Ehrenfest Theorem

The Ehrenfest Theorem ( Ehrenfest Equations ) is a statement on the form of equations of quantum mechanics for the average values of the observed quantities of Hamiltonian systems . These equations were first obtained by Paul Ehrenfest in 1927 .

The statement of the theorem ^[1] :

In quantum mechanics, the average values of the coordinates and momenta of a particle, as well as the force acting on it, are interconnected by equations similar to the corresponding equations of classical mechanics , that is, when a particle moves, the average values of these quantities in quantum mechanics change as the values of these quantities change classical mechanics.

A complete analogy takes place only if a number of requirements ^[2] ^{[3] are} met.

The Ehrenfest equation for the average value of the quantum observable Hamiltonian system has the form

{\frac {d}{dt}}\langle A\rangle ={\frac {1}{i\hbar }}\langle [A,H]\rangle +\left\langle {\frac {\partial A}{\partial t}}\right\rangle ,

{\ displaystyle {\ frac {d} {dt}} \ langle A \ rangle = {\ frac {1} {i \ hbar}} \ langle [A, H] \ rangle + \ left \ langle {\ frac {\ partial A} {\ partial t}} \ right \ rangle,}

{\ frac {d} {dt}} \ langle A \ rangle = {\ frac {1} {i \ hbar}} \ langle [A, H] \ rangle + \ left \ langle {\ frac {\ partial A} {\ partial t}} \ right \ rangle,

Where $\ A$ ${\ displaystyle \ A}$ $\ A$ - quantum observable, $\ H$ ${\ displaystyle \ H}$ $\ H$ - the Hamilton operator of the system, angle brackets indicate the taking of the average value, and square brackets indicate the commutator . This equation can be derived from the Heisenberg equation .

In the particular case, the mean values of the coordinate $\ q$ ${\ displaystyle \ q}$ $\ q$ and momentum $\ p$ ${\ displaystyle \ p}$ $\ p$ particles are described by equations

{\frac {d}{dt}}\langle q\rangle ={\frac {1}{m}}\langle p\rangle ,

{\ displaystyle {\ frac {d} {dt}} \ langle q \ rangle = {\ frac {1} {m}} \ langle p \ rangle,}

{\ frac {d} {dt}} \ langle q \ rangle = {\ frac {1} {m}} \ langle p \ rangle,

{\frac {d}{dt}}\langle p\rangle =-\left\langle {\frac {\partial U}{\partial q}}\right\rangle ,

{\ displaystyle {\ frac {d} {dt}} \ langle p \ rangle = - \ left \ langle {\ frac {\ partial U} {\ partial q}} \ right \ rangle,}

{\ frac {d} {dt}} \ langle p \ rangle = - \ left \ langle {\ frac {\ partial U} {\ partial q}} \ right \ rangle,

Where $\ m$ ${\ displaystyle \ m}$ $\ m$ Is the mass of the particle, $\ U(q)$ ${\ displaystyle \ U (q)}$ $\ U (q)$ Is the operator of the potential energy of the particle.

The Ehrenfest equations for mean coordinates and momenta are quantum analogues of the system of canonical Hamilton equations and define a quantum generalization of Newton’s second law .

Notes

↑ Matveev A.N. Atomic Physics, - M .: Higher School, 1989. p. 125.
↑ Ehrenfest theorems // Physical Encyclopedia : [in 5 vol.] / Ch. ed. A.M. Prokhorov . - M .: Great Russian Encyclopedia, 1999. - T. 5: Stroboscopic devices - Brightness. - S. 636-637. - 692 p. - 20,000 copies. - ISBN 5-85270-101-7 .
↑ Blokhintsev D.I. Fundamentals of quantum mechanics. 8th ed. - M .: URSS, 2014 .-- 664 s (paragraph 34, p. 136-138)

Literature

Ehrenfest P. Relativity. Quanta. Statistics. Collection of articles , - M .: Nauka, 1972. (Article "Comment on the approximate justice of classical mechanics in the framework of quantum mechanics" p. 82-84)
Blokhintsev D.I. Fundamentals of quantum mechanics. 5th ed. - M .: Nauka, 1976. - 664 s (paragraph 32, p. 130-133)
Matveev A.N. Atomic Physics, - M .: Higher School, 1989. - 439 p. (Pp. 124-126)
Messiah A. Quantum Mechanics. In 2 volumes / Ed. L.D. Fadeeva. Translation from French V.T. Khozyainova .. - M .: Nauka, 1978. - T. 1. - S. 307. (VI.2. P. 214-216)
Borisov A.V. Fundamentals of quantum mechanics , - Faculty of Physics, Moscow State University, 1998 ( Ehrenfest Theorems )

[1] Matveev A.N. Atomic Physics, - M .: Higher School, 1989. p. 125.

[2] Ehrenfest theorems // Physical Encyclopedia : [in 5 vol.] / Ch. ed. A.M. Prokhorov . - M .: Great Russian Encyclopedia, 1999. - T. 5: Stroboscopic devices - Brightness. - S. 636-637. - 692 p. - 20,000 copies. - ISBN 5-85270-101-7 .

[3] Blokhintsev D.I. Fundamentals of quantum mechanics. 8th ed. - M .: URSS, 2014 .-- 664 s (paragraph 34, p. 136-138)

Ehrenfest Theorem

Notes

Literature

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