I Quantum mechanics and gravity of Greenberger

[exponent137]

I read paper https://arxiv.org/ftp/arxiv/papers/1011/1011.3719.pdf .
I do not understand figure 2. Such double phase space (x-p) can be also for a harmonic oscillator. But, at a harmonic oscillator we cannot have two ellipses (or a circle and an ellipse) which touch on some points, but have the same center.

Maybe Greenberger thought something more general than Harmonic oscillator?

 

[PeterDonis]

Maybe Greenberger thought something more general than Harmonic oscillator?

I think in Fig. 2 he's assuming something more specific, and different--a pair of classical particles, one with $K$ times the mass of the other. He's just using this as an illustration, so I don't think he intends it to be fully general.

 

[exponent137]

If we look at the equation for a harmonic oscillator, it forms an ellipse in phase space of x and momentum. If we change maximal momentum, maximal x is changed, thus it is not possible an example where maximal x stays the same, but momentum is enlarged. Thus example in figure 2 is not possible. But I think that he ignored this aspect and he concentrates only on the equivalence principle.

He wrote also about Bohr model of an atom, but this one does not form ellipse in phase space of x and momentum. It is valid: $v^2\propto 1/r$.

 

[PeterDonis]

If we look at the equation for a harmonic oscillator, it forms an ellipse in phase space of x and momentum. If we change maximal momentum, maximal x is changed, thus it is not possible an example where maximal x stays the same, but momentum is enlarged.

Did you read my post #2? The circle and the ellipse are not describing two different proposed states for the same particle (or oscillator). They are describing two different particles (or oscillators), one with $K$ times the mass of the other.

Thus example in figure 2 is not possible.

It is if you understand what it means. See above.

 

[exponent137]

Yes, I overlooked this relation: $p_0=mx_0\omega$.
It is valid $p_0=mx_0\omega$ as a relation between amplitudes of momentum and locations.
Thus, new ellipse has the same $x_0$ amplitude when only mass is enlarged.
Thanks for help, although not for a direct answer, because you do not know what I think wrongly.

 

[exponent137]

1) I am still not sure if I understand eqs. (19) and (20). I think that the right sides of eqs. (20) means that in correspondence limit, ($\rightarrow c_l$), we have only one $c_l$, thus only one frequency?
https://arxiv.org/ftp/arxiv/papers/1011/1011.3719.pdf

2) That means, we can respect a harmonic oscillator and thus (19) and (20) can also be for a harmonic oscillator?

3) He mentioned gravitational and nongravitational forces. Does Harmonic oscillator can be described for both types of forces, in this paper?