Thought experiment
Let us have a thought experiment. At first, we fixate some configuration
for atom or molecule with N electrons, choose two electrons with
same spins and give them indices “1” and “2”. Let the wave function
for this configuration have some value Ψ0 and Ψ0 > 0 for
definiteness (the sign of Ψ0 does not matter). Then the wave function
must change sign and turn to –Ψ0 when we change places of electrons
indices “1” and “2”. We take with fingers or tweezers electrons
“1” and “2” and move them in some arbitrary way changing their
places. The other electrons and atom nuclei stay still. In this
situation electron “1” runs some trajectory and electron “2” runs
another trajectory in physical space. The two trajectories in
three-dimensional physical space can be described by one line in
six-dimensional configuration space named by us the interchange
trajectory. The spins can be ignored because they coincide.
It turns out that any choice of the interchange trajectory gives Ψ=0 at
some moment. (For example, we can move electron “1” far away in atomic
scale, electron “2” put on its place and then choose some intricate
way to move electron “1” to the former place of electron “2” to
avoid symmetrical situations). It follows from continuity of the wave
functions and the well-known mathematical analysis theorem that proves a
continuous function determined on some segment and having opposite signs
on its ends turn to zero at least at one point of the segment. We
consider the interchange trajectory as such segment. There can be more
than one zero points but their number will be odd necessarily. The nodal
surfaces of the wave function in configurational space consist of these
points.