In quantum mechanics, states are supposed to be specified by vectors in Hilbert space. However, students become confused about the representation of states and the meaning of “state” itself. We discuss consequences of the fact that quantum mechanics is intrinsically a probabilistic theory, and the ubiquitous confusion over whether quantum states, when specified as well as nature permits, are described by state vectors or rays.

## REFERENCES

1.

I am using the word

*deterministic*in the sense that*the state of any autonomous system at one time uniquely determines its state at any future time*. The statement “this theory is deterministic” is to be distinguished from a claim that it is*causal*. Causality, in my use of the word, means that every*event*A is connected with another*event*B such that B is the cause of A, or A is the effect of B, though I will not specify exactly what this connection is. Note that my usage of the two words “causal” and “deterministic” differs from that of some other authors.2.

Newtonian mechanics is deterministic despite the fact that for most such systems, sensitivity to initial conditions results in making them practically nondeterministic.

3.

I am employing the Schrödinger picture. In the Heisenberg picture the state vector remains fixed and the Heisenberg equations of motion uniquely determine the development of all the dynamical variables, so that here, too, the theory is deterministic.

4.

The superposition principle implies not just correlations in phase but also in magnitude; the usual shorthand for this effect is “phase correlations.”

5.

The book by Kurt Gottfried,

*Quantum Mechanics*(Benjamin, New York, 1966), which is excellent in many respects, explicitly states on p. 213 of Vol. 1 that “*all vectors belonging to a ray represent the same physical state*” (author’s italics). He then goes on to define rays in Hilbert space, albeit incorrectly.6.

See, for example, H. Reichenbach,

*Philosophic Foundations of Quantum Mechanics*(University of California, Berkeley, 1965).7.

However, some aspects of entanglement are simply the effects of quantum mechanics being a probabilistic theory, as we have seen in the Jack and Jill example.

8.

P. A. M. Dirac,

*The Principles of Quantum Mechanics*, 3rd ed. (Clarendon, Oxford, 1947), p. 12.9.

Roger G. Newton,

*Quantum Physics: A Text for Graduate Students*(Springer-Verlag, New York, 2002).10.

Gottfried describes in some detail (pp. 177–178 of Ref. 5) how “the phases are destroyed” in a measurement.

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© 2004 American Association of Physics Teachers.

2004

American Association of Physics Teachers

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