Systems and Subsystems, Multiple Particles
Up to this point, we have considered the state of a quantum system to be described by a unit vector in the corresponding Hilbert space, or more properly, an equivalence class of unit vectors under the equivalence relation \(\psi\) ∼ e i θ \(\psi\) . We wi
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19.1 Introduction Up to this point, we have considered the state of a quantum system to be described by a unit vector in the corresponding Hilbert space, or more properly, an equivalence class of unit vectors under the equivalence relation ψ ∼ eiθ ψ. We will see in this section that this notion of the state of a quantum system is too limited. We will introduce a more general notion of the state of a system, described by a density matrix. The special case in which the system can be described by a unit vector will be called a pure state. One way to see the inadequacy of the notion of state as a unit vector is to consider systems and subsystems. We will examine this topic in greater detail in Sect. 19.5, but for now let us consider the example of a system of two spinless “distinguishable” particles moving in R3 . (For now, the reader need not worry about the notion of distinguishable particles; just think of them as being two different types of particles, with, say, different masses or charges.) Let us assume the combined state of the two particles can be described by a unit vector in the corresponding Hilbert space, which is (according to Sect. 3.11) L2 (R6 ). We have, then, a wave function ψ(x, y), where x is the position of the first particle and y is the position of the second particle. Given a wave function ψ(x, y) for the combined system, what is the wave function describing the state of the first particle only? If the wave function of the combined system happens to be a product, say, ψ(x, y) = B.C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, DOI 10.1007/978-1-4614-7116-5 19, © Springer Science+Business Media New York 2013
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19. Systems and Subsystems, Multiple Particles
ψ1 (x)ψ2 (y), then, naturally, we would say that the state of the first particle is simply ψ1 . Of course, one might object that we could rewrite ψ as ψ(x, y) = [cψ1 (x)][ψ2 (y)/c] for any constant c, but this only affects the wave function for the first particle by a constant, which does not affect the physical state. In general, however, the wave function of the combined system need not be a product. Already when ψ is a linear combination of two products, ψ(x, y) = ψ1 (x)ψ2 (y) + φ1 (x)φ2 (y), it is unclear what the correct wave function is for the first particle. At first glance, it might seem natural to try ψ1 (x) + φ1 (x), but upon closer examination, this is not an unambiguous proposal. After all, we can just as well write ψ(x, y) = [c1 ψ1 (x)][ψ2 (y)/c1 ]+ [c2 φ1 (x)][φ2 (y)/c2 ], but then the resulting wave functions for the first particle, ψ1 (x) + ψ2 (x) and c1 ψ1 (x) + c2 ψ2 (x), are not scalar multiples of one another. For a general unit vector ψ in L2 (R6 ), the situation is even worse. The conclusion is this: There does not seem to be any way to associate to ψ a general unit vector ψ in L2 (R3 ) such that ψ could sensibly be described as “the state of the first particle.” Although we cannot associate with ψ a wave function ψ for the first particle, there is no difficulty in taking expectation value
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