Recently I've been studying Angular Momentum in Quantum Mechanics and I have a doubt about the eigenstates of orbital angular momentum in the position representation and the relation to the spherical harmonics. First of all, we consider the angular momentum operators [itex]L^2[/itex] and [itex]L_z[/itex]. We know that they commute so that we can find a basis of eigenstates commom to those two operators.(adsbygoogle = window.adsbygoogle || []).push({});

Also, we also know from the general theory of Angular Momentum that the eigenvalues of [itex]L^2[/itex] are of the form [itex]l(l+1)\hbar^2[/itex] with integral or half-integral [itex]l[/itex], while fixing [itex]l[/itex] the eigenvalues of [itex]L_z[/itex] are of the form [itex]m\hbar[/itex] with [itex]m\hbar[/itex] being restricted to the numbers [itex]-l,-l+1,\dots,l-1,l[/itex].

In that case the eigenvalue equations for the two operators in the position representation using spherical coordinates are:

[tex]L^2 \psi(r,\theta,\phi) = l(l+1)\hbar^2 \psi(r,\theta,\phi), \\ L_z\psi(r,\theta,\phi) = m\hbar \psi(r,\theta,\phi).[/tex]

The book then states the following:

I have several doubts regarding that. First of all, in the way the book presents, the spherical harmonics are In the eigenvalue equations, [itex]r[/itex] does not appear in any differential operator, so we can consider it to be a parameter and take into account only the [itex]\theta[/itex]- and [itex]\phi[/itex]-dependence of [itex]\psi[/itex]. Thus, we denote by [itex]Y_l^m(\theta,\phi)[/itex] a common eigenfunction of [itex]L^2[/itex] and [itex]L_z[/itex] which corresponds to the eigenvalues [itex](l+1)\hbar^2[/itex] and [itex]m\hbar[/itex].

The eigenvalue equations give the [itex]\theta[/itex]- and [itex]\phi[/itex]-dependence of the eigenfunctions of [itex]L^2[/itex] and [itex]L_z[/itex]. Once the solutions [itex]Y^m_l(\theta,\phi)[/itex] of these equations has been found, these eigenfunctions will be obtained in the form:

[tex]\psi_{l,m}(r,\theta,\phi)=f(r)Y^m_l(\theta,\phi)[/tex]

where [itex]f(r)[/itex] is a function of [itex]r[/itex] which appears as na integration constant for the partial differential equation.definedas eigenfunctions of the operators in question. I do know that [itex]r[/itex] appears just as a parameter, but this doesn't mean the solutions do not depend on this variable. The spherical harmonics are functions of [itex]\theta[/itex] and [itex]\phi[/itex] alone, so how can they be eigenfunctions of these operators if they do not depend on the parameter [itex]r[/itex]?

And the most important question: how the author concludes that the eigenfunctions are all of the form [itex]\psi_{l,m}(r,\theta,\phi)=f(r)Y^m_l(\theta,\phi)[/itex]? I mean, first he defined [itex]Y^m_l[/itex] as the eigenfunctions, later he says that the eigenfunctions have this specific form. How does one following the author's reasoning find out that the eigenfunctions have this form with this exact dependence on the parameter [itex]r[/itex]?

Of course we can solve the equations. In that way we indeed get what the spherical harmonics are. But then we see that this is the form of the separable solutions: if we use separation of variables we arrive at this form of eigenfunctions. But why there are no other solutions?

In general I'm quite confused in the way this is being presented and why the eigenstates of orbital angular momentum have that form in the position representation.

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# Eigenstates of Orbital Angular Momentum

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