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8.1: Derivation of Radial Equation

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    15771
  • Now, we have seen that the Cartesian components of the momentum, \({\bf p}\), can be represented as (see Section [s7.2]) \[p_i = -{\rm i}\,\hbar\,\frac{\partial}{\partial x_i}\] for \(i=1,2,3\), where \(x_1\equiv x\), \(x_2\equiv y\), \(x_3\equiv z\), and \({\bf r}\equiv (x_1, x_2, x_3)\). Likewise, it is easily demonstrated, from the previous expressions, and the basic definitions of the spherical coordinates [see Equations ([e8.21])–([e8zz])], that the radial component of the momentum can be represented as

    \[\label{e9.4} p_r \equiv \frac

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    {r} = -\rm i \hbar\frac{\partial}{\partial r}.\]

    Recall that the angular momentum vector, \({\bf L}\), is defined

    \[\label{e9.5} {\bf L} = {\bf r}\times {\bf p}.\]

    [See Equation ([e8.0]).] This expression can also be written in the following form: \[\label{e9.6} L_i = \epsilon_{ijk}\,x_j\,p_k.\] Here, the \(\epsilon_{ijk}\) (where \(i,j,k\) all run from 1 to 3) are elements of the so-called totally anti-symmetric tensor . The values of the various elements of this tensor are determined via a simple rule: \[\label{e9.7} \epsilon_{ijk} = \left\{ \begin{array}{rcl} 0 &\mbox{\hspace{1cm}}&\mbox{if $i,j,k$ not all different}\\[0.5ex] 1 &&\mbox{if $i,j,k$ are cyclic permutation of $1,2,3$}\\[0.5ex] -1 &&\mbox{if $i,j,k$ are anti-cyclic permutation of $1,2,3$} \end{array}\right. .\] Thus, \(\epsilon_{123}=\epsilon_{231}=1\), \(\epsilon_{321}=\epsilon_{132}=-1\), and \(\epsilon_{112}=\epsilon_{131}=0\), et cetera. Equation ([e9.6]) also makes use of the Einstein summation convention, according to which repeated indices are summed (from 1 to 3) . For instance, \(a_i\,b_i\equiv a_1\,b_1+a_2\,b_2+a_3\,b_3\). Making use of this convention, as well as Equation ([e9.7]), it is easily seen that Equations ([e9.5]) and ([e9.6]) are indeed equivalent.

    Let us calculate the value of \(L^2\) using Equation ([e9.6]). According to our new notation, \(L^2\) is the same as \(L_i\,L_i\). Thus, we obtain \[\label{e9.8} L^2 = \epsilon_{ijk}\,x_j\,p_k\,\epsilon_{ilm}\,x_l\,p_m = \epsilon_{ijk}\,\epsilon_{ilm}\,x_j\,p_k\,x_l\,p_m.\] Note that we are able to shift the position of \(\epsilon_{ilm}\) because its elements are just numbers, and, therefore, commute with all of the \(x_i\) and the \(p_i\). Now, it is easily demonstrated that \[\label{e9.9} \epsilon_{ijk}\,\epsilon_{ilm}\equiv \delta_{jl}\,\delta_{km}-\delta_{jm}\,\delta_{kl}.\] Here \(\delta_{ij}\) is the usual Kronecker delta, whose elements are determined according to the rule \[\delta_{ij} = \left\{ \begin{array}{rcl} 1 &\mbox{\hspace{1cm}}&\mbox{if $i$ and $j$ the same}\\[0.5ex] 0 &&\mbox{if $i$ and $j$ different}\\[0.5ex] \end{array}\right. .\] It follows from Equations ([e9.8]) and ([e9.9]) that \[\label{e9.11} L^2 = x_i\,p_j\,x_i\,p_j - x_i\,p_j\,x_j\,p_i.\] Here, we have made use of the fairly self-evident result that \(\delta_{ij}\,a_i\,b_j \equiv a_i\,b_i\). We have also been careful to preserve the order of the various terms on the right-hand side of the previous expression, because the \(x_i\) and the \(p_i\) do not necessarily commute with one another.

    We now need to rearrange the order of the terms on the right-hand side of Equation ([e9.11]). We can achieve this goal by making use of the fundamental commutation relation for the \(x_i\) and the \(p_i\): \[\label{e9.12} [x_i,p_j] = {\rm i}\,\hbar\,\delta_{ij}.\] [See Equation ([commxp]).] Thus, \[\begin{aligned} L^2 &= x_i\left(x_i\,p_j - [x_i,p_j]\right) p_j - x_i\,p_j\,\left(p_i\,x_j+[x_j,p_i]\right)\nonumber\\[0.5ex] &=x_i\,x_i\,p_j\,p_j - {\rm i}\,\hbar\,\delta_{ij}\,x_i\,p_j -x_i\,p_j\,p_i\,x_j - {\rm i}\,\hbar\,\delta_{ij}\,x_i\,p_j\nonumber\\[0.5ex] &=x_i\,x_i\,p_j\,p_j -x_i\,p_i\,p_j\,x_j - 2\,{\rm i}\,\hbar\,x_i\,p_i.\end{aligned}\] Here, we have made use of the fact that \(p_j\,p_i=p_i\,p_j\), because the \(p_i\) commute with one another. [See Equation ([commpp]).] Next, \[L^2 = x_i\,x_i\,p_j\,p_j - x_i\,p_i\left(x_j\,p_j - [x_j,p_j]\right) - 2\,{\rm i}\,\hbar\,x_i\,p_i.\] Now, according to Equation ([e9.12]), \[[x_j,p_j]\equiv [x_1,p_1]+[x_2,p_2]+[x_3,p_3] = 3\,{\rm i}\,\hbar.\] Hence, we obtain \[L^2 = x_i\,x_i\,p_j\,p_j - x_i\,p_i\,x_j\,p_j + {\rm i}\,\hbar\,x_i\,p_i.\] When expressed in more conventional vector notation, the previous expression becomes \[\label{e9.17} L^2 = r^{\,2}\,p^{\,2} - ({\bf r}\cdot{\bf p})^2 + {\rm i}\,\hbar\,{\bf r}\cdot{\bf p}.\] Note that if we had attempted to derive the previous expression directly from Equation ([e9.5]), using standard vector identities, then we would have missed the final term on the right-hand side. This term originates from the lack of commutation between the \(x_i\) and \(p_i\) operators in quantum mechanics. Of course, standard vector analysis assumes that all terms commute with one another.

    Equation ([e9.17]) can be rearranged to give \[p^{\,2} = r^{\,-2}\left[({\bf r}\cdot{\bf p})^2- {\rm i}\,\hbar\,{\bf r}\cdot{\bf p}+L^2\right].\] Now, \[{\bf r}\cdot{\bf p} = r\,p_r = -{\rm i}\,\hbar\,r\,\frac{\partial}{\partial r},\] where use has been made of Equation ([e9.4]). Hence, we obtain \[p^{\,2} = -\hbar^{\,2}\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r\,\frac{\partial}{\partial r}\right) + \frac{1}{r}\frac{\partial}{\partial r}- \frac{L^2}{\hbar^{\,2}\,r^{\,2}}\right].\] Finally, the previous equation can be combined with Equation ([e9.2]) to give the following expression for the Hamiltonian: \[\label{e9.21} H = -\frac{\hbar^{\,2}}{2\,m}\left(\frac{\partial^{\,2}}{\partial r^{\,2}} + \frac{2}{r}\frac{\partial}{\partial r}- \frac{L^2}{\hbar^{\,2}\,r^{\,2}}\right) +V(r).\]

    Let us now consider whether the previous Hamiltonian commutes with the angular momentum operators \(L_z\) and \(L^2\). Recall, from Section [s8.3], that \(L_z\) and \(L^2\) are represented as differential operators that depend solely on the angular spherical coordinates, \(\theta\) and \(\phi\), and do not contain the radial coordinate, \(r\). Thus, any function of \(r\), or any differential operator involving \(r\) (but not \(\theta\) and \(\phi\)), will automatically commute with \(L^2\) and \(L_z\). Moreover, \(L^2\) commutes both with itself, and with \(L_z\). (See Section [s8.2].) It is, therefore, clear that the previous Hamiltonian commutes with both \(L_z\) and \(L^2\).

    According to Section [smeas], if two operators commute with one another then they possess simultaneous eigenstates. We thus conclude that for a particle moving in a central potential the eigenstates of the Hamiltonian are simultaneous eigenstates of \(L_z\) and \(L^2\). Now, we have already found the simultaneous eigenstates of \(L_z\) and \(L^2\)—they are the spherical harmonics, \(Y_{l,m}(\theta,\phi)\), discussed in Section [sharm]. It follows that the spherical harmonics are also eigenstates of the Hamiltonian. This observation leads us to try the following separable form for the stationary wavefunction: \[\label{e9.22} \psi(r,\theta,\phi) = R(r)\,Y_{l,m}(\theta,\phi).\] It immediately follows, from Equation ([e8.29]) and ([e8.30]), and the fact that \(L_z\) and \(L^2\) both obviously commute with \(R(r)\), that \[\begin{aligned} L_z\,\psi &= m\,\hbar\,\psi,\\[0.5ex] L^2\,\psi&= l\,(l+1)\,\hbar^{\,2}\,\psi.\label{e9.24}\end{aligned}\] Recall that the quantum numbers \(m\) and \(l\) are restricted to take certain integer values, as explained in Section [slsq].

    Finally, making use of Equations ([e9.1]), ([e9.21]), and ([e9.24]), we obtain the following differential equation which determines the radial variation of the stationary wavefunction: \[-\frac{\hbar^{\,2}}{2\,m}\left[\frac{d^{\,2}}{d r^{\,2}} + \frac{2}{r}\frac{d}{d r}- \frac{l\,(l+1)}{r^{\,2}}\right]R_{n,l} +V\,R_{n,l} = E\,R_{n,l}.\] Here, we have labeled the function \(R(r)\) by two quantum numbers, \(n\) and \(l\). The second quantum number, \(l\), is, of course, related to the eigenvalue of \(L^2\). [Note that the azimuthal quantum number, \(m\), does not appear in the previous equation, and, therefore, does not influence either the function \(R(r)\) or the energy, \(E\).] As we shall see, the first quantum number, \(n\), is determined by the constraint that the radial wavefunction be square-integrable.

    Contributors

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

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)