QMGreensite_merged

326 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

Aningeneral,anystatewhoseeigenfunctionisψ(x)inthex-representationwillhave
aneigenfunctionψ(p)inthep-representationgivenby

ψ(p) = <p|ψ>

= <p|Iψ>

= <p|

{∫

dx|x><x|

}

|ψ>

=

∫

dx<p|x><x|ψ> (21.141)

or,usingtheresultabovefor<p|x>,

ψ(p)=

1

√

2 πh ̄

∫

dxψ(x)e−ipx/ ̄h (21.142)

Inotherwords,themomentum-space(P-representation)wavefunctionistheFourier
transformoftheposition-space(X-representation)wavefunction.
Next,wecomputematrixelementsofoperators.Forthemomentumoperator

<p 1 |P|p 2 >=p 2 <p 1 |p 2 >=p 1 δ(p 1 −p 2 ) (21.143)

i.e. themomentumoperatorisdiagonal(= 0 forp 1 +=p 2 )inthemomentumrepre-
sentation.Writing
<p|P|p′>=p ̃δ(p−p′) (21.144)

weseethat
p ̃=p (21.145)

Forthepositionoperator,weagainusetheidentityoperation

<p 1 |X|p 2 > = <p 1 |IXI|p 2 >

= <p 1 |

{∫

dx|x><x|

}

X

{∫

dy|y><y|

}

|p 2 >

=

∫

dxdy <p|x><x|X|y><y|p 2 >

=

1

2 π ̄h

∫

dxdyxδ(x−y)e(ip^2 y−ip^1 x)/ ̄h

=

1

2 π ̄h

∫

dxxei(p^2 −p^1 )/ ̄h

= i ̄h

∂

∂p 1

1

2 πh ̄

∫

dxei(p^2 −p^1 )/ ̄h

= i ̄h

∂

∂p 1

δ(p 1 −p 2 ) (21.146)

or
<p|X|p′>=x ̃δ(p−p′) (21.147)