Exercise 44 gives an antisymmetric multiparticle state for two particles in a
box with opposite spins. Another antisymmetric state with spins opposite and
the same quantum numbers is
$$
\psi_{n}\left(x_{1}\right) \downarrow \psi_{n}\left(x_{2}\right)
\uparrow-\psi_{n}\left(x_{1}\right) \uparrow \psi_{n}\left(x_{2}\right)
\downarrow
$$
Refer to these states as \(\mid\) and 11 . We have tended to characterize
exchange symmetry as to whether the state's sign changes when we swap particle
labels. but we could achieve the same result by instead swapping the
particles' stares, specifically the \(n\) and \(n^{\prime}\) in equation \((8-22)\).
In this exercise. we look at swapping only parts of the state-spatial or spin.
(a) What is the exchange symmeiry - symmetric (unchanged). antisymmetric
(switching sign), or neither-of multiparticle states 1 and \(\mathrm{II}\) with
respect to swapping spatial states alone? (b) Answer the same question. but
with respect to swapping spin states/arrows alone.
(c) Show that the algebraic sum of states 1 and \(\mathrm{II}\) may be written
\(\left(\psi_{n}\left(x_{1}\right)
\psi_{n}\left(x_{2}\right)-\psi_{n}\left(x_{1}\right)
\psi_{n}\left(x_{2}\right)\right)(\downarrow T+\uparrow \downarrow)\)
where the left arrow in any couple represents the spin of particle 1 and the
right arrow that of particle 2
(d) Answer the same questions as in parts \((a)\) and
(b). but for this algebraic sum. (e) Is the sum of states I and 11 still antis
ymmetric if we swap the particles? total-spatial plus spin -states? (f) If the
two particles repel each other, would any of the three multiparticle states -
l. II, and the sum - be preferred? Explain.
