When adding a spin-1 to a spin-1/2, we have a six-dimensional Hilbert space spanned by the factorized states $$ \left \{ \left | j_1=1, m_1 \right \rangle \otimes \left |j_2= \frac{1}{2}, m_2 \right \rangle \right \},$$ where $$m_1= \{-1,0,1 \},$$ and $$m_2=\{-\frac{1}{2}, \frac{1}{2} \}.$$

Equivalently, the same Hilbert space is spanned by the coupled states, resulting from adding the angular momenta, $$ \left \{ \left | j,\ m, \ j_1, \ j_2 \right \rangle \right \},$$ where $$j=|j_1-j_2|,\ |j_1-j_2|+1, \ ...\ ,j_1+j_2, $$ and $$m=-j , \ ..., \ j.$$ Now, we can write the coupled bases in terms of the factorized ones, since they span the same Hilbert space. As a concrete example, we can write $$ \left | j=\frac{3}{2},\ m=\frac{1}{2}, \ j_1=1, \ j_2=\frac{1}{2} \right \rangle = \frac{1}{\sqrt{3}} \left( \left | j_1=1, m_1=1 \right \rangle \otimes \left |j_2= \frac{1}{2}, m_2=- \frac{1}{2} \right \rangle \right) +\frac{2}{\sqrt{3}} \left( \left | j_1=1, m_1=0 \right \rangle \otimes \left |j_2= \frac{1}{2}, m_2=\frac{1}{2} \right \rangle \right).$$

The coefficients are just the Clebsch-Gordan coefficients, and they (squared) represent the probability of finding the system in the corresponding factorized state when a measurement occurs. My question is: why are they not equally probable? My understanding is that each of the factorized basis is equally as good the others.

I know how to get the coefficient mathematically using the ladder operators, but I guess I am missing the physical reason why the probabilities are distributed this way.


1 Answer 1


This is a good question which does not have a good answer, except to say that "it's in the math".

The least bad analogy that one can find is to realize that the values of $\hat L_\pm$ depend on $M$ and $J$ when the operators act on $\vert JM\rangle$ states. Thus, starting from the $\vert J,J\rangle$ state, the process of lowering to $\vert J,J-1\rangle$ will not produce states with the same proportions of each $\vert j_1m_1\rangle$ and $\vert j_2m_2\rangle$ since these proportions are given by the matrix elements of $L_-^{(1)}$ and $L_-^{(2)}$ acting on each of $\vert j_1m_1\rangle$ and $\vert j_2m_2\rangle$ respectively. This is an entirely mathematical argument.

In Section 27 of his book, Wigner provides a geometrical interpretation to $\vert\langle j_1m_1;j_2m_2\vert JM\rangle\vert^2$ in terms of the area of triangle with sides $\vec j_1,\vec j_2$ and $\vec J$ with projections $m_1,m_2$ and $M$, assuming large values of the quantities so some sort of classical interpretation can be reached. The model is based on a precession argument, with different vectors $\vec j_1$ and $\vec j_2$ precessing at different speeds on account of their different lengths and projections along $\vec J$.

Wigner's result is really an average and not a terribly good estimate of $\vert\langle j_1m_1;j_2m_2\vert JM\rangle\vert^2$, as you can see from Fig 2b of this paper:

Schulten K, Gordon RG. Semiclassical approximations to 3/-and 6/-coefficients for quantum-mechanical coupling of angular momenta. Journal of Mathematical Physics. 1975 Oct;16(10)

enter image description here

Schulten and Gordon improve on Wigner by converting a recursion relation to a differential equation, which they then solve using WKB techniques.

It's not that hard to illustrate why Wigner's physical argument is off: the values of the CGs actually oscillates significantly with $m_1$ and $m_2$ for fixed $M$, as illustrated in the graph below for $\langle 12,m; 17,-m\vert 24,0\rangle$. (The value of $m$ is on the horizontal axis, and the value of the CG on the vertical axis). In particular, for $m=0$, the CG is actually $0$ so there is probability $0$ of finding the factorized state $\vert 12,0\rangle\vert 17,0\rangle$ in the state $\vert 24,0\rangle$.

enter image description here

The oscillation clearly visible in the above figure can also be understood by (approximately) converting a recursion relation for CGs to a differential equation, which this time is recognized to be the same as the quantum harmonic oscillator, shifted from the origin by some amount which depends on the CG. There's details of this trick in

Rowe DJ, de Guise H. The shifted harmonic approximation and asymptotic SU (2) and SU (1, 1) Clebsch–Gordan coefficients. Journal of Physics A: Mathematical and Theoretical. 2010 Nov 25;43(50):505307.

and there is an arXiv version of this paper if you cannot access the journal version.

This paper just displaces the question you ask to "why should this be related to a harmonic oscillator" but the authors do not provide an answer to this newer question.


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