Hamiltonians, do they need to be positive definite?

[mtak0114]

Hi
(I'm not sure where this question belongs)
I had a general question about hamiltonians, do they need to be positive definite?
is this required in QM, or is this a relativistic requirement?

cheers

M

 

[Timo]

Hamiltonians do not need to be positive definite. The eigenvalues of the Hamiltonian are the energies. In standard notation, the energy of the ground state of the Hydrogen atom is about -13.6 eV, i.e. negative => one example of a negative eigenvalue => my statement.

I'm not completely sure that I understood your question, hence the explanation.

 

[the_house]

Timo is correct. However, you typically demand that a Hamiltonian is bounded from below. Otherwise there is no ground state.

Once you have a such a Hamiltonian, you could just trivially add a constant to the Hamiltonian (which doesn't change the physics) such that the lowest energy state (the ground state) has zero energy. It's often the convention to label a state that is not the ground state as zero energy (such as for the mentioned hydrogen atom). However, these Hamiltonians can always be chosen to be positive definite by a constant energy shift. This is why you may have seen some sort of general argument for something that assumes a positive definite Hamiltonian (assuming this is why you asked the question).

 

[element4]

This might be a little irrelevant, but I want to mention that if one requires supersymmetry then it is true that the energy will always be positive.

 

[mtak0114]

thanks for the replies

this agrees with what I was thinking
but from a relativistic perspective this
wouldn't the hamiltonian need to be positive definite
as the 4-momentum operator must have time like eigenvalues?
i.e. timelike four-momenta

as another question
what is the reason for a hamiltonian to be bounded from below?
I understand that this is required if one would like to avoid having an infinite energy source.
But is there a precise mathematical reason or is this simply a constraint that we as physicists impose?

thanks again

Mark

 

[element4]

as another question
what is the reason for a hamiltonian to be bounded from below?
I understand that this is required if one would like to avoid having an infinite energy source.
But is there a precise mathematical reason or is this simply a constraint that we as physicists impose?

thanks again

Mark

I don't think that there are any mathematical constraint that restricts the Hamiltonian to be bounded from below, this is a physical requirement. Take for example a potential of the form $$V(x) = x^3$$, then the Hamiltonian will not be bounded from below (and there is no mathematical reason for this not to be allowed)! I think that when this is solved, the particle will go to infinity in finite time, and therefore not so physical.