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## Main Question or Discussion Point

Can anybody explain what is meant by positive definite Hamiltonian? All I know is that if a Hamiltonian can be factorized as

[tex]

H={Q}^{\dagger}Q

[/tex]

then that Hamiltonian is one such example. But I am not sure if that is the definition of a positive definite Hamiltonian. In the quantum mechanical context what is meant by a positive definite operator, that also is not clear to me.

After Google search I found there is something called positive definite operator in mathematics, defined as: the operator A is a positive definite operator if for all non-null vector x,

[tex]

xAx>0

[/tex]

But clearly the definition/concept used in quantum mechanics is a bit different.

[tex]

H={Q}^{\dagger}Q

[/tex]

then that Hamiltonian is one such example. But I am not sure if that is the definition of a positive definite Hamiltonian. In the quantum mechanical context what is meant by a positive definite operator, that also is not clear to me.

After Google search I found there is something called positive definite operator in mathematics, defined as: the operator A is a positive definite operator if for all non-null vector x,

[tex]

xAx>0

[/tex]

But clearly the definition/concept used in quantum mechanics is a bit different.