# Ground state wavefunction real?

Hi..

In a textbook, the ground-state wavefunction for any general Hamiltonian was under consideration. Then, a statement was made that this wave function is real since it is the ground state.

Is it true that one can always choose the ground state wave function to be real?

I understand that absolute phases dont matter in quantum physics, but relative phases do. Is it that one can choose the ground state wavefunction to be real and, as a consequence, lose the freedom of choosing the phases for the other wavefunctions?

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A. Neumaier
2019 Award
Hi..

In a textbook, the ground-state wavefunction for any general Hamiltonian was under consideration. Then, a statement was made that this wave function is real since it is the ground state.

Is it true that one can always choose the ground state wave function to be real?
This is always the case if the Hamiltonian has real coefficients and no first order spatial derivatives. In this case, for any eigenfunction psi of H to the eigenvalue E, Re psi and
Im psi have the same property and are real. In particular, if the ground state is nondegenerate, Re psi and Im psi must be proportional, which means that the ground state is a constant multiple of a real function.

I understand the rest of it...

But why should there be no first order derivatives?

A. Neumaier
2019 Award
why should there be no first order derivatives?
For simplicity, and since this holds for many concrete Hamiltonians.

The more general statement is that everything holds when the Hamiltonian, expressed in terms of the quantum position and momentum operators, has only real coefficients.

I take it you mean the solution to the time independent schordinger equation(TISE). If that is the case, then not just the ground state, any solution can be taken to be real. If psi is a solution of the TISE, then so is its complex conjugate psi* (do it and check it, this is so because the potential is taken to bereal). Since the differential equation is linear, you can now take any linear combination of psi and psi* as it will also be a solution to TISE.

So you make your life easier by choosing psi+psi* and choosing a real wavefunction.