Correspondence principle

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In physics, the correspondence principle is a 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. Ehrenfest's theorem later justified and quantified this principle by showing how Newton's laws emerged from post-1925 or the "new" quantum mechanics.

The rules of quantum mechanics are highly successful in describing microscopic objects, such as atoms and elementary particles. On the other hand, experiments reveal that a variety of macroscopic systems (springs, capacitors, and so forth) can be accurately described by classical theories such as classical mechanics and classical electrodynamics. However, it is reasonable 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.

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[edit] Other uses of the term

The term "correspondence principle" is also used in a more general philosophical sense to mean the reduction of a new hypothesized scientific theory to another scientific theory (usually a precursor to the former) which 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 theory of special relativity satisfies the correspondence principle, as it reduces to classical mechanics in the limit of small velocities in comparison to the speed of light (example below). Also, general relativity reduces to Newtonian gravitation in the limit of weak gravitational fields.

In the field of Public Finance, the correspondence principle is used to define the provision of public goods and services by the geographic range of the positive and negative externalities caused by those goods and services. It is used in conjunction with the principle of subsidiarity, which devolves responsibility for providing public goods and services to the lowest level of government that can efficiently and equitably provide them.

[edit] Examples

[edit] 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 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 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 = 1kg, \omega\, = 1 rad/s, and A = 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 \omega\, = 1 rad/s, the difference between each energy level is \hbar \omega\approx 1.05\times 10^{-34}J, well below what we can detect.

[edit] 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 Newtonian expression for kinetic energy.

[edit] Quotation

Every theory is killed sooner or later... But if the theory has good in it, that good is embodied and continued in the next theory. —Albert Einstein

[edit] References

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