Talk:Ehrenfest theorem

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Ehrenfest Principle was redirected here based on the similarity of its definition here: to the main article Ehrenfest theorem. Also there was no reference to Ehrenfest Principle that I could find at www.britannica.com. Diverman 03:37, 22 March 2006 (UTC) (B.App.Sc.)

I will have to write an article on Ehrenfest's Principle - it is indeed a different subject than the one in this article and should not redirect here.--J S Lundeen 11:54, 13 April 2006 (UTC)

Shouldn't the derivatives of Phi be partial derivatives? Outside the integral they can be regular ones but inside the integral they should be partial ones, because Phi is a function of x and t.


 * I think you're right. Also there's no need to assume A is time-independent, indeed it often is time dependent (e.g. H(x,p,t) if there a changing external potential V(x,t)).--Michael C. Price talk 22:33, 7 January 2007 (UTC)

This seems too simplistic
Surely putting expectatation values around Heisenbergs time evolution equations, does not a new theorem make. I'm sure the real content of Ehrenfest's theorem is not this. But I'll see what comments others make. —Preceding unsigned comment added by 65.2.102.44 (talk) 11:38, 26 March 2008 (UTC)


 * Actually this is the real content--- the theorem is obvious in Heisenberg mechanics. It is called a theorem because it was first proven in Schrodinger's picture, where it was not anywhere near as obvious, and was actually suprising because wavepackets spread out. What Ehrenfest showed was that even though wavefunctions spread, the expected values obey Newton's laws. Note that the expected value of the position could be halfway between two big lumps of wavefunction which are nozero very far away from the central point. Nevertheless, the expected value of the position and the expected value of the momentum obey Newton's law. Again, the theorem follows from the equivalence of Heisenberg and Schrodinger's formulation, but in Heisenberg's formulation it is built in.Likebox (talk) 06:31, 9 May 2008 (UTC)

In my introductory textbooks (Mandl,Griffiths) and lecture notes, and a few pages I checked on the internet, they do not call the first equation of this article "Ehrenfest's theorem"; it's just given as an "ordinary equation". Rather, the final equations for the time derivative of the expectation values of x and p (given later in this article) are called "Ehrenfest's theorem". I.e. these two equations:
 * $$ \frac{d}{dt}\langle \mathbf{r}\rangle = \frac{1}{m}\langle \mathbf{r}\rangle$$


 * $$ \frac{d}{dt}\langle \mathbf{p}\rangle = -\langle \mathbf{\nabla}V(\mathbf{r})\rangle$$

If you want to call that particular equation "Ehrenfest's theorem" then you need to justify it historically, so I've added a citation needed tag. At the least I would hope that his original paper is cited. It may be easier to just describe what the theorem means in simple words in the intro (e.g. that it shows the quantum mechanical equations of motion are the same as the classical equations of motion, but with quantum averages, or that the expectation values obey the classical equations of motion). The various mathematical definitions and derivations can then be given later in the meat of the article. 157.82.168.201 (talk) 09:13, 26 April 2012 (UTC)


 * Merzbacher's 3rd Edition "Quantum Mechanics" uses $$ \frac{d}{dt}\langle \mathbf{p}\rangle = -\langle \mathbf{\nabla}V(\mathbf{r})\rangle$$ as "Ehrenfest's theorem". Ballentine's "Quantum Mechanics" introduces the term "Ehrenfest's theorem" and shortly thereafter derives an equation very similar. Eisberg's and Resnick's "Quantum Physics" derives the same equation but never names it or uses the term "Ehrenfest's theorem". I have the same suspicion that you do that this article is currently using an equation for "Ehrenfest's theorem" that is not the one used most commonly. Jason Quinn (talk) 03:17, 17 June 2012 (UTC)


 * There is a reference in Henrik Smith's book "Introduction to Quantum Mechanics" displaying the exact equation used in the article. This equation is more general and can be used to derive many other equations--LaoChen (talk)14:19, 18 June 2012 (UTC)

Lindblad equation
This is really a red herring--- it describes dissipative systems. I don't even think that the Eherenfest theorem is strictly true when the Lindblad operators are nonzero.Likebox (talk) 06:31, 9 May 2008 (UTC)

The importance of the Ehrenfest theorem is that it is the most realistic, the most ontological description of the real system you encounter in experiments. The Ehrenfest system describes the raw diagonal or spiral motion of classical and quantum particles which is Helmholtz decomposed as two orthogonal parts. They are Heisenberg and Shrödinger systems.

The Heisenberg system describes the height (the energy) of the system (the diagonal Ehrenfest trajectory) and the Shrödinger system describes the cyclic (or horizonal) phase shifting. These two systems are fundamentally incomplete descriptions of the real system, because they are two orthogonal elements of the Helmholtz decomposition of an arbitrary operator.

The ideal Born rule is an incomplete rule because it is strictly tied to the horizonal Shrödinger system. — Preceding unsigned comment added by 202.224.127.137 (talk) 08:14, 14 May 2020 (UTC)

From the POVM point of view, the Lindblad equation is deeply related to the diagonal Ehrenfest supersystem, even if it describes the horizonal energy exchange in Shrödinger subspace. — Preceding unsigned comment added by 202.224.127.137 (talk) 08:34, 14 May 2020 (UTC)

Far too complicated
For anyone who knows any quantum mechanics, this is all obvious and hardly needs an explanation. The non-obvious part is the historical context, which should be made more prominent.

More importantly, a lay reader who's clicked on this page because it's listed as one of the fundamental concepts of quantum mechanics will be horribly lost from the first sentence.

The lead should probably start with something like "The Ehrenfest theorem, named after Paul Ehrenfest, the Austrian physicist and mathematician, describes how classical mechanics apply to the expected values of quantum wavepackets, despite not applying to the wavepackets themselves." Follow with the sentence about Heisenberg mechanics, then add some of the info from Likebox's answer to "Seems too simplistic" above, and the connection to the correspondence principle. After that, go with the rest of the existing lead. --32.154.153.34 (talk) 12:10, 21 January 2010 (UTC)

Quantum Motions inserts
This is not the place to define and promote off-mainstream thinking on "quantum motions". You might try arguing about them in Method_of_quantum_characteristics, but it is a crepuscular area. An introductory article for the novice is not the place to parse out the limitations of the concept of "quantum trajectory". It is evident that for velocity-dependent potentials all the classical correspondences you have in mind are seriously undermined. The obvious "Hamilton's equations" correspondence you may have in mind is already self-evident in the introduction, and the textbooks cited. Please desist promoting off-mainstream references in wikipedia articles. Cuzkatzimhut (talk) 16:36, 28 February 2013 (UTC)


 * Rezarahemi seems to insist that "The theorem shows that the classical and the quantum motions are equivalent when the position and momentum vectors are replaced by their corresponding expectation values."  "Equivalent" is too strong and vague a term, and distinctly unhelpful to the novice reader. No, QM and classical mechanicanics are not always equivalent, even in the expectation value sense.  In the examples underneath, the expressions for the evolution of ⟨x⟩ and ⟨p⟩ are given as a special case for elementary independent potentials, and more verbiage or references are superfluous, especially in the introduction. I cannot imagine what about the odd references insisted upon is essential for that. But, as indicated, for hamiltonians involving products of p's and x's the situation grows increasingly ambiguous. I see no point in abusing the introductory article as a surrogate of a physics newsgroup, or a public whiteboard to discuss this week's inspirations. Please take your notions elsewhere, perhaps the wiki suggested above. Cuzkatzimhut (talk) 19:36, 28 February 2013 (UTC)

Ehrenfest's professional label
Ehrenfest is not a mathematician. His wife was. This is something in the physics community that is constantly said incorrectly. — Preceding unsigned comment added by 2610:E0:A040:CDFD:D483:ACB8:5C24:8018 (talk) 16:44, 14 November 2013 (UTC)
 * I tweaked his professional label to "theoretical physicist", to at least comport with his wiki bio. The ambiguity arises from the dual status of theoretical physicists in the early 20th century. A theoretical physicist was regarded as a borderline mathematician, and you may note his 1905 Habilitationsschrift on the calculus of variations, published in a math journal, today would be classified as strictly math. Still, given his subsequent role and impact on science, today he can be safely pigeonholed as a theoretical physicist, not even a mathematical physicist. I thus acted on your implicit advice. Cuzkatzimhut (talk) 18:50, 14 November 2013 (UTC)