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AngularMomentum - Angular Momentum Operator Algebra Michael...

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Angular Momentum Operator Algebra Michael Fowler 10/29/07 Preliminaries: Translation and Rotation Operators As a warm up to analyzing how a wave function transforms under rotation, we review the effect of linear translation on a single particle wave function ( ) x ψ . We have already seen an example of this: the coherent states of a simple harmonic oscillator discussed earlier were (at t = 0) identical to the ground state except that they were centered at some point displaced from the origin. In fact, the operator creating such a state from the ground state is a translation operator. The translation operator T ( a ) is defined at that operator which when it acts on a wave function ket () x gives the ket corresponding to that wave function moved over by a , that is, ( ) ( ) ( ) , Ta x x a ψψ =− so, for example, if x is a wave function centered at the origin, T ( a ) moves it to be centered at the point a . We have written the wave function as a ket here to emphasize the parallels between this operation and some later ones, but it is simpler at this point to just work with the wave function as a function, so we will drop the ket bracket for now. The form of T ( a ) as an operator on a function is made evident by rewriting the Taylor series in operator form: ( ) () () 22 2 2! . d a dx da d xa x x dx dx ex x −= + = = Now for the quantum connection: the differential operator appearing in the exponential is in quantum mechanics proportional to the momentum operator ( ˆ / pi d d x = − = ) so the translation operator ( ) ˆ / . iap Ta e = = An important special case is that of an infinitesimal translation, ( ) ˆ / ˆ 1/ ip Te i p ε εε == = = . The momentum operator is said to be the generator of the translation. ˆ p
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2 ( A note on possibly confusing notation : Shankar writes (page 281) ( ) . Tx x ε =+ Here x denotes a delta-function type wave function centered at x . It might be better if he had written () 00 Txx , then we would see right away that this translates into the wave function transformation () ( ) ( ) xx x δδ −= , the sign of now obviously consistent with our usage above.) It is important to be clear about whether the system is being translated by a , as we have done above or whether, alternately, the coordinate axes are being translated by a, that latter would result in the opposite change in the wave function. Translating the coordinate axes, along with the apparatus and any external fields by a relative to the wave function would of course give the same physics as translating the wave function by + a . In fact, these two equivalent operations are analogous to the time development of a wave function being described either by a Schrödinger picture, in which the bras and kets change in time, but not the operators, and the Heisenberg picture in which the operators develop but the bras and kets do not change. To pursue this analogy a little further, in the “Heisenberg” case () () [ ] ˆˆ 1/ / ˆ ˆ ˆ ˆ ,/ ip xT x T e x e x i p x x εε ˆ −− = == = and is unchanged since it commutes with the operator. So there are two possible ways to deal
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AngularMomentum - Angular Momentum Operator Algebra Michael...

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