Ground state wavefunction real?

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Discussion Overview

The discussion revolves around the nature of the ground state wavefunction in quantum mechanics, specifically whether it can always be chosen to be real. Participants explore the implications of Hamiltonians with real coefficients and the conditions under which the ground state wavefunction is real.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that the ground state wavefunction can be chosen to be real, particularly when the Hamiltonian has real coefficients and no first order spatial derivatives.
  • Others argue that the absence of first order derivatives is a simplification and that the more general condition is that the Hamiltonian must have only real coefficients.
  • One participant suggests that if a wavefunction is a solution to the time-independent Schrödinger equation (TISE), then its complex conjugate is also a solution, implying that any linear combination of these can yield a real wavefunction.
  • A later reply questions the relevance of time dependence in the context of the ground state, emphasizing that the discussion is specifically about the ground state as mentioned by the author.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which the ground state wavefunction can be real, with no consensus reached on whether the absence of first order derivatives is necessary or if the statement applies more generally.

Contextual Notes

Limitations include the dependence on the specific form of the Hamiltonian and the assumptions regarding the nature of the wavefunctions involved. The discussion does not resolve the implications of time dependence in relation to the ground state wavefunction.

krishna mohan
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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 don't 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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krishna mohan said:
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 I am 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?
 
krishna mohan said:
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.
 

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