Is There a Definitive Answer to the Existence of Action in Quantum Mechanics?

[lugita15]

Jesus-christ-on-the-f-ing-cross, how many times do I have to tell you that you are wrong. That gradient does *not* give you the Lorentz force and electromagnetism is *not* conservative. You obviously do not understand the Lagrangian formulation of dynamics.

You keep saying that I'm wrong, but you're not giving me any actualproof that the net electromagnetic force is nonconservative.

 

[olgranpappy]

You keep saying that I'm wrong, but you're not giving me any actualproof [sic] that the net electromagnetic force is nonconservative.

Hmm... Miss Manners told me that if I don't have anything nice to say then I shouldn't say anything--I'm just going to quote jdstokes (post 25, I believe) in which he has already answered your question.

No, the electromagnetic (Lorentz) force is not always conservative. It is given by m\ddot{\mathbf{r}} = e[\mathbf{E} + \dot{\mathbf{r}}\times \mathbf{B}] which you can verify by plugging the Lagrangian into the E-L equations.

If you take a closed line integral along some current loop, you get e\left(\oint \mathbf{E} \cdot d\mathbf{l} + \oint \dot{\mathbf{r}}\times \mathbf{B} \cdot d\mathbf{l} \right). The second integral inside the brackets must vanish by perpendicularity of the magnetic force to the velocity field (which is tangential to the path of integration). The first term is nonzero whenever there is a changing magnetic field (nonzero curl). In either case the Lagrangian works fine.

James

 

[lugita15]

This Lagrangian is indeed correct, you can find it in Goldstein if you don't believe me :)

No, the electromagnetic (Lorentz) force is not always conservative. It is given by m\ddot{\mathbf{r}} = e[\mathbf{E} + \dot{\mathbf{r}}\times \mathbf{B}] which you can verify by plugging the Lagrangian into the E-L equations.

If you take a closed line integral along some current loop, you get e\left(\oint \mathbf{E} \cdot d\mathbf{l} + \oint \dot{\mathbf{r}}\times \mathbf{B} \cdot d\mathbf{l} \right). The second integral inside the brackets must vanish by perpendicularity of the magnetic force to the velocity field (which is tangential to the path of integration). The first term is nonzero whenever there is a changing magnetic field (nonzero curl). In either case the Lagrangian works fine.

James

Thank you for your proof, now I understand why the Lorentz force is nonconservative. However, this puzzles me: I had always that that at a fundamental level, all forces were conservative, and they only appeared nonconservative when degrees of freedom were ignored. For instance, friction seems to be nonconservative, but that is only because the motion of the individual molecules is not taken into account. So how can the electromagnetic force be truly nonconservative?

 

[olgranpappy]

Thank you for your proof, now I understand why the Lorentz force is nonconservative. However, this puzzles me: I had always that that at a fundamental level, all forces were conservative, and they only appeared nonconservative when degrees of freedom were ignored. For instance, friction seems to be nonconservative, but that is only because the motion of the individual molecules is not taken into account. So how can the electromagnetic force be truly nonconservative?

The integrals that jdstokes presented are equal to the work done as a charge traverses a closed path in space... but why would a particle ever do such a thing? In general such a closed path is not the same as the path a particle would travel along if it moved only under the forces of the electric fields. Some external agent has to grab the particle and move it along the path! If the net work done by the field is positive then the net work done by the external agent is negative; if the net work done by the field is negative then the net work done by the external agent is positive.

The total mechanical and electromagnetic energy of a closed system will remain constant even though the lorentz force (as we have just seen) is not a conservative force. Just like how if you take into account the heat generated by a friction force you can retain conservation of energy.

...if there is radiation it is hard to "close" the system, but the energy that departs from the system through its boundries can be taken into account as well via the Poynting vector.

Cheers.

 

[lugita15]

The total mechanical and electromagnetic energy of a closed system will remain constant even though the lorentz force (as we have just seen) is not a conservative force. Just like how if you take into account the heat generated by a friction force you can retain conservation of energy.

Of course there is always conservation of energy. I wasn't talking about conservation of energy. I was talking about the fact that fundamentally, there is no such thing as a nonconservative force, and forces seem nonconservative only when degrees of freedom are neglected.
As it says in the Feynman Lectures on Physics:
"We have spent a considerable time discussing conservative forces; what about nonconservative forces? We shall take a deeper view of this than is usual, and state that there are no nonconservative forces! As a matter of fact, all the fundamental forces in nature appear to be conservative. This is not a consequence of Newton's laws. In fact, so far as Newton himself knew, the forces could be nonconservative, as friction apparently is. When we say friction apparently is, we are taking a modern view, in which it has been discovered that all the deep forces, the forces between particles at the most fundamental level, are conservative."
Also, it says in a Wikipedia article on force, "However, for any sufficiently detailed description, all forces are conservative."
In a Wikipedia article on conservative forces, it says:
"Nonconservative forces arise due to neglected degrees of freedom. For instance, friction may be treated without resorting to the use of nonconservative forces by considering the motion of individual molecules; however that means every molecule's motion must be considered rather than handling it through statistical methods. For macroscopic systems the nonconservative approximation is far easier to deal with than millions of degrees of freedom."
This is why I am so surprised that the electromagnetic force is nonconservative, as it seems to contradict these quotes.

 

[olgranpappy]

Of course there is always conservation of energy. I wasn't talking about conservation of energy. I was talking about the fact that fundamentally, there is no such thing as a nonconservative force, and forces seem nonconservative only when degrees of freedom are neglected.
As it says in the Feynman Lectures on Physics:
"We have spent a considerable time discussing conservative forces; what about nonconservative forces? We shall take a deeper view of this than is usual, and state that there are no nonconservative forces! As a matter of fact, all the fundamental forces in nature appear to be conservative. This is not a consequence of Newton's laws. In fact, so far as Newton himself knew, the forces could be nonconservative, as friction apparently is. When we say friction apparently is, we are taking a modern view, in which it has been discovered that all the deep forces, the forces between particles at the most fundamental level, are conservative."

...

This is why I am so surprised that the electromagnetic force is nonconservative, as it seems to contradict these quotes.

Apparently when Feynman uses the phrase "conservative" in reference to forces he does *not* mean "are the gradient of a scalar potential." If that had been what he meant then, as we have seen already in great detail, he would be wrong. But that is not what he means. Of course, he probably did not have electromagnetism in mind when he was giving these lecture... hence the misunderstanding.

What he means (what he must mean) is just what we have already discussed in regards to friction--that energy is conserved. And not that the force is derived as the gradient of a scalar.

 

[lugita15]

Apparently when Feynman uses the phrase "conservative" in reference to forces he does *not* mean "are the gradient of a scalar potential." If that had been what he meant then, as we have seen already in great detail, he would be wrong. But that is not what he means. Of course, he probably did not have electromagnetism in mind when he was giving these lecture... hence the misunderstanding.

What he means (what he must mean) is just what we have already discussed in regards to friction--that energy is conserved. And not that the force is derived as the gradient of a scalar.

He did mean conservative force in the sense of a gradient of a scalar field. This is because he says:
"If we calculate how much work is done by a force in moving an object from one point to another along some curved path, in general the work depends upon the curve, but in special cases it does not. If it does not depend upon the curve, we say that the force is a conservative force." So he did mean "conservative force" in the usual sense of the term.

 

[olgranpappy]

okay, then he was wrong.

 

[smallphi]

I am reading "The variational principles of mechanics" by Cornelius Lanczos (4th ed.) and I bumped on something that may clarify the polemics about system of charges in external (possibly time dependent) electromagnetic field.

page 114:

Hamilton's principle holds for arbitrary mechanical systems which are characterized by monogenic forces and holonomic auxiliary conditions.

The definition of 'monogenic' forces acting on the system is explained on pages 30/31 and is less restrictive than 'conservative' forces. The monogenic forces must be derivable from a work function U that depends on generalized coordinates, velocities and time:

U = U(q_1, ..., q_n; \dot{q_1}, ..., \dot{q_n}; t)

F_i = \frac{\partial U}{\partial q_i} - \frac{d}{dt} \frac{\partial U}{\partial \dot{q_i}}

Conservative forces are particular case of monogenic forces when the work function U depends only on coordinates but not on velocities or time. For conservative forces, the work function U is the usual potential energy with minus sign.

The Lorentz forces acting on the charges of the system from the external EM field must be derivable from a work function. In this case the external Lorentz forces are monogenic but not conservative and according to Lanczos, the system can be treated with action methods.

Another case is a closed system consisting of charges AND the field of those charges. The external forces acting on that system are zero i. e. trivially conservative.

 

[olgranpappy]

I am reading "The variational principles of mechanics" by Cornelius Lanczos (4th ed.) and I bumped on something that may clarify the polemics about system of charges in external (possibly time dependent) electromagnetic field.

page 114:


The definition of 'monogenic' forces acting on the system is explained on pages 30/31 and is less restrictive than 'conservative' forces. The monogenic forces must be derivable from a work function U that depends on generalized coordinates, velocities and time:



Conservative forces are particular case of monogenic forces when the work function U depends only on coordinates but not on velocities or time. For conservative forces, the work function U is the usual potential energy with minus sign.

The Lorentz forces acting on the charges of the system from the external EM field must be derivable from a work function. In this case the external Lorentz forces are monogenic but not conservative and according to Lanczos, the system can be treated with action methods.

interesting

 

[smallphi]

Another quote from "The variational principles of mechanics" by Cornelius Lanczos relevant to the original question:

page 351:

... the basic feature of the differential equations of wave-mechanics [quantum mechanics of Schroedinger] is their self-andjoint character, which means that they are derivable from a variational principle.