Quantum Mechanics for Mathematicians
Definition(Intertwining operator).If(π 1 ,V 1 ),(π 2 ,V 2 )are two representations of a groupG, an intertwining operator between ...
20.2 Constructing intertwining operators The method we will use to construct the intertwining operatorsUkis to find a solution t ...
20.3 Explicit calculations As a balance to the abstract discussion so far in this chapter, in this section we’ll work out explic ...
the quadratic functionμLthat satisfies μL, q 1 q 2 p 1 p 2 =LT q 1 q 2 p 1 p 2 ...
The case of the groupSO(2)⊂Sp(4,R) can be generalized to a larger subgroup, the groupGL(2,R) of all invertible linear transforma ...
(Note that for such phase space rotations, we are making the opposite choice for convention of the positive direction of rotatio ...
20.3.3 The Fourier transform as an intertwining operator For another indication of the non-trivial nature of the intertwining op ...
Note that in the Schr ̈odinger representation −i 1 2 (QP+PQ) =−i(QP− i 2 ) 1 =−q d dq − 1 2 1 The operator will have as eigenfun ...
For a semi-direct productNoK, we will have an automorphism ΦkofN for eachk∈K. From this action onN, we get an induced action on ...
Definition(Stabilizer group or little group).The subgroupKα⊂Kof elements k∈Ksuch that ̂Φk(α) =α for a givenα∈N̂is called the sta ...
For another treatment of these operators along the lines of this chapter, see section 14 of [37]. For a concise but highly insig ...
Chapter 21 Central Potentials and the Hydrogen Atom When the Hamiltonian function is invariant under rotations, we then expect e ...
erator for a particle moving in a potentialV(q 1 ,q 2 ,q 3 ) will be H= 1 2 m (P 12 +P 22 +P 33 ) +V(Q 1 ,Q 2 ,Q 3 ) =− ~^2 2 m ...
will span a 2l+ 1 dimensional (sincem=−l,−l+ 1,...,l− 1 ,l) space of energy eigenfunctions forHof eigenvalueE. For a general pot ...
scattering states (E >0) bound states (E <0) n= 1 n= 2 n= 3 n= 4 l= 0 l= 1 l= 2 l= 3 1 s 2 s 3 s 4 s 2 p 3 p 4 p 3 d 4 d 4 ...
gnl(r) =glEn(r) the solutions are of the form gnl(r)∝e− r na 0 ( 2 r na 0 )l L^2 nl++1l ( 2 r na 0 ) where theL^2 nl++1l are cer ...
Kepler’s second law for such motion comes from conservation of angular momentum, which corresponds to the Poisson bracket relati ...
• {wj,wk}=jklll ( − 2 h m ) This is the most surprising relation, and it has no simple geometrical explanation (although one ca ...
If we now restrict attention to the subspaceHE⊂ Hof energy eigenstates of energyE, on this space we can define rescaled operator ...
The relation between the Hamiltonian and the Casimir operatorsM^2 and N^2 is 2 H(K^2 +L^2 +~^21 ) = 2H(2M^2 + 2N^2 +~^21 ) = 2H( ...
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