I have two electrons which are in a (symmetric) triplet state where one of them has spin pointing up ($m_s=1/2$) and the other one has spin pointing down ($m_s=-1/2$), so we have
$|1,0\rangle =\;(\uparrow\downarrow + \downarrow\uparrow)/\sqrt2$
According to Pauli's exclusion principle, the two electrons must not have the same set of quantum numbers. However, as the two spin directions differ, is it possible that the corresponding spatial wave functions for the electrons are both in ground state ($n=1,l=0,m_l=0$)? Or is it a contradiction to the fact that it is a triplet state?
Best Answer
The spin wave function is symmetric with respect to the exchange of particles. Therefore the spacial wave function has to be antisymmetric. I.e. at least one of the quantum numbers has to be different.
The wave function may look as if the electrons have opposite spin, but actually the spins are the same if measured at an axis 90° from z.
EDIT:
The spin eigenvectors of different axes are not independent from each other. $$ \begin{align} \left|\strut\uparrow\right\rangle &= \frac{1}{\sqrt 2}\left(\left|\strut x_+\right\rangle + \left|\strut x_-\right\rangle \right)\\ \left|\strut\downarrow\right\rangle &= \frac{1}{\sqrt 2}\left(\left|\strut x_+\right\rangle - \left|\strut x_-\right\rangle \right) \end{align} $$ Substitute that into your definition of $\left|\strut\psi\right\rangle$ and you will get $$ \begin{align} \left|\strut\psi\right\rangle &= \frac{1}{2\sqrt 2}\left[\left(\left|\strut x_+\right\rangle + \left|\strut x_-\right\rangle \right)\otimes\left(\left|\strut x_+\right\rangle - \left|\strut x_-\right\rangle \right) + \left(\left|\strut x_+\right\rangle - \left|\strut x_-\right\rangle \right)\otimes\left(\left|\strut x_+\right\rangle + \left|\strut x_-\right\rangle \right)\right]\\ &=\frac{1}{2\sqrt 2}\left[% \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_+\right\rangle\left|\strut x_-\right\rangle% + \left|\strut x_-\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% + \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% + \left|\strut x_+\right\rangle\left|\strut x_-\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% \right]\\ &=\frac{1}{2\sqrt 2}\left[% \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% + \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% \right]\\ &=\frac{1}{\sqrt 2}\left[% \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% \right]\\ \end{align} $$
To check for the symmetry you don't need this calculation. It is sufficient to check that $$ \frac{1}{\sqrt2}\left( \left|\strut\uparrow\right\rangle\left|\strut\downarrow\right\rangle + \left|\strut\downarrow\right\rangle\left|\strut\uparrow\right\rangle \right) = \frac{1}{\sqrt2}\left( \left|\strut\downarrow\right\rangle\left|\strut\uparrow\right\rangle + \left|\strut\uparrow\right\rangle\left|\strut\downarrow\right\rangle \right) $$
The Pauli principle says that wave functions have to be negated when swapping any two Fermions.