437(1): Development of the Separation of Variables Method
I checked all my calculations, this note gives more details of the separation of variables method, and at the end of the usual calculation of all the textbooks transforms into m space. This is where the transformation into m space should take place. In a stationary metric dm(r) / dt = 0 (Carroll notes). The normalized wavefunction is Eq. (31), and using the so called Schwarzschild metric as an example, the result (35) is obtained, a small shift in the energy levels due to m space. This is a simple example of a Lamb type shift. In the flat space limit r to infinity the result is the usual one <E> = E. So everything seems OK.