Quantum Mechanics for Mathematicians

Definition(Momentum operators). For a quantum system with state space
H=L^2 (R^3 )given by complex-valued functions of position variablesq 1 ,q 2 ,q 3 ,
momentum operatorsP 1 ,P 2 ,P 3 are defined by

P 1 =−i~

∂

∂q 1

,P 2 =−i~

∂

∂q 2

,P 3 =−i~

∂

∂q 3

These are given the name “momentum operators” since we will see that their
eigenvalues have an interpretation as the components of the momentum vector
for the system, just as the eigenvalues of the Hamiltonian have an interpretation
as the energy. Note that while in the case of the Hamiltonian the factor of~kept
track of the relative normalization of energy and time units, here it plays the
same role for momentum and length units. It can be set to one if appropriate
choices of units of momentum and length are made.
The differentiation operator is skew-adjoint since, using integration by parts^3
one has for each variable, forψ 1 ,ψ 2 ∈H

∫+∞

−∞

ψ 1

(

d

dq

ψ 2

)

dq=

∫+∞

−∞

(

d

dq

(ψ 1 ψ 2 )−

(

d

dq

ψ 1

)

ψ 2

)

dq

=−

∫+∞

−∞

(

d

dq

ψ 1

)

ψ 2 dq

(assuming that theψj(q) go to 0 at±∞). ThePjare thus self-adjoint operators,
with real eigenvalues as expected for an observable operator. Multiplying by
−ito get the corresponding skew-adjoint operator of a unitary Lie algebra
representation we find

−iPj=−~

∂

∂qj

Up to the~factor that depends on units, these are exactly the Lie algebra
representation operators on basis elements of the Lie algebra, for the action of
R^3 on functions onR^3 induced from translation:

π(a 1 ,a 2 ,a 3 )f(q 1 ,q 2 ,q 3 ) =f(q 1 −a 1 ,q 2 −a 2 ,q 3 −a 3 )

π′(a 1 ,a 2 ,a 3 ) =a 1 (−iP 1 )+a 2 (−iP 2 )+a 3 (−iP 3 ) =−~

(

a 1

∂

∂q 1

+a 2

∂

∂q 2

+a 3

∂

∂q 3

)

Note that the convention for the sign choice here is the opposite from the
case of the Hamiltonian (−iP=−~dqd vs. −iH=~dtd). This means that the
conventional sign choice we have been using for the Hamiltonian makes it minus
the generator of translations in the time direction. The reason for this comes
from considerations of special relativity (which will be discussed in chapter 40),
where the inner product on space-time has opposite signs for the space and time
dimensions.

(^3) We are here neglecting questions of whether these integrals are well-defined, which require
more care in specifying the class of functions involved.