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Two Spin One-Half Particles
Consider a system consisting of two spin one-half particles. Suppose
that the system does not possess any orbital angular momentum.
Let
and
be the spin angular momentum operators
of the first and second particles, respectively, and let
![\begin{displaymath}
{\bf S} ={\bf S}_1 + {\bf S}_2
\end{displaymath}](img2029.png) |
(839) |
be the total spin angular momentum operator. By analogy
with the previous analysis, we conclude that it is possible to simultaneously measure either
,
,
, and
, or
,
,
,
, and
.
Let the quantum numbers associated with measurements of
,
,
,
,
, and
be
,
,
,
,
, and
, respectively.
In other words, if the spinor
is
a simultaneous eigenstate of
,
,
,
and
, then
![$\displaystyle S_1^{ 2} \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$](img2039.png) |
![$\textstyle =$](img122.png) |
![$\displaystyle s_1 (s_1+1) \hbar^2
\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},$](img2040.png) |
(840) |
![$\displaystyle S_2^{ 2} \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$](img2041.png) |
![$\textstyle =$](img122.png) |
![$\displaystyle s_2 (s_2+1) \hbar^2
\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},$](img2042.png) |
(841) |
![$\displaystyle S_{1z} \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$](img2043.png) |
![$\textstyle =$](img122.png) |
![$\displaystyle m_{s_1} \hbar
\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},$](img2044.png) |
(842) |
![$\displaystyle S_{2z} \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$](img2045.png) |
![$\textstyle =$](img122.png) |
![$\displaystyle m_{s_2} \hbar
\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)},$](img2046.png) |
(843) |
![$\displaystyle S_z \chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}$](img2047.png) |
![$\textstyle =$](img122.png) |
![$\displaystyle m_s \hbar
\chi_{s_1,s_2;m_{s_1},m_{s_2}}^{(1)}.$](img2048.png) |
(844) |
Likewise, if the spinor
is
a simultaneous eigenstate of
,
,
,
and
, then
Of course, since both particles have spin one-half,
, and
. Furthermore, by analogy with previous
analysis,
![\begin{displaymath}
m_s = m_{s_1}+ m_{s_2}.
\end{displaymath}](img2060.png) |
(849) |
Now, we saw, in the previous section, that when spin
is added
to spin one-half then the possible values of the total angular momentum
quantum number are
. By analogy, when spin one-half
is added to spin one-half then the possible values of the
total spin quantum number are
. In other words,
when two spin one-half particles are combined, we either obtain
a state with overall spin
, or a state with overall spin
. To be more exact, there are
three possible
states (corresponding to
, 0, 1), and
one possible
state (corresponding to
). The three
states
are generally known as the triplet states, whereas the
state is known as the singlet state.
Table 4:
Clebsch-Gordon coefficients for adding spin one-half to
spin one-half. Only non-zero coefficients are shown.
|
The Clebsch-Gordon coefficients for adding spin one-half to
spin one-half can easily be inferred from Table 2 (with
),
and are listed in Table 4. It follows from this table that the
three triplet states are:
where
is shorthand for
,
etc. Likewise, the singlet
state is written:
![\begin{displaymath}
\chi^{(2)}_{0,0} = \frac{1}{\sqrt{2}}\left(\chi^{(1)}_{-1/2,1/2}-\chi^{(1)}_{1/2,-1/2}\right).
\end{displaymath}](img2085.png) |
(853) |
Subsections
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Up: Addition of Angular Momentum
Previous: Angular Momentum in the
Richard Fitzpatrick
2010-07-20