What is Ehrenfest's theorem: Definition and 21 Discussions

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




{\displaystyle F=-V'(x)}
on a massive particle moving in a scalar potential


{\displaystyle V(x)}
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
where A is some quantum mechanical operator and ⟨A⟩ is its expectation value. This more general theorem was not actually derived by Ehrenfest (it is due to Werner Heisenberg).It is most apparent in the Heisenberg picture of quantum mechanics, where it is just the expectation value of the Heisenberg equation of motion. It provides mathematical support to the correspondence principle.
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. 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. This makes the operator expectation values obey corresponding classical equations of motion, provided the Hamiltonian is at most quadratic in the coordinates and momenta. Otherwise, the evolution equations still may hold approximately, provided fluctuations are small.

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  1. uxioq99

    Time Independence of the Momentum Uncertainty for a Free Particle Wave

    Mine is a simple question, so I shall keep development at a minimum. If a particle is moving in the absence of a potential (##V(x) = 0##), then ##\frac{\langle\hat p \rangle}{dt} = \langle -\frac{\partial V}{\partial x}\rangle=0## will require that the momentum expectation value remains...
  2. Faizan Samad

    Calculate the expectation value of V from Ehrenfest's theorem

    Homework Statement I have a general question how I calculate the expectation value of V (potential energy) with Ehrenfest’s theorem. Do I have to integrate d<p>/dt with respect to d<x>. Also if the potential is symmetric (even) would that mean the expectation value of the potential is 0...
  3. mjda

    I Ehrenfest Theorem: Enunciate & Implications for Classical/Quantum Mechanics

    This may seem rather silly, but how would I go about enunciating Ehrenfest’s theorem? Also, does anyone know what this theorem implies for the relation between classical and quantum mechanics? Any suggestions or help is greatly appreciated!
  4. D

    Constants of motion in quantum mechanics

    Homework Statement A particle of mass m and spin s, it's subject at next central potential: ## \begin{equation*} V(\mathbf{r})= \begin{cases} 0\text{ r<a}\\ V_0\text{ a<r<b}\\ 0\text{ r>b} \end{cases} \end{equation*} ## Find the constants of motion of the system and the set of...
  5. M

    Classical limit of atomic motion

    Hello, I need same help with the following exercise: (1a)Recall Ehrenfest’s theorem and state the conditions for classicality of the trajectory of a quantum particle. (1b) Consider an atom whose state is described by a wavepacket with variance ∆x^2 in position and ∆p^2 in momentum. The atom...
  6. weezy

    I What is a time-dependent operator?

    While studying Ehrenfest's theorem I came across this formula for time-derivatives of expectation values. What I can't understand is why is position/momentum operator time-independent? What does it mean to be a time-dependent operator? Since position/momentum of a particle may change...
  7. Q

    Tannor Quantum Mechanics derivative of variance of position

    0http://stackoverflow.com/questions/34833391/tannor-quantum-mechanics-derivative-of-variance-of-position# In the Tannor textbook Introduction to Quantum Mechanics, there is a second derivative of chi on p37. It looks like this: χ"(t) = d/dt ( (1/m) * (<qp + pq> - 2<p><q> ) (Equation...
  8. Logan Rudd

    When can you apply Ehrenfest's theorem?

    I know when the initial state (##\Psi(x,0)##) is given, ##\frac{d<x>}{dt} \not=<p>##. I thought you can only apply Ehrenfest's theorem when ##\Psi## is a function of x and t, however it seems like you can also apply it to the time-independent part (##\psi(x)##) by itself as well. Can someone...
  9. U

    2D Harmonic Oscillator and Ehrenfest's Theorem

    Homework Statement Part (a): Derive Ehrenfest's Theorem. What is a good quantum number? Part (b): Write down the energy eigenvalues and sketch energy diagram showing first 6 levels. Part (c): What's the symmetry of the new system and what happens to energy levels? Find a new good quantum...
  10. B

    The rotational analog of Ehrenfest's Theorem

    Homework Statement Show \frac{d}{dt}\langle\bf{L}\rangle = \langle \bf{N} \rangle where \bf{N} = \bf{r}\times(-\nabla V) 2. Homework Equations . \frac{d}{dt}\langle A \rangle = \frac{i}{\hbar} \langle [H, A] \rangle The Attempt at a Solution I get to this point...
  11. J

    Shankar 14.4.1 using Ehrenfest's Theorem

    Homework Statement Show that if H = -\gamma\mathbf{L\cdot B}, and B is position independent, \frac{\mathrm{d} \left \langle \mathbf{L} \right \rangle}{\mathrm{d} t}=\left \langle \boldsymbol\mu \times\mathbf{B} \right \rangle=\left \langle \boldsymbol\mu \right \rangle\times\mathbf{B} Here H...
  12. SamRoss

    Not following one step for Ehrenfest's theorem

    I was looking at this proof of Ehrenfest's theorem http://farside.ph.utexas.edu/teaching/qmech/lectures/node35.html I'm confused about equation 158. It looks like the first term under the integral sign in the first expression is vanishing to obtain the second expression but I don't know why...
  13. C

    Proving Ehrenfest's Theorem: Diff. vs. Partial Diff.

    In proving the Ehrenfest Theorem This is the typical first line: \frac{d }{dt}<O> = \frac{\partial}{\partial t} <\psi|O|\psi> = <\dot{\psi}|O|\psi> + <\psi|O|\dot{\psi}>+<\psi|\dot{O}|\psi> My question is how can the exact differential \frac{d }{dt}<O> be changed the partial...
  14. 1

    Solve Momentum Expectation Change w/ Ehrenfest's Thm

    If I had a hamiltonian of the form iA(p + c), where A is a constant matrix, p the momentum operator and c an ordinary constant how do I find the time rate of change of the expectation momentum value? I've tried using Ehrenfest's theorem but I don't understand whether in [p,H], I should treat p...
  15. R

    Ehrenfest's Theorem Homework: Solve for d<p>/dt

    Homework Statement Show that \frac{d<p>}{dt} = < - \frac{\partial V}{\partial x}> Homework Equations The Attempt at a Solution I am trying to repeat the derivation that griffiths gives for deriving <p>, but it doesn't seem to give me anything that would indicate this proof is correct. <p> =...
  16. K

    Ehrenfest Theorem: Significance & Relation to Space-Time

    Is there any physical significance of this theorem? Can we make some kind of conclusion about space and time because the derivative of the expectation value of momentum with respect to time is equal to the negative of the expectation value of the derivative of potential energy w.r.t. space...
  17. D

    Quantum Mechanics - Ehrenfest's Theorem

    We have to apply Ehrenfest's theorem and I don't think it was ever explained well to us. I have read that expectation values of measurable quantities behave according to classical physics equations ie. M\frac{d\left<x(t)\right>}{dt} = \left<p(t)\right> I think I must be applying this idea...
  18. S

    Solve Griffith's Problem 1.12: Ehrenfest's Theorem

    Homework Statement Griffith's problem 1.12 Calculate d\left<p\right>/dt. Answer \frac{d\left<p\right>}{dt} = \left<\frac{dV}{dx}\right> 2. The attempt at a solution so we know that \left<p\right> = -i\hbar \int \left(\Psi^* \frac{d\Psi}{dx}\right) dx so then...
  19. E

    Rotational Analog to Ehrenfest's Theorem

    Hi, I'm trying to prove that for a particle in a potential V(r), the rate of change of the expectation value of the orbital angular momentum L is equal to the expectation value of the torque: \frac{d}{dt}<L> = <N> where N = r \times (-\bigtriangledown{V}) Basically, I'm having problems...
  20. Y

    Ehrenfest's Theorem: Quantum Mechanics Explained

    Can anyone tell me what is the Ehrenfest's Theorm in quantum mechanic, I don't understand how it can provide an example for correspondence principle?
  21. M

    Solve Ehrenfest's Theorem: Find Position Expectation Value

    I have been asked to "find a solution to Ehrenfest's Theorem" (in this case for the expectation value of position, of a particle confined to a circle). What does this mean - what kind of answer should i find?