Description of Two particle system of electrons

Consider a two particle system consisting of two electrons. The complete state of the electron includes its position wave function and also a spinor describing the orientation of its spin: $$\psi(r) \otimes \chi(s).$$ Why does it follows that for the the two particle system that if we have an anti-symmetric spin state of the two electrons such as the singlet state $$\frac{1}{\sqrt{2}}(| \uparrow \rangle \otimes | \downarrow \rangle - | \downarrow \rangle \otimes | \uparrow \rangle)$$ then this has to be joined with a symmetric spatial function (and similarly if we have a symmetric state of two electrons such as $| \uparrow \rangle \otimes |\uparrow \rangle$ then this has to be joined by an anti-symmetric spatial wave function?

Also if two electrons occupy the singlet spin state then the spatial wave function describing the two particle state would be symmetric, but I thought that for identical particles which are fermions (such as electrons), the spatial wave function is always antisymmetric?

Thanks for any assistance.

• An odd state multipled by an even state gives you an odd state. It's been a while but I think it is analogous to that with symmetery, to preserve anti symmetry. Anyway, I am sure someone will sort us both out. – user140606 Jan 16 '17 at 18:48

• Thanks for your answer. Do you mean that say the spatial wave function $\psi(x_{1}, x_{2})$ of a two electron system is symmetric under the exchange operator then we require that the spin would be in the singlet state $| 0 0 \rangle =\frac{1}{\sqrt{2}}(| \uparrow \rangle \otimes | \downarrow \rangle - | \downarrow \rangle \otimes | \uparrow \rangle)$ hence we would have a total wave function $$\Psi = |\psi \rangle \otimes |0 0 \rangle?$$ – Alex Jan 16 '17 at 20:30
• Okay thanks. One other query... Would I be right in stating the following: Given that the state of the two particle system is $$\Psi = | \psi \rangle \otimes |00 \rangle$$ could I then project the tensor product onto the position basis by using the linear operator $\langle x_1, x_2 | \otimes I$, thus resulting in the tensor product $$\psi(x_1, x_2) \otimes |00\rangle$$, since the left hand side of the tensor product is a complex number and the right side is a matrix, we can consider the state $$\psi(x_1,x_2)|00 \rangle$$ as simply matrix times a complex scalar? – Alex Jan 17 '17 at 10:32