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# Motion in Central Field

To further study the motion of an electron in a central field, whose Hamiltonian is

 (11.128)

it is convenient to transform to polar coordinates. Let

 (11.129)

and

 (11.130)

It is easily demonstrated that

 (11.131)

which implies that in the Schrödinger representation

 (11.132)

Now, by symmetry, an energy eigenstate in a central field is a simultaneous eigenstate of the total angular momentum

 (11.133)

Furthermore, we know from general principles that the eigenvalues of are , where is a positive half-integer (because , where is the standard non-negative integer quantum number associated with orbital angular momentum.) (See Chapter 6.)

It follows from Equation (11.101) that

 (11.134)

However, because is an angular momentum, its components satisfy the standard commutation relations

 (11.135)

(See Section 4.1.) Thus, we obtain

 (11.136)

However, , so

 (11.137)

Further application of Equation (11.101) yields

 (11.138) (11.139)

However, it is easily demonstrated from the fundamental commutation relations between position and momentum operators that

 (11.140)

(See Section 2.2.) Thus,

 (11.141)

which implies that

 (11.142)

Now, . Moreover, commutes with , , and . Hence, we conclude that

 (11.143)

Finally, because commutes with and , but anti-commutes with the components of , we obtain

 (11.144)

where

 (11.145)

If we repeat the previous analysis, starting at Equation (11.138), but substituting for , and making use of the easily demonstrated result

 (11.146)

we find that

 (11.147)

Now, commutes with , as well as the components of and . Hence,

 (11.148)

Moreover, commutes with the components of , and can easily be shown to commute with all of the components of . It follows that

 (11.149)

Hence, Equations (11.128), (11.144), (11.148), and (11.149) imply that

 (11.150)

In other words, an eigenstate of the Hamiltonian is a simultaneous eigenstate of . Now,

 (11.151)

where use has been made of Equation (11.137), as well as . It follows that the eigenvalues of are . Thus, the eigenvalues of can be written , where is a non-zero integer.

Equation (11.101) implies that

where use has been made of Equations (11.130) and (11.145).

It is helpful to define the dimensionless operator , where

 (11.153)

Moreover, it is evident that

 (11.154)

Hence,

 (11.155)

where use has been made of Equation (11.24). It follows that

 (11.156)

We have already seen that commutes with and . Thus,

 (11.157)

Because commutes with and , and , as well as , we obtain

 (11.158)

However, and , so, multiplying through by , we get

 (11.159)

Equation (11.131) then yields

 (11.160)

Equation (11.152) implies that

 (11.161)

Making use of Equations (11.148), (11.153), (11.154), and (11.156), we get

 (11.162)

Hence, the Hamiltonian (11.128) becomes

 (11.163)

Now, we wish to solve the energy eigenvalue problem

 (11.164)

where is the energy eigenvalue. However, we have already shown that an eigenstate of the Hamiltonian is a simultaneous eigenstate of the operator belonging to the eigenvalue , where is a non-zero integer. Hence, the eigenvalue problem reduces to

 (11.165)

which only involves the radial coordinate, . It is easily demonstrated that anti-commutes with . Hence, given that takes the form (11.32), and that , we can represent as the matrix

 (11.166)

Thus, writing in the spinor form

 (11.167)

and making use of Equation (11.132), the energy eigenvalue problem for an electron in a central field reduces to the following two coupled radial differential equations:

 (11.168) (11.169)

Next: Fine Structure of Hydrogen Up: Relativistic Electron Theory Previous: Electron Spin
Richard Fitzpatrick 2016-01-22