D.22 Positive molecular ion wave function

Since the assumed Hamiltonian is real, taking real and imaginary parts of the eigenvalue problem $H\psi=E\psi$ shows that the real and imaginary parts of $\psi$ each separately are eigenfunctions with the same eigenvalue, and both are real. So you can take $\psi$ to be real without losing anything.

The expectation value of the energy for $\vert\psi\vert$ is the same as that for $\psi$, assuming that an integration by parts has been done on the kinetic energy part to convert it into an integral of the square gradients of $\psi$. Therefore $\vert\psi\vert$ must be the same function as $\psi$ within a constant, assuming that the ground state of lowest energy is nondegenerate. That means that $\psi$ cannot change sign and can be taken to be positive.

(Regrettably this argument stops working for more than two electrons due to the antisymmetrization requirement of chapter 5.6. It does keep working for bosons, like helium atoms in a box, [17, p. 321])

With a bit more sophistication, the above argument can be be inverted to show that the ground state must indeed be unique. Assume that there would be two different (more precisely: not equal within a constant) ground state eigenfunctions, instead of just one. Then linear combinations of the two would exist that crossed zero. The absolute value of such a wave function would, again, have the same expectation energy as the wave function itself, the ground state energy. But the absolute value of such a wave function has kinks of finite slope at the zero crossings. (Just think of the graph of $\vert x\vert$.) If these kinks are locally slightly smoothed out, i.e. rounded off, the kinetic energy would decrease correspondingly, since kinetic energy is the integral of the square slope and the slope has been reduced nontrivially in the immediate vicinity of the zero crossings. However, there would not be a corresponding increase in potential energy, since the potential energy depends on the square of the wave function itself, not its slope, and the square of the wave function itself is vanishingly small in the immediate vicinity of a zero crossing. If the kinetic energy goes down, and the potential energy does not go up enough to compensate, the energy would be lowered. But that contradicts the fact that the ground state has the lowest possible energy. The contradiction implies that the original assumption of two different ground state eigenfunctions cannot be right; the ground state must be unique.