11.6: Motion in Central Field

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

\( \mbox{\boldmath\)\( r = (x^{2}+y^{2}+z^{2})^{1/2},\) \ref{1219}

and

\( [r,p_r] = {\rm i} \,\hbar,\) \ref{1221}

which implies that in the Schrödinger representation

\( {\bf J} = {\bf L} + \frac{\hbar}{2}\,\)\( .\) \ref{1223}

Furthermore, we know from general principles that the eigenvalues of \( j\,(j+1)\,\hbar^2\) , where \( j=\vert l+1/2\vert\) , where \( (\)\( \cdot {\bf L})\,(\)\( \cdot{\bf L}) = L^{\,2} + {\rm i}\,\)\( \times ({\bf L}\times {\bf L}).\) \ref{1224}

However, because \( {\bf L}\) is an angular momentum, its components satisfy the standard commutation relations

\( (\)\( \cdot {\bf L})\,(\)\( \cdot{\bf L}) = L^{\,2} -\hbar\,\)\( \cdot{\bf L} = J^{\,2} - 2\,\hbar\,\)\( \cdot{\bf L} -\frac{\hbar^2}{4}\,\Sigma^{\,2}.\) \ref{1226}

However, \( (\)\( \cdot {\bf L} + \hbar)^2 = J^{\,2}+\frac{1}{4}\,\hbar^2.\) \ref{1227}

Further application of \ref{1192} yields

\( \mbox{\boldmath\)\( \mbox{\boldmath\)\( = {\bf L}\cdot {\bf p} + {\rm i}\,\)\( \cdot{\bf L}\times {\bf p}= {\rm i}\,\)\( \cdot{\bf L}\times{\bf p},\) \ref{1228} \( \mbox{\boldmath\)\( \mbox{\boldmath\)\( = {\bf p}\cdot {\bf L} + {\rm i}\,\)\( \cdot{\bf p}\times {\bf L}= {\rm i}\,\)\( \cdot{\bf p}\times{\bf L},\) \ref{1229}

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

\( (\)\( \cdot {\bf L})\,(\)\( \cdot{\bf p}) + (\)\( \cdot {\bf p})\,(\)\( \cdot{\bf L}) =-2\,\hbar\,\)\( \cdot{\bf p},\) \ref{1231}

which implies that

\( \mbox{\boldmath\)\( \mbox{\boldmath\)\( \gamma^5\,\)\( \Sigma\) \( =\) \( \gamma^5\) commutes with \( {\bf L}\) , and \( \Sigma\) . Hence, we conclude that

\( \mbox{\boldmath\)\( \mbox{\boldmath\)\( \beta\) commutes with \( {\bf L}\) , but anti-commutes with the components of \( [\zeta,\)\( \cdot {\bf p}] = 0,\) \ref{1234}

where

\( {\bf x}\) for \( {\bf L}\times {\bf x} + {\bf x}\times{\bf L} = 2\,{\rm i}\,\hbar\,{\bf x},\) \ref{1236}

we find that

\( \mbox{\boldmath\)\( r\) commutes with \( \Sigma\) and \( {\bf L}\) . Hence,

\( \beta\) commutes with the components of \( {\bf L}\) , and can easily be shown to commute with all components of \( \Sigma\) . It follows that

\( [\zeta, H] =0.\) \ref{1240}

In other words, an eigenstate of the Hamiltonian is a simultaneous eigenstate of \( \zeta^{\,2} = [\beta\,(\)\( \cdot{\bf L}+\hbar)]^{\,2} = (\)\( \cdot{\bf L}+\hbar)^2 = J^{\,2}+\frac{1}{4}\,\hbar^2,\)

\ref{1241}

where use has been made of Equation \ref{1227}, as well as \( \zeta^{\,2}\) are \( \zeta\) can be written \( k=\pm(j+1/2)\) is a non-zero integer.

Equation \ref{1192} implies that

\( \mbox{\boldmath\)\( \mbox{\boldmath\)\( = {\bf x}\cdot{\bf p} + {\rm i}\,\)\( \cdot{\bf x}\times {\bf p} = r\,p_r + {\rm i}\,\)\( \cdot{\bf L}\) \( \epsilon\), where

\( \mbox{\boldmath\)\( [\epsilon,r] = 0.\) \ref{1244}

Hence,

\( \mbox{\boldmath\)\( \epsilon^2 = 1.\) \ref{1246}

We have already seen that \( \alpha\) \( \cdot{\bf x}\) and \( r\) . Thus,

\( (\)\( \cdot{\bf x})\,({\bf x}\cdot{\bf p}) - ({\bf x}\cdot{\bf p})\,(\)\( \cdot{\bf x}) =\) \( \cdot\left[{\bf x}\,({\bf x}\cdot{\bf p})- ({\bf x}\cdot{\bf p})\,{\bf x}\right] = {\rm i}\,\hbar\,\)\( \cdot{\bf x},\) \ref{1248}

where use has been made of the fundamental commutation relations for position and momentum operators. However, \( \Sigma\) \( \gamma^5\), we get

\( [\epsilon,p_r]= 0.\) \ref{1250}

Equation \ref{1242} implies that

\( \mbox{\boldmath\)\( \mbox{\boldmath\)\( \mbox{\boldmath\)\( H= - e\,\phi(r) + c\,\epsilon\,(p_r-{\rm i}\,\hbar/r) + {\rm i}\,c\,\epsilon\,\beta\,\zeta/r + \beta\,m_e\,c^2.\) \ref{1253}

Now, we wish to solve the energy eigenvalue problem

\( E\) is the energy eigenvalue. However, we have already shown that an eigenstate of the Hamiltonian is a simultaneous eigenstate of the \( k\,\hbar\) , where \( \left[- e\,\phi(r) + c\,\epsilon\,(p_r-{\rm i}\,\hbar/r) + {\rm i}\,c\,\hbar\,k\,\epsilon\,\beta/r + \beta\,m_e\,c^2\right]\psi = E\,\psi,\) \ref{1255}

which only involves the radial coordinate \( r\) . It is easily demonstrated that \( \beta\) . Hence, given that \( \epsilon^2=1\) , we can represent \( \epsilon = \left(\begin{array}{rr}0&-{\rm i}\\ [0.5ex]{\rm i}&0\end{array}\right).\)

\ref{1256}

Thus, writing \( \psi = \left(\begin{array}{c} \psi_a(r)\\ [0.5ex]\psi_b(r)\end{array}\right),\) \ref{1257}

and making use of \ref{1222}, the energy eigenvalue problem for an electron in a central field reduces to the following two coupled radial differential equations:

\( = 0,\) \ref{1258} \( = 0.\) \ref{1259}

Contributors

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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