Does the Equivalence principle holds in QM.

[Karlisbad]

Does the "Equivalence principle" holds in QM.

If we know that in QM under an "small" height so [tex] z<< R_{earth} [/tex] (radius of the earth) so the QM equation is (SE):

[tex] -D^{2}\Phi (z) +2m^{2}gz\Phi (z) =2mE_{n} \Phi (n) [/tex]

SO the wave function is just a "Airy function" and [tex] | \Phi (z) |^{2} [/tex] is an "intensity" near a caustic (then does it means that the problem of "gravity" is somehow related to optics ??)

- from the ODE above , we can guess that in QM "galileo was wrong", sice a Hamer and a feather won't fall at the same speed (If Bohmian mechanics is right you get the same answer due to the "Bohmian potential" )

- Then, was Einstein Wrong?, in the sense that inertial and gravitational mass are not the same ? Ehrenfrest's theorem produces:

[tex] \frac{<F_g >}{<a_g>}= \frac{<F_i>}{<a_i>} [/tex]

but only as a "mean value" in the practice perhaps in an experiment you could get different inertial and gravitational masses due to QM effect, then how does it affect to GR?.

 

[starkind]

Gravity and light affect each other...as gravity bends light and light (as an energy) curves space, hence affects gravitation.

The equivalence principle applies to conditions away from a large gravitational field. Due to spatial curvature, the part of an object closer to the gravitating object is compressed, loosely speaking, as compared to the part of the object further from the gravitating object, assuming of course that the object has three spatial dimensions. Presumably a being in a closed box close to a large gravitating object could in principle determine the presense of the object with appropriate instrumentation, without having to refer to outside objects.

I welcome corrections.

S.

 

[lalbatros]

Karlisbad,

Please consider that in the classical limit quantum mechanics reproduces exactly classical mechanics. The Bohmian representation of QM just emphasizes this relation between QM and CM.

I think that if you look closer at this question, using the time-dependent Schrödinger equation, you will see that there is no violation of the equivalence principle. The time-dependence of the average position will not depend of the mass of the particle. Likely, will the spread of the particle probability distribution depend on the mass. (note: the stationary SE is of no use for discussing the equivalence principle since the equivalence principle deals with motion)

It is well known that the spread of the wavefunction depends on the mass of theparticle. This provides one possible illustration of the transition from QM to CM, by letting the mass increase. However, this spread is not related to the gravitation field, it is a pure quantum effect. Therefore, one cannot say that difference of spreading would violate the equivalence principle. This spreading occurs in all circumstances as a manifestation of quantum mecanics.

The equivalence principle applies to the motion but not the the spreading of a particle, so to speak.

Michel

 

[lalbatros]

starkind,

No correction only a comment.
Back to the basics first!
You will enjoy those basics even more and get a chance to go further.

Michel

 

[starkind]

Thank you Lalbatros.

Do you know The Rhyme Of The Ancient Mariner by Coleridge? Or perhaps the work of Baudelaire, Les Fleurs du Mal? A large white ocean-going bird figures prominently in those. I do not just now recall the name of Baudelaire's poem, but will find it if you are interested.

Yes, it is good to have a base. My chief interests are the same as before, but I do not mind thanking you for your advice, nor admireing your technical expertese. But do you mean I should reach back even further before attempting to make comment here? I still need help interpreting the maths of these papers.

S.

 

[lalbatros]

starkind,

I had two reasons to choose this name. One is the poem by Beaudelaire.

I had the day to think to the question by Karlisbad.
I find it now extremely interresting, much more than at first sight.
It goes beyond the basics!

Cordially,

Michel


L'ALBATROS
Les fleurs du mal / Charles Baudelaire

Souvent, pour s'amuser, les hommes d'équipage
Prennent des albatros, vastes oiseaux des mers,
Qui suivent, indolents compagnons de voyage,
Le navire glissant sur les gouffres amers.

A peine les ont-ils déposés sur les planches,
Que ces rois de l'azur, maladroits et honteux,
Laissent piteusement leurs grandes ailes blanches
Comme des avirons traîner à côté d'eux.

Ce voyageur ailé, comme il est gauche et veule !
Lui, naguère si beau, qu'il est comique et laid !
L'un agace son bec avec un brûle-gueule,
L'autre mime, en boîtant, l'infirme qui volait !

Le Poète est semblable au prince des nuées
Qui hante la tempête et se rit de l'archer ;
Exilé sur le sol au milieu des huées,
Ses ailes de géant l'empêchent de marcher.

And see also: http://www.albatros.eu.com/

 

[lalbatros]

Karlisbad,

Your question is extremely interresting.

However, I do not see that it could in any way put GR into trouble.
In the end, we can forget a lot of the discussions about the equivalence principle.
To my knowledge it translates very shortly in GR:

any gravitational field can be canceled locally by a change of coordinates​

However, I can translate your question as follows:

why is it that gravitational mass governs the spreading of a wavepacket​

or

why is it that inertial mass governs the spreading of a wavepacket​

or

why is it that mass governs the spreading of a wavepacket​

May be "going back to the basics" will provide us an answer.
May be it is going to be more!
What's your idea?

Michel

 

[pmb_phy]

The equivalence principle applies to conditions away from a large gravitational field. Due to spatial curvature, the part of an object closer to the gravitating object is compressed, loosely speaking, as compared to the part of the object further from the gravitating object, assuming of course that the object has three spatial dimensions.

The equivalence principle holds under all conditions. There are several forms of the equivalence principle, one of which states that a uniform g-field is equivalent to a uniformly accelerated frame of reference. Another states the same thing but the term "Locally" is applied which makes a the statement true even in curved spacetime. The last is the "comma goes to semi-colon" rule which states that from going from a locally Lorentzian system to some other coordinate system for which the metric is not trival [i.e. not (1,-1,-1,-1) ] one changes from partial derivatives (colon) with covariant derivatives (semi-colon).

Pete