In previous post I mentioned that the First Commutation Relation and Second Commutation Relation are equivalent. Here is a proof from Classical Perspective.

We assume one of them to be true. For example, we assume First Commutation Relation to be true. i.e.

[\hat{x}, \hat{p}] = i\hbar\rightarrow \hat{p} = -i\hbar\frac{\partial{}}{\partial{x}}

Classical Hamiltonian is given by

\hat{H} = \frac{\hat{p}^2}{2m} + \hat{V} = -\frac{\hbar^2}{2m}\frac{\partial{}^2}{\partial{x^2}} + V

We next compute the commutation between Hamiltonian and Position

[\hat{H},\hat{x}] = [\frac{\hat{p}^2}{2m} + \hat{V},\hat{x}]=\frac{1}{2m}[\hat{p}^2, x] =i\hbar\frac{\hat{p}}{m}

We see the operator in the last term is precisely the velocity operator \hat{v}

\hat{v}\psi = v\psi

Therefore from definition of velocity

\hat{v}\psi = \frac{\partial{x}}{\partial{t}}\psi = [\frac{\partial{}}{\partial{t}}, \hat{x}]\psi

So we conclude that

[\hat{H},\hat{x}]\psi = i\hbar[\frac{\partial{}}{\partial{t}}, \hat{x}]\psi

\hat{H} = i\hbar\frac{\partial{}}{\partial{t}}

It then follows

[\hat{H},\hat{t}] = [i\hbar\frac{\partial{}}{\partial{t}}, \hat{t}] = i\hbar

Equivalence proved.

(Note it’s only proved for non-relativistic hamiltonian, i still haven’t worked out the most general case at the moment)

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3 thoughts on “Two Commutation Relations

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