  In this paper, we study the implications of conformal invariance in momentum
space for correlation functions in quantum mechanics. We find that three point
functions of arbitrary operators can be written in terms of the $_2 F_1$
hypergeometric function. We then show that generic four-point functions can be
expressed in terms of Appell's generalized hypergeometric function $F_2$ with
one undetermined parameter that plays the role of the conformal cross ratio in
momentum space. We also construct momentum space conformal partial waves, which
we compare with the Appell $F_2$ representation. We test our expressions
against free theory and DFF model correlators, finding an exact agreement. We
then analyze five, six, and all higher point functions. We find, quite
remarkably, that $n$-point functions can be expressed in terms of the
Lauricella generalized hypergeometric function, $E_A$, with $n-3$ undetermined
parameters, which is in one-to-one correspondence with the number of conformal
cross ratios. This analysis provides the first instance of a closed form for
generic momentum space conformal correlators in contrast to the situation in
higher dimensions. Further, we show that the existence of multiple solutions to
the momentum space can be attributed to the Fourier transforms of the various
possible time orderings. Finally, we extend our analysis to theories with
$\mathcal{N}=1,2$ supersymmetry, where we find that the constraints due to the
superconformal ward identities are identical to identities involving
hypergeometric functions.
