sec6 - VI. Angular momentum Up to this point, we have been...

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VI. Angular momentum Up to this point, we have been dealing primarily with one dimensional systems. In practice, of course, most of the systems we deal with live in three dimensions and 1D quantum mechanics is at best a useful model . In this section, we will focus in particular on the quantum mechanics of 3D systems. Many of the elements we discovered for one dimensional problems will carry over directly to higher dimensions; however, we will encounter certain effects that are qualitatively new, and we will spend most of our time exploring these new phenomena. The first change comes in how we associate operators with classical observables. In one dimension, we had q q ˆ p p ˆ = i q In three dimensions the position and momentum are vectors and so we must substitute vector calculus for the single variable results: r r ˆ ( i x ˆ + j y ˆ + k z ˆ ) p p ˆ Δ « i p j p k ˆ + ˆ + p ˆ ÷ x y z Where r is the position vector and vector quantities will always be indicated in bold face . Note that the operators that correspond to different axes (i.e p ˆ x and z ˆ ) commute with one another, while the position and momentum along a given axis (i.e p ˆ x and x ˆ ) obey the normal commutation relation. We can summarize this in a few equations: r r i p r j ] = i Z δ [ p ˆ i , p ˆ j ] = 0 [ ˆ, ˆ ] = 0 [ ˆ, ˆ j i ij where i and j can take the value 1,2 or 3 to indicate the x ˆ , y ˆ and z ˆ components of each vector. a. Rotations The first difference between 3D and 1D is the possibility of performing a rotation of our system about one of the three axes. Let us denote a rotation of an angle θ about a unit vector n by R ( θ ) . Clearly, R ( ) n n
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is a matrix (it transforms vectors to vectors). Further, it is clear that R ( θ ) R ( ) R ( ) R ( ) (rotations about different axes do not n m m n commute). Note that this has nothing to do with quantum mechanics and everything to do with geometry! It is easy to verify that the rotation operators associated with the three Cartesian axes are: 1 0 0 Δ Δ Δ ÷ ÷ ÷ sin R x ( ) 0 cos θ = 0 sin cos « cos 0 sin Δ Δ Δ ÷ ÷ ÷ 0 cos R ( ) y 0 1 0 = sin « cos sin 0 Δ Δ Δ ÷ ÷ ÷ R z ( ) sin cos 0 = 0 0 1 « Note that the rotation matrices for x and y can be obtained from the z matrix by the cyclic permutation x y , y z , z x . This must always be the case, because our labeling of the x, y and z axes is totally arbitrary! The only thing we must be careful of is that the “triple
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sec6 - VI. Angular momentum Up to this point, we have been...

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