  We introduce an alternative expression for the ground state wave function of
the fermionic Tonks-Girardeau gas. Our wave function is constructed based on
the occupation numbers and natural orbitals of the one-body density matrix. We
demonstrate that the newly found wave function describes the ground state of
the fermionic Tonks-Girardeau gas under any external potential. By expressing
the proposed wave function in the framework of second quantization, we show
that the ground state of the fermionic Tonks-Girardeau gas is a
number-conserving Bardeen-Cooper-Schrieffer (BCS) state. We provide explicit
expressions for the corresponding coefficients that describe the fermionic
Tonks-Girardeau gas as a number-conserving BCS state. Additionally, the
suitable form of the proposed wave function in second quantization allows us to
derive the necessary expectation values to experimentally detect pairing in the
fermionic Tonks-Girardeau gas. With this, we prove and show how to detect that
the fermionic Tonks-Girardeau gas not only exhibits non-trivial quantum
correlations but is also a paired state.
