Exploring the WKB Wave and Energy Quantization in Higher Dimensions and QFT

[tpm]

If in 1-D the WKB wave and energy quantization are:

$$\Psi (x) = e^{iS(x)/\hbar}$$ and $$\oint_C dq p =2\pi (n+1/2) \hbar$$

My question is what happens with more than one dimension ?? (many body system or 3-D system), what happens with QFT ?? i know that as an analogy you could always put the WKB wavefunction in the form:

$$\Psi [\phi] = e^{iS[\phi]/\hbar}$$

but what happens with the energies??..i know this must/can be used when delaing Semiclassical Quantum Gravity won't it ??

 

[eljose79]

In higher dimensions, the WKB approximation and energy quantization still hold true, but the mathematical expressions become more complex. In a many-body system or 3-D system, the WKB wavefunction would be in the form of a multidimensional integral, and the energy quantization would be determined by the sum of the momenta in all dimensions.

In quantum field theory (QFT), the concept of energy quantization is still applicable, but the energy levels are determined by the quantization of the field itself, rather than the quantization of the particle's position and momentum. The WKB approximation can also be applied in QFT, but it becomes more complicated due to the infinite number of degrees of freedom in a field.

In semiclassical quantum gravity, the WKB approximation is used to approximate the wavefunction of the gravitational field, while the energy quantization is determined by the quantization of the spacetime geometry. This approach is still an active area of research and is not yet fully understood.

Overall, the WKB approximation and energy quantization are powerful tools in understanding quantum systems in higher dimensions, but the mathematical expressions become more complex and challenging to solve. Further research and development in this area will help us better understand the behavior of quantum systems in multiple dimensions.

 

[denian]

In higher dimensions, the WKB wave and energy quantization still hold true, but the equations become more complex. In a many-body or 3-D system, the WKB wavefunction would be in the form of:

\Psi(x_1,x_2,...,x_n) = e^{iS(x_1,x_2,...,x_n)/\hbar}

The energy quantization equation would also be modified to take into account the additional dimensions and particles. This would be in the form of:

\oint_C dq_1 dq_2 ... dq_n p_1 p_2 ... p_n = 2\pi (n+1/2) \hbar

where n is the number of dimensions and particles. This equation may look daunting, but it still follows the same principle as the 1-D case, where the integral of the momentum over a closed path is equal to a multiple of Planck's constant.

In quantum field theory (QFT), the WKB approximation can also be applied. The wavefunction would be in the form of:

\Psi[\phi] = e^{iS[\phi]/\hbar}

where \phi represents the field variables. The energy quantization in QFT is more complex and is described by the Hamiltonian operator, but the WKB approximation can still be used to approximate the solutions.

In the context of semiclassical quantum gravity, the WKB approximation is often used to study the behavior of quantum particles in a curved spacetime. This is because the WKB approximation allows for a semiclassical treatment of gravity, where the effects of gravity are taken into account while still treating particles as quantum objects. The energy quantization in this case would be modified to incorporate the effects of gravity on the particles.

Overall, the WKB wave and energy quantization are important concepts in higher dimensions and in QFT, and they can be applied in various fields of physics, including semiclassical quantum gravity.