# On the Time—Energy Uncertainty Relation

Chapter

## Abstract

There is only one well-known application for the time—energy uncertainty relation: the connection between the life-time and the energy-width of resonance states. The relation in question was commonly known even before quantum mechanics was established; its first quantum mechanical derivation was based on Dirac’s original theory of the interaction between matter and radiation.^{1} The point which should be noted is that the uncertainty relation does not apply to time and energy *in abstracto* but to the life-time of a definite state of a system. In the example referred to, this is the state in which an atom is in an excited state but there is no radiation present.

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## References

- 1.P. A. M. Dirac,
*Proc. Roy. Soc. (London)***A114**243, 710 (1927). The calculation was carried out by V. Weisskopf and E. Wigner,*Z. Physik***63**54 (1930). See also the article of the same authors,*ibid*.**65**18 (1930) and many subsequent discussions of the same subject and of resonance states decaying by the emission of particles rather than radiation.Google Scholar - 2.W. Heisenberg,
*Z. Physik***43**172 (1927). For a rigorous derivation, see H. P. Robertson,*Phys. Rev.***34**163 (1929). The derivation of section 5 is patterned on that of this article.Google Scholar - 3.G. R. Allcock,
*Ann. Phys. (N.Y.)***53**, 253, 286, 311 (1969). Section II of the first of these articles gives a very illuminating discussion of the ideas which underlie also the present section. It also contains a review of the literature of the time—energy uncertainty principle, making it unnecessary to give such a review here. The review also gives a criticism of some of the unprecise interpretations of the time—energy uncertainty relation which are widely spread in the literature. The later parts of the aforementioned articles arrive at a pessimistic view on the possibility of incorporating into the present framework of quantum mechanics time measurements as described by Allcock in section n or in the present section. This pessimism, which is not shared by the present writer, is expressed, however, quite cautiously and is mitigated by the various assumptions on which it is based.Google Scholar - 4.The notation of chapter xx of P. A. M. Dirac’s
*The Principles of Quantum Mechanics*(Oxford University Press, various editions) is used.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1997