# THE CENTRAL-FIELD APPROXIMATION

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**THE CENTRAL-FIELD APPROXIMATION**

A less drastic and more useful approximation is to think of all the electrons together as making up a charge cloud that is, on average, spherically symmetric. We can then think of each individual electron as moving in the total electric field due to the nucleus and this averaged-out cloud of all the other electrons. There is a corresponding spherically symmetric potential-energy function VCr). This picture is called the central-field approximation; it provides a useful starting point for the understanding of atomic structure. If you are disappointed that we have to make approximations at such an early stage in our discussion, keep in mind that we are dealing with problems that initially defied all attempts at analysis, with or without approximations. In the central-field approximation we can again deal with one-electron wave functions. The Schrodinger equation differs from the equation for hydrogen only in that the l/r potential-energy function is replaced by a different function VCr). But it turns out thatVCr) does not enter the differential equations for 8(9) and <I>(¢,), so those angular functions are exactly the same as for hydrogen, and the orbital angular-momentum states are also the same as before. The quantum numbers I, m.; and m, have the same meaning as before, and the magnitude and z-component of the orbital angular. momentum are again given by Eqs. (43-4) and (43-5). The radial wave functions and probabilities are different than for hydrogen because of the change in V(r), so the energy levels are no longer given by Eq. (43-3). We car still label a state using the four quantum numbers (n, I, ml, mJ In general, the energy OJ a state now depends on both n and I, rather than just on n as with hydrogen. The restrictions on values of the quantum numbers are the same as before.