Quantum Problem: Particle Motion Under Constraints and Christoffel Symbols

[tpm]

My question is if the QM problem of a classical particle under a potential V(x) so $$H=p^{2} +V(x)$$

Is the same of this problem of a particle constrained to move under a surface with the additional condition $$V(x)=0$$

Or a particle moving on a surface with $$\Gamma ^{i}_{jk}$$

So the potential is obtained from the 'Christoffel-Symbols'.

This problem is similar to the classic one by Einstein,.. where for 'weak field' you obtain Newton equation for the Potential..then my problem is if you can apply the same to QM

 

[Mentz114]

Hi tpm. You seem to jumping between theories too freely. The description in quantum terms of the bound particle is very different from the classical or GR scenario. They are not related in the way you imply. Have you studied QM ? There are no classical 'paths' or 'positions' in standard QM, and only probablities can be calculated.

 

[denian]

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I can provide a response to this content by stating that the problem of a classical particle under a potential V(x) and the problem of a particle constrained to move under a surface with the additional condition V(x)=0 are not exactly the same. In the first case, the particle is subject to a specific potential function, while in the second case, the potential is constrained to be zero. This constraint can significantly change the behavior of the particle and lead to different solutions.

Regarding the problem of a particle moving on a surface with Christoffel symbols, this is known as the geodesic equation in general relativity. In this case, the potential is obtained from the Christoffel symbols, which describe the curvature of the surface. This problem is different from the previous two, as it involves the effects of gravity and the curvature of the space-time on the motion of the particle.

In terms of applying this to quantum mechanics, it is possible to use the geodesic equation to describe the motion of a quantum particle on a curved surface. However, this would require incorporating quantum effects and principles, such as wave-particle duality and the uncertainty principle, into the equation. This is a complex and ongoing area of research in quantum gravity and quantum field theory.

In summary, while there may be similarities between the classical and quantum problems described, the addition of constraints and the use of Christoffel symbols introduce significant differences that must be taken into account. Further research and analysis are needed to fully understand and apply these concepts in the field of quantum mechanics.