Physics 6210/Spring 2007/Lecture 8
Lecture 8
Relevant sections in text:
§
1.6
Momentum
How shall we view momentum in quantum mechanics? Should it be “mass times ve
locity”, or what?
Our approach to the definition of momentum in quantum mechanics
will rely on a rather fundamental understanding of what is “momentum”.
To motivate
our definition, let me remind you that the principal utility of the quantity called “mo
mentum” is due to its conservation for a closed system.
One can then understand the
motion of interacting systems via an “exchange of momentum”. Next, recall the intimate
connection between symmetries of laws of physics and corresponding conservation laws.
In particular, symmetry under spatial translations corresponds to conservation of linear
momentum. In the Hamiltonian formulation of the classical limit of mechanics this cor
respondence becomes especially transparent when it is seen that the momentum is the
infinitesimal generator of translations, viewed as canonical transformations. In the Hamil
tonian framework, the conservation of momentum is identified with the statement that the
Hamiltonian is translationally invariant, that is, is unchanged by the canonical transfor
mation generated by the momentum. We shall see that this same logic applies in quantum
mechanics. Indeed, nowadays momentum is mathematically identified – by definition – as
the generator of translations. Let us see how all this works.
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 Spring '07
 M
 mechanics, Mass, Momentum, TA, Hilbert space, canonical commutation relations

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