QuantumPhysics.dvi

5.5 General position and momentum operators and eigenstates

Given self-adjoint position and momentum operators, denoted respectivelyXandP, which

satisfy the canonical commutation relation [X,P] = i ̄hI, there are a number of general

results that may be deduced without appeal to a specific Hamiltonian. These results will be

derived here in all generality.

• Thetranslation operatorT(a) is defined for any reala∈Rby,

T(a) = exp

(

−i

a

̄h

P

)

(5.53)

Self-adjointness ofPand reality ofaand ̄himply thatT(a) is unitary, and satisfies

T(a)†T(a) =T(a)T(a)†=I (5.54)

The properties of the exponential imply the followingU(1) group composition property,

T(a)T(b) =T(a+b) (5.55)

Using the Baker-Campbell-Haussdorff formula,eABe−A=eAdABwhere theadjoint mapis

defined byAdAB= [A,B], the definition ofT(a), and the canonical commutation relations,

it follows,

T(a)†XT(a) =X+aI (5.56)

More generally, we haveT(a)†f(X)T(a) =f(X+aI) for any analytic functionf. Takinga

infinitesimal,T(a)∼I−iaP/ ̄h, or equivalentlyP, generates infinitesimal translations inX.

• SinceXandP are self-adjoint,each operator separatelymay be diagonalized, with

real eigenvalues. The corresponding eigenvalues will be denotedxand ̄hkrespectively, and

the eigenstates will be denoted as follows,

X|x;X〉 = x|x;X〉

P|k;P〉 = ̄hk|k;P〉 (5.57)

Self-adjointness also implies that orthogonality of eigenstates associated with distinct eigen-

values, namely〈x′;X|x;X〉= 0 forx′ 6 =x, and〈k′;Pk;P〉= 0 fork′ 6 =k. The normalization

of the states may be fixed, in part, by choosing the following relations,

〈x′;X|x;X〉 = δ(x′−x)

〈k′;P|k;P〉 = 2πδ(k′−k) (5.58)