Starting with the Heisenberg equation of motion, we can derive Ehrenfest's theorem simply by projecting the Heisenberg equation onto from the right and from the left, or taking the expectation value, so. relations satisfied by the position and momentum operators This is because the ensemble is defined strictly as a function of a conserved quantity of the system (energy). Dirac's rule of thumb suggests that statements in quantum mechanics which contain a commutator correspond to statements in classical mechanics where the commutator is supplanted by a Poisson bracket multiplied by iħ. The reason is that Ehrenfest's theorem is closely related to Liouville's theorem of Hamiltonian mechanics, which involves the Poisson bracket instead of a commutator. but we recognize this as Newton's second law. H��W 4����F�l�dW� �4���r� KB�ȧ��I� F�mC���"3$�R���7 Abstract Ehrenfest’s Theorems provide a bridge between quantum and classical mechanics. Ehrenfest's theorem, named after the Austrian physicist Paul Ehrenfest, states that the classical laws of motion hold (approximately) for the quantum mechanical expectation values of observables. . They relate time derivatives of expectation values to expectation values of appropriate operators. Suppose some system is presently in a quantum state Φ. Thisis\Ehrenfest’s Theorem." Set A^ = ^qj. where A is some QM operator and ⟨A⟩ is its expectation value. conjugate momenta Otherwise, the evolution equations still may hold approximately, provided fluctuations are small. This more general theorem was not actually derived by Ehrenfest (it is due to Werner Heisenberg ). position coordinates throughtheQuantum-ClassicalcorrespondenceofEqu. The Ehrenfest theorem is a special case of a more general relation between the expectation of any quantum mechanical operator and the expectation of the commutator of that operator with the Hamiltonian of the system. Whence, the Schrödinger equation was derived from the Ehrenfest theorems by assuming the canonical commutation relation between the coordinate and momentum. The These are represented by three commuting position are. If we apply the Schrödinger equation, we find that, Note H = H ∗, because the Hamiltonian is Hermitian. The Heisenberg picture moves the time dependence of the system to operators instead of state vector. If we want to know the instantaneous time derivative of the expectation value of A, that is, by definition, where we are integrating over all space. This result is again in accord with the classical equation. Setting , the commutator equations can be converted into the differential equations[6][7], whose solution is the familiar quantum Hamiltonian. If one assumes that the coordinate and momentum commute, the same computational method leads to the Koopman–von Neumann classical mechanics, which is the Hilbert space formulation of classical mechanics. Heisenberg picture. However, the converse is also true: the Schrödinger equation can be inferred from the Ehrenfest theorems. %PDF-1.2 %���� operators Taking the expectation values of both sides with respect to a Heisenberg state ket that does not evolve in time, we obtain the so-called Ehrenfest theorem : (3.47) When written in terms of expectation values, this result is independent of whether we are using the Heisenberg or Schrödinger picture. It is most apparent in the Heisenberg picture of quantum mechanics, where it is just the expectation value of the Heisenberg equation of motion. [6] Therefore, this derivation as well as the derivation of the Koopman–von Neumann mechanics shows that the essential difference between quantum and classical mechanics reduces to the value of the commutator [x̂, p̂]. In this case d dt hqji = 1 i h h[qj;H]i = 1 i h hi h @ @pj Hi = h @H @pj i (11) This makes the operator expectation values obey corresponding classical equations of motion, provided the Hamiltonian is at most quadratic in the coordinates and momenta. 4 Simple Applications The following results are immediate: d dt hxi = h p m i d dt hpi = h @V @x i = hFi d dt hLi = hr Fi (10) The Hamiltonian equations of motion are obtained as follows. This is an example of the correspondence principle: the result manifests as Newton's second law in the case of having so many excitations superposed in the wavefunction that the net motion is given by the expectation value simulating a classical particle. ���ܩ�t��Q�=����}���=��y~��. If W does not depend explicitly on time, we have d dt hWi= ˙jWj + jWj ˙ (4) The time derivative of can be found from the Schrödinger equation: ˙ Since these identities must be valid for any initial state, the averaging can be dropped and the system of commutator equations for the unknown quantum generator of motion Ĥ are derived, Assuming that observables of the coordinate and momentum obey the canonical commutation relation [x̂, p̂] = iħ. , and three commuting momentum operators For example, for the expectation value of momentum and the expectation value of the gradient of the potential energy (force) holds Newton's second law: d p ^ d t = − ∇ V ( r →) = F ( r →) . classical physics: that is. [5] By expanding the right-hand-side, replacing p by −iħ∇, we get, After applying the product rule on the second term, we have. , respectively. From Infogalactic: the planetary knowledge core, Derivation of the Schrödinger equation from the Ehrenfest theorems, Although the expectation value of the momentum, derivation of the Koopman–von Neumann mechanics, "Remarks concerning the status & some ramifications of Ehrenfest's theorem", https://infogalactic.com/w/index.php?title=Ehrenfest_theorem&oldid=851971, Creative Commons Attribution-ShareAlike License, About Infogalactic: the planetary knowledge core. to our free particle Hamiltonian. commutation Similarly we can obtain the instantaneous change in the position expectation value. 180 0 obj << /Linearized 1 /O 182 /H [ 1288 2323 ] /L 251569 /E 54436 /N 37 /T 247850 >> endobj xref 180 44 0000000016 00000 n 0000001231 00000 n 0000003611 00000 n 0000003769 00000 n 0000003963 00000 n 0000004518 00000 n 0000004817 00000 n 0000005117 00000 n 0000013594 00000 n 0000014150 00000 n 0000015028 00000 n 0000015661 00000 n 0000016128 00000 n 0000016541 00000 n 0000019111 00000 n 0000019244 00000 n 0000019532 00000 n 0000019696 00000 n 0000021963 00000 n 0000022226 00000 n 0000022337 00000 n 0000025383 00000 n 0000034142 00000 n 0000034588 00000 n 0000035154 00000 n 0000042989 00000 n 0000043381 00000 n 0000043732 00000 n 0000044191 00000 n 0000044916 00000 n 0000045419 00000 n 0000045709 00000 n 0000046405 00000 n 0000046850 00000 n 0000047264 00000 n 0000047780 00000 n 0000051713 00000 n 0000052060 00000 n 0000052351 00000 n 0000052581 00000 n 0000052711 00000 n 0000054205 00000 n 0000001288 00000 n 0000003588 00000 n trailer << /Size 224 /Info 179 0 R /Root 181 0 R /Prev 247839 /ID[<1cd36237de792db8b6f1cd3fc082623e><1cd36237de792db8b6f1cd3fc082623e>] >> startxref 0 %%EOF 181 0 obj << /Type /Catalog /Pages 175 0 R >> endobj 222 0 obj << /S 2973 /Filter /FlateDecode /Length 223 0 R >> stream The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force F = −dV/dx on a massive particle moving in a scalar potential, Loosely speaking, one can thus say that "quantum mechanical expectation values obey Newton’s classical equations of motion". This page was last modified on 3 October 2015, at 01:07. Placing this into the above equation we have. Whence Jammer’s careful use of the word “analogy,” and of the (where The Hamiltonian is assumed to have the same form as in runs from 1 to 3), with three corresponding The expectation values are computed on quantum mechanical operators. In the Heisenberg picture, the derivation is trivial. Let us now add a potential Shankar treats Ehrenfest’s theorem a bit more generally. 4 Status of Ehrenfest’s Theorem that associate classically with systems of type H(x,p)= 1 2m p 2+ V(x). For an operator W we can use the product rule to state that d dt hWi= d dt h jWj i (2) = ˙jWj + jWj ˙ + W˙ (3) where a dot indicates a time derivative. It was established above that the Ehrenfest theorems are consequences of the Schrödinger equation. Consider a three-dimensional system characterized by three independent Cartesian For more general Hamiltonians H(X,P), the same method gives d dt This more general theorem was not actually derived by Ehrenfest (it is due to Werner Heisenberg). But except under special circumstances which favor the replacement V (x) −→V ( x ) (3) the systems (2) and (3) pose profoundly different mathematical and interpretive problems. Statistical equilibrium (steady state): A microcanonical ensemble does not evolve over time, despite the fact that every constituent of the ensemble is in motion. (This loose statement needs some caveats, see. in the [6] We begin from, Applications of the product rule leads to, into which we substitute a consequence of Stone's theorem, where ħ was introduced as a normalization constant to the balance dimensionality.

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