Quantum Theory by David Bohm

Author:David Bohm [Bohm, David]
Language: eng
Format: epub, pdf
ISBN: 9780486134888
Publisher: Dover Publications
Published: 2012-07-18T16:00:00+00:00

(5a)

For an attractive Coulomb force ∂V/∂r = e2/r2, so that we obtain

(5b)

We note that the equilibrium radius increases with the angular momen-tum, as we should expect. This radius is the one for which the attractive force just balances the centrifugal force. It is therefore the classical radius for a circular orbit.

Fio. 1

In general, if a particle is bound (E < 0), it will oscillate (classically) between some limits r = a, and r = b, as shown in Fig. 1. For example, in an elliptic orbit of a hydrogen atom, the radius oscillates periodically between inner and outer limits. Only for a circular orbit is there no oscillation.

5. Separation into Relative Co-ordinates. Thus far, we have been assuming that the potential has been a function of the distance r from a fixed point. In many problems, such as, for example, the hydrogen atom, the potential is a function of the relative distance, r1 - r2, between the electron and the proton (r1 is the radius vector of the electron, and r2 that of the proton). Nevertheless, as is possible in classical theory, the equations can be separated into two sets, one involving only r1 - r2 and the other involving only the position of the center of mass. To do this, we write the Hamiltonian for the two particles as follows:211

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