I have a problem understanding three arguments in quantum mechanic:
- When we talk about a particle in a central field we have this kind of Hamiltonian: $$H=\frac{p^2}{2m}+V(r)$$ if we use spherical coordinates we can write the Laplacian operator as: $$\nabla^2=\frac{\partial^2}{\partial r^2}+\frac{2}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}(\frac{\partial^2}{\partial \theta^2}+\frac{cos(\theta)}{sin(\theta)}\frac{\partial}{\partial \theta}+\frac{1}{sin^2(\theta)}\frac{\partial^2}{\partial \phi^2})$$ and so $p^2$ in Schordinger's representation as:
$$p^2=-\hbar^2(\frac{\partial^2}{\partial r^2}+\frac{2}{r}\frac{\partial}{\partial r})+\frac{L^2}{r^2}$$ so that the entire Schrodinger's equation becomes:
$$(\frac{\partial^2}{\partial r^2}+\frac{2}{r}\frac{\partial}{\partial r}-\frac{L^2}{\hbar^2r^2}-\frac{2mV(r)}{\hbar^2}+\frac{2mE}{\hbar^2})\psi(r,\theta,\phi)=0 \ (1)$$
Now the problem: my professore said that "
since a central Hamiltonian commutes with the operators $L^2$ and $L_z$ (I know why), we write the solution of equation (1) in the factorised form:
$$\psi(r,\theta,\phi)=F(\theta,\phi)R(r)$$
First question: Why, from the fact that a central Hamiltonian commutes with the operators $L^2$ and $L_z$, we come to write the eigenfunction in that "separated" form? I think the answer is related to the fact that if 2 operators commute with each other they have a complete set of simultaneous eigenfunctions, but I can't understand why this lead to the "separate" form of the Hamiltonian eigenfuntion.
- The other topic that causes me the same confusion is the following one:
talking about the Hydrogen atom we have $$H=\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+V(|q_1-q_2|)$$ which becomes, after a change of variables $$H=\frac{P^2}{2M}+\frac{p^2}{2\mu}+V(|q|) \ (2)$$
where $M=m_1+m_2$ and $\frac{1}{\mu}=\frac{1}{m_1}+\frac{1}{m_2}$
Now in my notes I've written that
the equation (2) is a separable variables equation, namely the sum of two terms $H_1$ which only depends from the center of mass variables and $H_2$ which only depends from relative variables, so $H_1$ and $H_2$ commute each other. Since the total Hamiltonian is separable we can search the total eigenfunction in the form $\psi(X,x)=\phi(X)\psi(x)$ (3) where $X$ refers to center of mass variable and $x$ to relative variables.
I can now write equation $(2)$ in Schrodinger's representation:
$$(\frac{-\hbar^2\nabla^2_X}{2M}-\frac{-\hbar^2\nabla^2_x}{2\mu}+V(|X|)\psi(X,x)=E\psi(X,x)) \ (4)$$
then I replace equation $(3)$ into equation $(4)$ obtaining:
$$(-\frac{\hbar^2}{2M}\nabla^2_X\phi(X))\psi(x)+((\frac{-\hbar^2}{2\mu}+V(|x|))\psi(x))\phi(X)=E\phi(X)\psi(x)$$
I can now divide by $\phi(X)\psi(x)$:
$$-\frac{\hbar^2}{2M}\frac{\nabla^2_X\phi(X)}{\phi(X)}-\frac{\hbar^2}{2\mu}\frac{\nabla^2_x\psi(x)}{\psi(x)}+V(|x|)=E$$
which leads to two separated eigenvalues equations:
$$-\frac{\hbar^2}{2M}\nabla^2_X\phi(X)=E_1\phi(X)$$
$$-\frac{\hbar^2}{2\mu}\nabla^2_x\psi(x)+V(|x|)\psi(x)=E_2\psi(x)$$
which I can solve separately.
Here the question is: Why the fact that $H=H_1+H_2$ and $H_1$ commutes with $H_2$ brings me to search the total eigenfunction in the form $\psi(X,x)=\phi(X)\psi(x)$?
- In a website (sorry, I can't find it anymore) I've read this when talking about spherical harmonics:
"if we want to determine simultaneous eigenfuntions of $L^2$ and $L_z$ we have to solve this system:
$$L^2|l,m\rangle=\hbar^2l(l+1)|l,m\rangle$$ $$L_z|l,m\rangle=\hbar m|l,m\rangle$$
in Schrodinger's representation:
$$-(\frac{1}{sin(\theta)}\frac{\partial}{\partial{\theta}}(sin(\theta)\frac{\partial}{\partial{\theta}})+\frac{1}{sin^2(\theta)}\frac{\partial^2}{\partial{\phi^2}})\psi_{lm}(r,\theta,\phi)=l(l+1)\psi_{lm}(r,\theta,\phi)$$
$$-i\frac{\partial}{\partial{\phi}}\psi_{lm}(r,\theta,\phi)=m\psi_{lm}(r,\theta,\phi)$$
we can note that operators $L^2$ and $L_z$ don't contain variable $r$ so the $\psi_{lm}(r,\theta,\phi)$ must be of the form $\psi_{lm}(r,\theta,\phi)=f(r)Y_{lm}(\theta,\phi)$.
Third question: Why if operators $L^2$ and $L_z$ don't contain variable $r$ the $\psi_{lm}(r,\theta,\phi)$ must be of the form $\psi_{lm}(r,\theta,\phi)=f(r)Y_{lm}(\theta,\phi)$?
The final question about all this is:
are the three problems I've written related in a certain way? because all three propose an eigenfunction written as a product of eigenfunctions but in the first case this is due to the fact that $H$, $L^2$, $L_z$ commute with each other. In the second case this is due to the fact that $H$ can be written as a sum of Hamiltonians that commute with each other and we speak of separation of variables. In the third case it is due to the fact that the variable $r$ does not appear in the operators $L^2$ and $L_z$. Do all three mean the same thing or are they three different things?
