Working in stochastic spin space and using POV measures as in the Davies and Lewis measurement scheme, we construct a formalism to describe the simultaneous measurement of incompatible spin components. The methods are illustrated with a new analysis of the Stern-Gerlach experiment, and with a discussion of spin dynamics in stochastic spin space. We also present a new short proof of a theorem on representations of spin-1/2 systems, find a joint spectral family for (noncommuting) spin components, and indicate the connection of our result with the Riesz extension theorem.
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J. von Neumann,Mathematical Foundations of Quantum Mechanics (Princeton Univ. Press, Princeton, N.J., 1955).
E. Prugovecki,Quantum Mechanics in Hilbert Space (Academic Press, N.Y., 1971), Chapter IV.
C. Y. She and H. Heffner,Phys. Rev. 152, 1103–1110 (1966); E. Prugovecki,Can. J. Phys. 45, 2173–2219 (1967); J. L. Park and H. Margenau,Int. J. Theor. Phys. 1, 211–283 (1968); E. B. Davies,J. Funct. Anal. 6, 318–346 (1970); E. B. Davies and J. T. Lewis,Comm. Math. Phys. 17, 239–260 (1970); E. Prugovecki,Found. Phys. 4, 9–18 (1974); S. T. Ali and G. G. Emch,J. Math. Phys. 15, 176 (1974); T. Louton,Pac. J. Math. 59, 147–159 (1975); E. Prugovecki,Found. Phys. 4, 557–571 (1975);J. Math. Phys. 17, 517–523 (1976); S. T. Ali and H. D. Doebner,J. Math. Phys. 17, 1105–1111 (1976); E. Prugovecki,Int. J. Theor. Phys. 16, 321–331 (1977).
F. E. Schroeck, Jr., inMathematical Foundations of Quantum Theory, A. R. Marlow, ed. (Academic Press, N.Y., 1978), pp. 299–327.
F. E. Schroeck, Jr., inClassical, Semiclassical and Quantum Mechanical Problems in Mathematics, Chemistry and Physics, K. Gustafson and W. P. Reinhardt, eds. (Plenum Press, 1981); F. E. Schroeck, Jr.,J. Math. Phys. 22, 2562–2572 (1981).
E. Prugovecki,J. Phys. A. 10, 543–549 (1977).
H. Boerner,Representations of Groups (North-Holland, Amsterdam, 1963), p. 273, Theorem 3.3.
W. Gerlach and O. Stern,Ann. Physik 9, 349–352 (1922); see also Taylor,Phys. Rev. 28, 576–582 (1926).
K. Gottfried,Quantum Mechanics, Vol. I (W. A. Benjamin, N.Y., 1966), pp. 165–190.
J. M. Jauch and C. Piron,Helv. Phys. Acta 40, 559–570 (1967).
F. Riesz and B. Nagy,Functional Analysis, (Frederick Ungar, N.Y., 1960), Appendix: Extensions of Linear Transformations in Hilbert Space Which Extend Beyond This Space.
C. W. Curtis,Linear Algebra, 3rd ed. (Allyn and Bacon, Boston, 1974), pp. 250–254.
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Schroeck, F.E. On the stochastic measurement of incompatible spin components. Found Phys 12, 479–497 (1982). https://doi.org/10.1007/BF00729996
- Simultaneous Measurement
- Short Proof
- Measurement Scheme
- Extension Theorem
- Spin Dynamic