11.6: Motion in Central Field
- Page ID
- 1258
To further study the motion of an electron in a central field, whose Hamiltonian is
and
which implies that in the Schrödinger representation
Furthermore, we know from general principles that the eigenvalues of \( j\,(j+1)\,\hbar^2\) , where \( j=\vert l+1/2\vert\) , where \( (\)\( \cdot {\bf L})\,(\)\( \cdot{\bf L}) = L^{\,2} + {\rm i}\,\)\( \times ({\bf L}\times {\bf L}).\) However, because \( {\bf L}\) is an angular momentum, its components satisfy the standard commutation relations However, \( (\)\( \cdot {\bf L} + \hbar)^2 = J^{\,2}+\frac{1}{4}\,\hbar^2.\) Further application of \ref{1192} yields However, it is easily demonstrated from the fundamental commutation relations between position and momentum operators that which implies that where we find that\ref{1224} \( (\)\( \cdot {\bf L})\,(\)\( \cdot{\bf L}) = L^{\,2} -\hbar\,\)\( \cdot{\bf L} = J^{\,2} - 2\,\hbar\,\)\( \cdot{\bf L} -\frac{\hbar^2}{4}\,\Sigma^{\,2}.\) \ref{1226} \ref{1227} \( \mbox{\boldmath\)\( \mbox{\boldmath\)\( = {\bf L}\cdot {\bf p} + {\rm i}\,\)\( \cdot{\bf L}\times {\bf p}= {\rm i}\,\)\( \cdot{\bf L}\times{\bf p},\) \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} \( (\)\( \cdot {\bf L})\,(\)\( \cdot{\bf p}) + (\)\( \cdot {\bf p})\,(\)\( \cdot{\bf L}) =-2\,\hbar\,\)\( \cdot{\bf p},\) \ref{1231} \( \mbox{\boldmath\)\( \mbox{\boldmath\)\( \gamma^5\,\)\( \Sigma\) \( =\) \( \gamma^5\) commutes with \( {\bf L}\) , and \( \Sigma\) . Hence, we conclude that commutes with \( {\bf L}\) , but anti-commutes with the components of \( [\zeta,\)\( \cdot {\bf p}] = 0,\) \( \mbox{\boldmath\)\( \mbox{\boldmath\)\( \beta\) \ref{1234} for \( {\bf L}\times {\bf x} + {\bf x}\times{\bf L} = 2\,{\rm i}\,\hbar\,{\bf x},\) \( {\bf x}\) \ref{1236} commutes with \( \Sigma\) and \( {\bf L}\) . Hence,\( \mbox{\boldmath\)\( r\)
In other words, an eigenstate of the Hamiltonian is a simultaneous eigenstate of \( \zeta^{\,2} = [\beta\,(\)\( \cdot{\bf L}+\hbar)]^{\,2} = (\)\( \cdot{\bf L}+\hbar)^2 = J^{\,2}+\frac{1}{4}\,\hbar^2,\)
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
Hence,
We have already seen that \( \alpha\) \( \cdot{\bf x}\) and \( r\) . Thus,
where use has been made of the fundamental commutation relations for position and momentum operators. However, \( \Sigma\) \( \gamma^5\), we get
Equation \ref{1242} implies that
Now, we wish to solve the energy eigenvalue problem
which only involves the radial coordinate \( r\) . It is easily demonstrated that \( \beta\) . Hence, given that \( \epsilon^2=1\) , we can represent \( \epsilon = \left(\begin{array}{rr}0&-{\rm i}\\ [0.5ex]{\rm i}&0\end{array}\right).\)
Thus, writing \( \psi = \left(\begin{array}{c} \psi_a(r)\\ [0.5ex]\psi_b(r)\end{array}\right),\) 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: Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)\ref{1257} \( = 0,\) \ref{1258} \( = 0.\) \ref{1259} Contributors