FOUNDATIONS OF QUANTUM MECHANICS

166 CHAPTER VIII. THE MEASUREMENT PROBLEM
VIII. 3
MEASUREMENT ACCORDING TO QUANTUM MECHANICS
The following schematic representation of the measurement process in quantum mechanics is
given by Von Neumann (1932).
Suppose that A is a physical quantity of the object system S, represented quantum mechanically
by the maximal operator A on Hilbert space H S , having a discrete spectrum a 1 , . . . , a N . Now
let S interact with a measuring apparatus M, where M is described quantum mechanically also.
For the measuring apparatus M to be able to function as a measuring apparatus, it has to have an
pointer quantity R, represented by the operator R on Hilbert space H M , having orthonormal eigenstates
|r 0 ⟩, . . . , |r N ⟩. These eigenstates have to be orthonormal since they correspond to pointer
readings which can be distinguished by the human eye. Let |r 0 ⟩ be the eigenstate in which the pointer
shows no deflection. The Hilbert space of this composite system S M is H = H S ⊗ H M with
dim H M = dim H S + 1, the basis of R including |r 0 ⟩, where that of A does not include |a 0 ⟩.
Prior to the measurement, the measuring apparatus M is in the eigenstate |r 0 ⟩. We want this state
to change, as a result of the measurement interaction, into the eigenstate |r j ⟩ which is indicative of the
value a j of A, thus, let S initially be in the eigenstate |a j ⟩ of A. Moreover, we want the measurement
to be ideal, so that the state |a j ⟩ of S does not change.
Von Neumann showed that this transition can indeed be brought about by a unitary transformation,
which means we have to find for the composite system SM a unitary evolution operator U, inducing
the transition
U ( |a j ⟩ ⊗ |r 0 ⟩ ) = |a j ⟩ ⊗ |r j ⟩, (VIII. 2)
where U describes the measurement interaction lasting some unspecified time interval.
EXERCISE 36. Show that the operator
U =
N∑ N∑
|a l ⟩ ⊗ |r [l+m] ⟩ ⟨a l | ⊗ ⟨r m | (VIII. 3)
l=1 m=0
(a) is unitary, and (b) induces the desired transition (VIII. 2). Here, [l + m] means l + m modulo
N + 1, i.e.: [N + 1] = 0, [N + 2] = 1, etc.
The formula (VIII. 2) strongly resembles the transition (VIII. 1). Apparently, everything we desired
concerning the ideal measurement process in quantum mechanics, including the requirement
that the value of A must not be disturbed, can be achieved using a unitary operator. At first sight,
there does not seem to be any problem with a completely quantum mechanical treatment of the measurement
interaction, taken as an ordinary physical process obeying Schrödinger’s equation. As in the
classical case, the method of measuring is not discussed. We also did not appeal to the measurement
or the projection postulate.