quantum mechanics - Why is the parity of the spatial wavefunction $(-1)^{ell}$?

Consider a composite particle state $|\psi\rangle$ (like a hadron or a meson) that is an eigenstate of some Hamiltonian (e.g. the QCD Hamiltonian). Since the Hamiltonian is invariant under rotations and parity this particle state is also an eigenstate of the angular momentum and parity operator:

$$L^2 |\psi\rangle = l(l+1)|\psi\rangle$$ $$P |\psi\rangle = (-1)^{a}|\psi\rangle$$

where $a$ is an integer number. Why is $a = l$?

For two particles one can use the 'trick' to transform into relative coordinates and then find that in relative coordinates the eigenfunction is $\sim Y_{lm}$. The parity of the spherical harmonics then leads to $(-1)^l$.

I don't see how to extend this to 3 or more particles.

EDIT: I had the following idea how to extend to 3 particles:

For 3 particles the Hamiltonian looks like:

$$H = \frac{p_1^2}{2m_1} + \frac{p_2^2}{2m_2} + \frac{p_3^2}{2m_3} + V_1(|x_2-x_1|) + V_2(|x_3-x_1|) + V_3(|x_3-x_2|)$$ .

Now choose new coordinates by

$$R = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{M}$$ $$y = x_2 - x_1$$ $$z = x_3 - x_1$$

The Hamiltonian becomes:

$$H = \frac{p_R^2}{2M} + \frac{p_y^2}{2\mu_{12}} + \frac{p_z^2}{2\mu_{13}} + \frac{p_y\cdot p_z}{m_1} + V_1(|y|) + V_2(|z|) + V_3(|z-y|)$$ where $\frac{1}{\mu_{ij}} = \frac{1}{m_i} + \frac{1}{m_j}$ are reduced masses

The total angular momentum is given by

$$L = x_1 \times p_1 + x_2 \times p_2 + x_3 \times p_3 = R \times p_R + y \times p_y + z \times p_z$$ .

The $l$ in parity $= (-1)^l$ is given by the internal angular momentum

$$L_i = y \times p_y + z \times p_z$$ which commutes with the Hamiltonian.

Therefore an eigenfunction is given by

$$|\psi(y,z)\rangle = |f(|y|,|z|)\rangle |L M\rangle_{\hat{y}\hat{z}}$$ . Using Clebsch-Gordan coefficients this angular can be written as: $$|L M\rangle_{\hat{y}\hat{z}} = \sum_{m m'} \langle lm,l'm'|LM\rangle Y_{lm}(\hat{y}) Y_{l'm'}(\hat{z})$$ for some $l$ and $l'$

The overall parity is given by $(-1)^{l + l'}$ which does not necessary equals $(-1)^L$. For example $l = l' = 1$ would lead to a (from my point of view valid) solution:

$$|10\rangle_{\hat{y}\hat{z}} = \frac{1}{\sqrt{2}} \left(Y_{11}(\hat{y})Y_{1-1}(\hat{z}) - Y_{1-1}(\hat{y})Y_{11}(\hat{z})\right)$$ with parity $(-1)^{l+l'} = (-1)^{1+1} = 1 \neq (-1)^1 = (-1)^L$.

There must be something which excludes such combinations. Why is this solution not valid?

Answer

$\def\bL{\mathbf L} \def\bl{\mathbf l} \def\bp{\mathbf p} \def\br{\mathbf r} \def\bP{\mathbf P} \def\bR{\mathbf R} \def\frac#1#2{{\textstyle {#1 \over #2}}} \def\half{\frac12}$ In order to reply your objections it's best for me to write some equations. As a general comment: you should not think of a change of reference frame, but only of expressing the original quantities (hamiltonian, angular momentum) in terms of new coordinates. Let's see how.

I'm going to assume all mases are equal, just to make formulas simpler. But you may verify that the argument is quite general. On the other hand, it's a time-honoured approach, known as Jacobi's coordinates and widely used in celestial mechnics more or less since mid-19th century.

Call $\br_1$, $\br_2$, $\br_3$, the position vectors of the three particles. Define

$$\bR = {\textstyle {1 \over 3}} (\br_1 + \br_2 + \br_3)$$ $$\br= \half (\br_1 + \br_2) - \br_3.$$ $$\br' = \br_2 - \br_1$$ Kinetic energy: $$K = \half\,m \left(3\,\dot\bR^2 + \frac23 \dot\br^2 + \half {\dot\br'}^2\right).$$ Conjugate momenta: $$\bP = 3\,m\,\dot\bR \qquad \bp = \frac23 m\,\dot\br \qquad \bp = \half m\,\dot\br'$$ $$K = {P^2 \over 6\,m} + {3\,p^2 \over 2\,m} + {{p'}^2 \over m}.$$ Angular momentum: $$\bL_{\mathrm{tot}} = \bR \times \bP + \br \times \bp + \br' \times \bp' = \bL + \bl + \bl'.$$ Commutators are those expected for canonical coordinateses $\bR$ with $\bP$, $\br$ with $\bp$, $\br'$ with $\bp'$.

Parity quantum number refers to the transformation

$$\br \to - \br \qquad \br' \to -\br'$$ and consequently $$\bp \to - \bp \qquad \bp' \to -\bp'.$$ Eigenstates of $\bl^2$, $\bl'^2$ have parity $(-1)^{l+l'}$ and I see that you say the same thing, even if with different coordinates (which, incidentally, gives rise to a non-separable hamiltonian).

What I cannot understand is why you are troubled with your example. You built a state of internal angular momentum 1, $z$-component 0, starting from states with $l=l'=1$, then parity +. What's wrong with that? It's a perfectly possible situation.