- #1

kemiisto

- 2

- 0

## Homework Statement

Find the operator for position [tex]x[/tex] if the operator for momentum p is taken to be [tex]\left(\hbar/2m\right)^{1/2}\left(A + B\right)[/tex], with [tex]\left[A,B\right] = 1[/tex] and all other commutators zero.

## Homework Equations

Canonical commutation relation

[tex]\left [ \hat{ x }, \hat{ p } \right ] = \hat{x} \hat{p} - \hat{p} \hat{x} = i \hbar[/tex]

## The Attempt at a Solution

Using [tex]c = \left(\hbar/2m\right)^{1/2}[/tex]

[tex]\hat{x} \hat{p} f - \hat{p} \hat{x} f = i \hbar[/tex]

[tex]\hat{x} c \left(\hat{A} + \hat{B}\right) f - c \left(\hat{A} + \hat{B}\right) \hat{x} f = i \hbar[/tex]

[tex]c \hat{x} \left(\hat{A} + \hat{B}\right) f - c \left(\hat{A} + \hat{B}\right) \hat{x} f = i \hbar[/tex]

[tex]\hat{x} \hat{A} f + \hat{x} \hat{B} f - \hat{A} \hat{x} f - \hat{B} \hat{x} f = i \hbar / c[/tex]

[tex]\hat{x} \hat{A} f - c \hat{A} \hat{x} f + \hat{x} \hat{B} f - \hat{B} \hat{x} f = i \hbar / c[/tex]

[tex]\left [ \hat{x}, \hat{A} \right ] + \left [ \hat{x}, \hat{B} \right ] = i \hbar / c[/tex]

"all other commutators zero"

[tex]0 + 0 = i \hbar / c[/tex]

Problem 1.2 from http://www.oup.com/uk/orc/bin/9780199274987/" .

Last edited by a moderator: