Abstract
In the limit of large quantum excitations, the classical and quantum probability distributions for a Schrödinger equation can be compared by using the corresponding WKBJ solutions whose rapid oscillations are averaged. This result is extended for one-dimensional Hamiltonians with a non-usual kinetic part. The validity of the approach is tested with a Hamiltonian containing a relativistic kinetic energy operator.
Highlights
By many aspects, quantum theory is a very strange theory with numerous non-intuitive predictions
It can be shown that both functions approach each other, in the limit of large quantum excitations, once the rapid oscillations of the quantum density are averaged
There is no relevant differences between the results obtained by the two procedures, since the Fourier transform of the wave-function obtained by the WKBJ method in the position space is equal to the wave-function obtained by the WKBJ method in the momentum space up to a term O(α2)
Summary
Quantum theory is a very strange theory with numerous non-intuitive predictions. An interesting approach for stationary quantum states is to compare the probability density given by the square modulus of the wave-function with a “classical probability distribution” obtained from the corresponding classical equations of motion. The classical probability distribution can be compared directly with the explicit (analytical or numerical) corresponding quantum distribution for some particular Hamiltonians. This is done, for instance, in [1, 2] for one-dimensional Schrodinger equations. The notion of classical probability distribution for the usual Schrodinger equation is recalled in Sec. III, and extended to the case of more general Hamiltonians.
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