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22 - Physics 6210/Spring 2007/Lecture 22 Lecture 22...

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Physics 6210/Spring 2007/Lecture 22 Lecture 22 Relevant sections in text: § 3.1, 3.2 Rotations in quantum mechanics Now we will discuss what the preceding considerations have to do with quantum me- chanics. In quantum mechanics transformations in space and time are “implemented” or “represented” by unitary transformations on the Hilbert space for the system. The idea is that if you apply some transformation to a physical system in 3-d, the state of the system is changed and this should be mathematically represented as a transformation of the state vector for the system. We have already seen how time translations and spatial translations are described in this fashion. Following this same pattern, to each rotation R we want to define a unitary transformation, D ( R ), such that if | ψ is the state vector for the system, then D ( R ) | ψ represents the state vector after the system has undergone a rotation characterized by R . The key requirement here is that the pattern for combining two rotations to make a third rotation is “mimicked” by the unitary operators. For this we require that the unitary operators D ( R ) depend continuously upon the rotation axis and angle and satisfy D ( R 1 ) D ( R 2 ) = e 12 D ( R 1 R 2 ) , where ω 12 is a real number, which may depend upon the choice of rotations R 1 and R 2 , as its notation suggests. This phase freedom is allowed since the state vector D ( R 1 R 2 ) | ψ cannot be physically distinguished from e 12 D ( R 1 R 2 ) | ψ . If we succeed in constructing this family of unitary operators D ( R ), we say we have constructed a “unitary representation of the rotation group up to a phase”, or a “projective unitary representation of the rotation group”. You can think of all the ω parameters as simply specifying, in part, some of the freedom one has in building the unitary represen- tatives of rotations. (If the representation has all the ω parameters vanishing we speak simply of a “unitary representation of the rotation group”.) This possible phase freedom in the combination rule for representatives of rotations is a purely quantum mechanical possibility and has important physical consequences. Inci- dentally, your text book fails to allow for this phase freedom in the general definition of representation of rotations. This is a pedagogical error, and an important one at that. This error is quite ironic: the first example the text gives of the D ( R ) operators is for a spin 1/2 system where the phase factors are definitely non-trivial, as we shall see.
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