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A Note on the Equations of Motion in Non-Relativistic Quantum Mechanics

Published online by Cambridge University Press:  24 October 2008

F. C. Powell
F. C. Powell*
Affiliation:
Gonville and Caius College
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Extract

It is well known that the motion of a dynamical system can be pictured in two distinct ways, which Dirac names the Heisenberg picture and the Schrödinger picture. The equations of motion take different forms in the two pictures, but of course have identical physical consequences, since the motion of the system does not depend in any way on which picture we choose. It should therefore be possible to express the equations in an invariant form (independent, that is, of the picture used). It will be shown in this note that not only can this be done (equation (3)), but it can be done in such a way that it is not even necessary to introduce a picture at all (equation (3′)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 31 , Issue 2 , April 1935 , pp. 291 - 294
Copyright
Copyright © Cambridge Philosophical Society 1935

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References

* See his Quantum Mechanics, second edition.

† See Dirac's Quantum Mechanics, first edition, Chaps. ii and iii.

‡ More strictly, we give physical significance to the numbers ø1 ξψ 2.

* The existence of a correspondence with this property follows from the isomorphism of the Hilbert space of the ψ's with that of the ψ+'s.

* I.e. a complete set of independent ψ's in terms of which any ψ is expanded, the coefficients in the expansion being the representatives of ψ.