|
| related topics |
| {field, particle, equation} |
| {equation, function, exp} |
| {classical, space, random} |
| {energy, gaussian, time} |
| {particle, mechanics, theory} |
| {let, theorem, proof} |
| {phase, path, phys} |
| {time, wave, function} |
| {measurement, state, measurements} |
| {level, atom, field} |
|
On the conection between the Liouville equation and the Schrodinger
equation
Edelver Carnovali, Humberto M. Franca
abstract: We derive a classical Schrodinger type equation from the classical Liouville
equation in phase space. The derivation is based on a Wigner type Fourier
transform of the classical phase space probability distribution, which depends
on an arbitrary constant $\alpha$ with dimension of action. In order to achieve
this goal two requirements are necessary: 1) It is assumed that the classical
probability amplitude $\Psi(x,t)$ can be expanded in a complete set of
functions $\Phi_n(x)$ defined in the configuration space; 2) the classical
phase space distribution $W(x,p,t)$ obeys the Liouville equation and is a real
function of the position, the momentum and the time. We show that the constant
$\alpha$ appearing in the Fourier transform of the classical phase space
distribution, and also in the classical Schrodinger type equation, has its
origin in the spectral distribution of the vacuum zero-point radiation, and is
identified with the Planck's constant $\hbar$.
- oai_identifier:
- oai:arXiv.org:quant-ph/0512049
- categories:
- quant-ph
- comments:
- Submitted to Physics Letters A. 16 pages
- arxiv_id:
- quant-ph/0512049
- created:
- 2005-12-06
- updated:
- 2006-04-17
Full article ▸
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