- #1

- 726

- 9

## Main Question or Discussion Point

It is obvious to me how

[tex]

\hat {x} = x; \hspace{5 mm}

\hat {p}_x = -i \hbar \frac {\partial} {\partial x}

[/tex]

implies

[tex]

[ \hat {x} , \hat {p}_x ] = i \hbar

[/tex]

and I can accept that these two formulations are mathematically equivalent, but I do not know how in general (or even in this specific case) to go in the opposite direction, that is, to start with operators defined solely in terms of their commutation relations and change them into forms which show what they do to wave functions in the position basis.

Is this always possible, and if so, how?

[tex]

\hat {x} = x; \hspace{5 mm}

\hat {p}_x = -i \hbar \frac {\partial} {\partial x}

[/tex]

implies

[tex]

[ \hat {x} , \hat {p}_x ] = i \hbar

[/tex]

and I can accept that these two formulations are mathematically equivalent, but I do not know how in general (or even in this specific case) to go in the opposite direction, that is, to start with operators defined solely in terms of their commutation relations and change them into forms which show what they do to wave functions in the position basis.

Is this always possible, and if so, how?