Undergrad Is there any way to get a geometrical description of QM

[nashed]

This is maybe one of my greatest gripes with QM, I have never seen a geometrical description of it.

What I mean by geometrical, is a description of the given object in the 3D world we live in, not a description in Hilbert Space, is such a description even possible in principle? I've been studying about Bell inequalities and they seem to suggest that maybe not, but I figured I might as well ask.

 

[A. Neumaier]

The closest known geometric description is in terms of geometric quantization, but this may not be what you are looking for.

 

[martinbn]

This is maybe one of my greatest gripes with QM, I have never seen a geometrical description of it.

What I mean by geometrical, is a description of the given object in the 3D world we live in, not a description in Hilbert Space, is such a description even possible in principle? I've been studying about Bell inequalities and they seem to suggest that maybe not, but I figured I might as well ask.

Can you have such a description in classical mechanics?

 

[dextercioby]

In classical mechanics in the Newtonian formulation the "real" physical space in which the motion of material objects (such as me or the OP) is "seen" is mapped onto R^n (n=1,2 or 3) as an affine space.

 

[martinbn]

In classical mechanics in the Newtonian formulation the "real" physical space in which the motion of material objects (sich as me or the OP) "seen" is mapped onto R^n (n=1,2 or 3) as an affine space.

But the description of the objects in the real space, say a particle, requires you to use a different space, for a particle it's $\mathbb R^6$.

 

[dextercioby]

Yes, as you put velocities as well. I only referred to positions.

 

[nashed]

But the description of the objects in the real space, say a particle, requires you to use a different space, for a particle it's $\mathbb R^6$.

Thing is, velocity is closely related to position, and eve if it's not, you've got the Newtonian formulation which happens entirely in 3D space... I'm wondering if such a formulation is even theoretically possible for QM.

 

[bhobba]

Can you have such a description in classical mechanics?

Symplectic Geometry?
https://www3.nd.edu/~eburkard/Talks/GSS%20Talk%20110413.pdf

I think the problem is the OP may not quite understand the modern conception of geometry.

Thanks
Bill

 

[secur]

This is maybe one of my greatest gripes with QM, I have never seen a geometrical description of it. What I mean by geometrical, is a description of the given object in the 3D world we live in, not a description in Hilbert Space ...

You want a way to visualize electrons, protons etc? Classically they would be little spheres moving around under the influence of forces, like billiard balls, or planets. But in QM that picture is wrong (although not totally useless). Instead there's a wave function associated with the particle which is impossible to visualize entirely. But it can be attempted. The best examples are from chemistry. You've seen pictures of the atomic orbits of hydrogen atom, for instance? They look like spheres, barbells, etc. There are animations of such things. See this other current thread, https://www.physicsforums.com/threads/gluon-field-fluctuations.898565/, for pictures of "gluon fluctuation", a similar idea. This type of graphics represents QM functions visually (3-d). It's far from complete, leaves out essential info like (for instance) the complex nature of the wave. But it's probably about the best you can do.

 

[houlahound]

Doesn't the position basis do what the OP wants? The position of a particle is determined by the potential well in physical space.

Not sure I get the question, the most newb problems are defined and solved in real space, the complex numbers disappear when the complex conjugate it taken to get position, expectation value...even the integrals are defined with real space limits.

 

[Unknowns]

The uncertainty principle can be derived by using geometrical arguments, i.e. slit experiment.

 

[PeterDonis]

a description of the given object in the 3D world we live in, not a description in Hilbert Space, is such a description even possible in principle?

Not for QM in general. Under conditions where the physics can be approximated classically, such a description can (usually) be given. But under conditions where a classical approximation breaks down, the ability to give a description of the kind you are talking about breaks down too. That is to be expected: QM was developed in large part because this classical type of description simply didn't work for certain phenomena.

 

[Zafa Pi]

This is maybe one of my greatest gripes with QM, I have never seen a geometrical description of it.

What I mean by geometrical, is a description of the given object in the 3D world we live in, not a description in Hilbert Space, is such a description even possible in principle? I've been studying about Bell inequalities and they seem to suggest that maybe not, but I figured I might as well ask.

I don't know if this helps (it's a bit more complicated than need be): http://www.techlib.com/science/bells_inequality.htm