Why can't iℏ∂/∂t be considered the Hamiltonian operator?

klepkowy7c 2022-07-19 Answered
Why can't i t be considered the Hamiltonian operator?
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Answers (1)

Marshall Mcpherson
Answered 2022-07-20 Author has 11 answers
If one a priori wrongly declares that the Hamiltonian operator H ^ is the time derivative i t , then the Schrödinger equation
(1) H ^ Ψ   =   i Ψ t
would become a tautology. Such trivial Schrödinger equation could not be used to determine the future (nor past) time evolution of the wavefunction Ψ ( r , t )
On the contrary, the Hamiltonian operator H ^ is typically a function of the operators r ^ and p ^ , and the Schrödinger equation
(2) H ^ Ψ   =   i Ψ t
One may then ask why is it then okay to assign the momentum operator as a gradient
(3) p ^ k   =   i r k   ?
(This is known as the Schrödinger representation.) The answer is because of the canonical commutation relations
(4) [ r ^ j , p ^ k ]   =   i   δ k j   1 ^ .
On the other hand, the corresponding commutation relation for time t is
(5) [ H ^ , t ]   =   0 ,
because time t is a parameter not an operator in quantum mechanics. Note that in contrast
(6) [ i t ,   t ]   =   i ,
which also shows that one should not identify H ^ and i t
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