0\; =\; m\; c^2\; \backslash$

For energy when the body moves at the speed v compared to the observer $E\; =\; \backslash \; frac\; \{m\; c^2\}\; \{\backslash \; sqrt\; \{1\; -\; v^2/c^2\}\}\; \backslash$

When speed compared to the observer is nonnull, energy exceeds energy at rest of a quantity defined as being the kinetic energy:

$T\; =\; E\; -\; E\_0\; =\; \backslash \; frac\; \{m\; c^2\}\; \{\backslash \; sqrt\; \{1\; -\; v^2/c^2\}\}\; \backslash \; -\; \backslash \; m\; c^2\; \backslash$

$T\; \backslash \; approx\; m\; c^2\; \backslash \; left\; ((1\; -\; (-\; \backslash \; begin\; \{matrix\}\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; end\; \{matrix\})\; v^2/c^2)\; -\; 1\; \backslash \; right)\; =\; \backslash \; begin\; \{matrix\}\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; end\; \{matrix\}\; m\; v^2\; \backslash$

One finds well the traditional expression of the kinetic energy.

See too

Related articles

• Mechanical quantum

• Mechanical relativist

References

Random links:

Estables | -2026 | Rene Sliding gauge | Matter Vispa | Jean Mallet | Eloise_Klein_Healy

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org