Chapter 8, Problem 45E

Exercise 44 gives an antisymmetricmultiparticle state for two particles in a box with opposite spins. Another antisymmetric state with spins opposite and the same quantum numbers is
${\psi }_n\left(x_1\right)\downarrow {\psi }_{n^{\prime }}\left(x_2\right)\uparrow -{\psi }_{n^{\prime }}\left(x_1\right)\uparrow {\psi }_n\left(x_2\right)\downarrow$

Refer to these states as I and II. 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 swap ping the particles’ states, specifically the n and n’ in equation (8−22). In this exercise, we look at swapping only parts of the state—spatial or spin. (a) What is the exchange symmetry—symmetric (unchanged), antisymmetric (switching sign), or neither—of multiparticle states I and 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 I and II may be written
$\left({\psi }_n\left(x_1\right){\psi }_{n^{\prime }}\left(x_2\right)-{\psi }_{n^{\prime }}\left(x_1\right){\psi }_n\left(x_2\right)\right)\left(\downarrow \uparrow +\uparrow \downarrow \right)$

where the left arrow in any couple represents the spin of particle I 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 II still antisymmetric if we swap the particles’ total—spatial plus spin—states? (f) if the two particles repel each other, would any of the three muitiparticle states−I, II, and the sum−be preferred?
Explain.