Can the WKB Approach Predict Quantum Gravity Energies?

[Karlisbad]

If we have a 1-dimensional problem so for big n "Energies" can be found in the form:

$$2 \int_{a}^{b}dx \sqrt (E_{n} - V(x) ) = (n+1/2) \hbar$$

where "a" and "b" are the turning points, then could we writte the equation for energies (where a>c>b using Mean-value theorem for integrals )


$$2 \sqrt (E_{n} - V(c) )(b-a) = (n+1/2) \hbar$$

for finite a,b,c ?


-Another question, when dealing with Semiclasical Quantum Gravity, do the "Energies" satisfy the same WKB constraint?, in particular if we define:


$$\pi _{ab}$$ as the "momenta" conjugate to the metric then the

"Energies" of quantum gravity for big n satisfy


$$\oint dV\pi _{ab}(x,y,z) = (n+1/2) \hbar$$ ?..

 

[Jhero]

Thank you for your interesting post about 1-dimensional problems and the application of the Mean-value theorem for integrals. I would like to provide some clarification and additional information on the topics you have mentioned.

Firstly, the equation you have written for the energies (2 \int_{a}^{b}dx \sqrt (E_{n} - V(x) ) = (n+1/2) \hbar) is known as the Bohr-Sommerfeld quantization condition. It is used to determine the allowed energy levels for a particle in a 1-dimensional potential well. The turning points, a and b, represent the points where the potential energy is equal to the total energy of the particle.

In the case of finite a, b, and c, the Mean-value theorem for integrals can be used to approximate the integral in the Bohr-Sommerfeld quantization condition. However, this is only an approximation and may not give accurate results for all cases. It is important to note that the turning points a and b should still be chosen carefully to ensure that the potential energy is equal to the total energy of the particle at those points.

Moving on to your question about the WKB constraint in Semiclassical Quantum Gravity, it is important to understand that the WKB (Wentzel-Kramers-Brillouin) approximation is a method used to approximate solutions to the Schrödinger equation in the semiclassical regime. It is not specific to any particular field of study, such as quantum gravity. Therefore, the WKB constraint would not necessarily apply to the "Energies" of quantum gravity.

I hope this helps to clarify some of the concepts you have mentioned. If you have any further questions, please do not hesitate to ask.