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How do you properly use Bra-Ket notation for products of functions like they appear in Slater determinant type wavefunctions. For example

$\Psi(x_1,x_2)=\frac{1}{\sqrt2}(\phi_1(x_1)\phi_2(x_2) - \phi_2(x_1)\phi_1(x_2)) $

How can this be written in Bra-Ket notation ?

Is $$|\Psi \rangle= \frac{1}{\sqrt2}|\phi_1\rangle |\phi_2\rangle - \frac{1}{\sqrt2}|\phi_2\rangle |\phi_1\rangle$$ correct ?

And is the following the same $|\Psi \rangle= \frac{1}{\sqrt2}|\phi_1 \phi_2\rangle - \frac{1}{\sqrt2}|\phi_2\phi_1\rangle = \frac{1}{\sqrt2}|\phi_1\rangle \otimes |\phi_2\rangle - \frac{1}{\sqrt2}|\phi_2\rangle\otimes |\phi_1\rangle$ ? Are all three variants equivalent or are there some differences ?

I also assumed i should be able to write something like $$|\Psi\rangle =\sum _i c_i|\varphi_i\rangle$$

but i can't see how that would work.

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    $\begingroup$ Yes, the tensor product is implicitly omitted. Then to derive the wave function you take the scalar product against $\langle x|_1 \otimes \langle x|_2$ where the indices (1,2) represent the corresponding Hilbert spaces which in turn form the general space by tensor product. $\endgroup$ – gented May 28 at 14:39

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