Quantum Mechanics for Mathematicians
The space of all functions onMis far too big, allowing states localized in both position and coordinate variables in the caseM= ...
Chapter 18 Semi-direct Products The theory of a free particle is largely determined by its group of symmetries, the group of sym ...
If one then acts on the result with (a 1 ,R 1 ) one gets (a 1 ,R 1 )·((a 2 ,R 2 )·v) = (a 1 ,R 1 )·(a 2 +R 2 v) =a 1 +R 1 a 2 +R ...
Definition(Semi-direct product group).Given a groupK, a groupN, and an actionΦofKonNby automorphisms Φk:n∈N→Φk(n)∈N the semi-dir ...
wherek∈Sp(2d,R), then the automorphism Φkthat defines the Jacobi group is given by the one studied in section 16.2 Φk ((( cq cp ...
whereXis an antisymmetricdbydmatrix anda∈Rd. Exponentiating such matrices will give elements ofE(d). The Lie bracket is then giv ...
In terms of such pairs, the Lie bracket is given by [((( cq cp ) ,c ) ,L ) , ((( c′q c′p ) ,c ) ,L′ )] = (( L ( c′q c′p ) −L′ ( ...
Chapter 19 The Quantum Free Particle as a Representation of the Euclidean Group The quantum theory of a free particle is intimat ...
representation, where thePjare differentiation operators, this will be a second- order differential operator, and the eigenvalue ...
provides a unitary Lie algebra representation on the state spaceHof functions of the position variablesq 1 ,q 2. This will be gi ...
The position space wavefunctions can be recovered from the Fourier inversion formula ψ(q,t) = 1 2 π ∫ R^2 eip·qψ ̃(p,t)d^2 p Sin ...
depending on two variablesθ,p. To put this delta-function in a more useful form, recall the discussion leading to equation 11.9 ...
are multiplication operators and, taking the Fourier transform of 19.2 gives the differentiation operator Γ ̃′S(l) =− ( p 1 ∂ ∂p ...
Although we won’t prove it here, the representations constructed in this way provide essentially all the unitary irreducible rep ...
or, in components l 1 =q 2 p 3 −q 3 p 2 , l 2 =q 3 p 1 −q 1 p 3 , l 3 =q 1 p 2 −q 2 p 1 The Euclidean groupE(3) is a subgroup of ...
and so the constant characterizing an irreducible will be the energy 2mE. Our ir- reducible representation will be on the space ...
The groupSO(3) acts on momentum vectors by rotation, with orbit of the group action the sphere of momentum vectors of fixed ene ...
whereLjacts onHEandSj=ρ′(lj) acts onC^2 s+1. This tensor product representation will not be irreducible, but its irreducible com ...
Chapter 20 Representations of Semi-direct Products In this chapter we will examine some aspects of representations of semi-direc ...
representations ofN, together with the irreducible representations of certain subgroups ofK. The reader should be warned that mu ...
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