Symmetries and Their Consequences
In Chapter 2, we explored the consequences of the symmetries of the Hamiltonian.
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11.2. Translational Invariance in Quantum Theory Consider a single particle in one dimension. How shall we define translational invariance? Since a particle in an arbitrary state has neither a well-defined position nor a well-defined energy, we cannot define translational invariance to be the invariance of the energy under an infinitesimal shift in the particle position. Our previous experience, however, suggests that in the quantum formulation the expectation values should play the role of the classical variables. We therefore make the correspondence shown in Table 11.1. Having agreed to formulate the problem in terms of expectation values, we still have two equivalent ways to interpret the transformations: (X)-+ (X) + t:
(11.2.la)
(P)-+(P)
(11.2.lb)
:j: It may be worth refreshing your memory by going through Sections 2.7 and 2.8.
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280 CHAPTER 11
Table 11.1. Correspondence between Classical and Quantum Mechanical Concepts Related to Translational Invariance Concept Translation Translational invariance Conservation law
Classical mechanics
Quantum mechanics
X--->x+c: p--->p Xf--->Yt
(X)--->(X)+ c: (P)--->(P) (H)--->(H) (P)=O (anticipated)
p=O
The first is to say that under the infinitesimal translation, each state I If/) gets modified into a translated state, I If!.) such that ( 11.2.2a)
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