Are these two functions equal?

How does one evaluate the inner product between two antisymmetrized gaussian functions?

• I have an n-dimensional simple Gaussian function.  f(x) = exp(-a* |x - k|^2)  where a is a scalar, x is a vector, k is a vector.  k is the center of the Gaussian. Assume I have two such Gaussian functions with different values of a and k.  Assume I apply the Antisymmetrizer operator (http://en.wikipedia.org/wiki/Antisymmetrizer) to each Gaussian and then try to compute the inner product between these two functions. 1> What is the expression for the inner product of 2 anti-symmetrized Gaussian functions. 2> How does the value of the inner product depend on the dimension of the vector space of the domain of the gaussian functions? How would you approach this problem?

• Answer:

  Let [math]|\Psi(x_1, x_2)\rangle[/math]  be [math]\frac{1}{\sqrt{2}}\left( |\psi_1(x_1)\rangle|\psi_2(x_2)\rangle - |\psi_2(x_1)\rangle|\psi_1(x_2)\rangle\right) [/math] and similarly for [math]|\Psi'(x_1,x_2)\rangle[/math]. The inner product is [math] \langle \Psi'(x_1,x_2)| \Psi(x_1,x_2)\rangle[/math] which you can probably guess is [math] \frac{1}{2}\left( \langle \psi'_1(x_1)| \psi_1(x_1)\rangle \langle \psi'_2(x_2)| \psi_2(x_2)\rangle + \cdots \right)[/math] where the three remaining terms from the FIFO expansion of the product in the dots.