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## Homework Statement

Using [x,e

^{iap}]=-ħae

^{iap}show that x

^{n}e

^{iap}= e

^{iap}(x-ħa)

^{n}

## Homework Equations

[x,e

^{iap}]=-ħae

^{iap}

From which it follows that,

xe

^{iap}= e

^{iap}(x-ħa)

## The Attempt at a Solution

[x

^{n},e

^{iap}] = [xx

^{n-1},e

^{iap}]

= [x,e

^{iap}]x

^{n-1}+ x[x

^{n-1},e

^{iap}]

= -ħae

^{iap}x

^{n-1}+ x(x

^{n-1}e

^{iap}-e

^{iap}x

^{n-1})

= -ħae

^{iap}x

^{n-1}+ x

^{n}e

^{iap}- e

^{iap}(x-ħa)x

^{n-1}

Expanding the original commutator on the LHS and moving the second term to the RHS gives,

x

^{n}e

^{iap}= -ħae

^{iap}x

^{n-1}+ x

^{n}e

^{iap}- e

^{iap}(x-ħa)x

^{n-1}+ e

^{iap}x

^{n}

= -ħae

^{iap}x

^{n-1}+ x

^{n}e

^{iap}+ ħae

^{iap}x

^{n-1}

x

^{n}e

^{iap}= x

^{n}e

^{iap}Grrrrrrrrrrrrrrrr