Symmetrization Principle for Identical Particles with Different Spin

[center o bass]

Hey I'm tring to grasp the symmetrization principle for identical particles. If the particles are indistinguishble, for example two electrons (not regarding spin), then $$\psi(x_1,x_2)$$ must correspond to the same physical situation as $$\psi(x_2,x_1)$$. It follows that

$$|\psi(x_1,x_2)|^2 = |\psi(x_2,x_1)|^2$$

and from this the symmetrization principle follows as

$$\psi(x_1,x_2) = \pm \psi(x_2,x_1)$$.

This is all okay... However, if we had two electrons 1 and 2 with differents spins along the z-axis m1 and m2. Then how could the physical situation of 1 being at x1 with z-spin m1 and 2 being at x2 with spin m2 be the same as the situation of 2 being at x1 with spin m2 while 1 is at x2 with spin m1?

That is how could the physical situation of

$$\psi_{m_1,m_2}(x_1,x_2)$$
be the same as the situation

$$\psi_{m_2,m_1}(x_2,x_1)?$$

Could'nt we just mesure the z-spin of the particle at x1 in the two situations and determine the difference?

 

[Bill_K]

It's more like this... ψ(x₁,m₁; x₂,m₂) = -ψ(x₂,m₂; x₁,m₁). In both cases the electron at x₁ has spin m₁, and the electron at x₂ has spin m₂. What is being interchanged is the identity of the particles.