WKB approximation for Tolman's problem (Collapsing dust)

The quantization of the motion of a spherically symmetric dust cloud is considered in Wheeler's approach. A WKB approximation is obtained for the wave functions and energy spectrum, the well-known problem of the ordering of the noncommuting factors in the given Hamiltonian being solved. An estimate of the tunneling of the surface of the cloud beyond the classical turning point (in particular, beyond the event horizon) is estimated.