Principles and Concepts
of Quantum Mechanics
Implications of
Quantum Mechanics

9B. Measurement Theory. Perception of
Eigenvalues of Operators.

Summary
Quantum mechanics implies the results of a measurement can only correspond to the eigenvalues of the measured operator.

In the early years after quantum mechanics was discovered, certain axioms were postulated to codify the relation between observation and the mathematics of quantum mechanics. This was the content of measurement theory. It turns out, however, that these need not be given the status of axioms because they can all be derived from the principles of quantum mechanics. We are concerned here with the derivation of the following principle:

[P8A] If a measurement is made on a physical quantity corresponding to some operator, the measurement will always yield one of the eigenvalues of the operator.

As a first illustration of how this follows from the mathematics of quantum mechanics, we consider the measurement of the z-component of spin of a spin particle. The initial state the measurement is being conducted on is  (9B-1)
and the operator corresponding to the measured quantity is  (9B-2)
After the particle goes through the Stern-Gerlach apparatus, the wave function will separate into two parts, a part traveling slightly upwards and a part traveling slightly downwards. The particle wave function is then  (9B-3)
Detectors, D(up) and D(down) are put on the two paths, and an observer perceives the readings on the two detectors. The full wave function is then  (9B-4)
So an observer sees (yes up, no down) or (no up, yes, down). Because the two versions of reality are isolated, she never sees anything besides these two results, each of which corresponds to one of the two eigenvalues. No ‘average’ value for the observed value is perceived, nor is a ‘double exposure.’

As another example, suppose a single hydrogen atom is in an excited state. It can then emit, let’s say, a photon (photon-like wave function) of wavelength or a photon of wavelength or it can be in a state corresponding to emitting some combination of the two. The photon wave function is put through a grating and the two different-wavelength parts travel on two different paths, so the property being measured corresponds to an ‘operator’ which separates the different wavelengths of light. The photon wave function after emission is  (9B-5)
The detector is a strip of film that the light from the grating impinges on. When the film grains are examined, the observer will perceive either one grain exposed at a position corresponding to or one grain exposed at a position corresponding to . In accord with principle [P8A], and again taking into account the non-communication of the ‘universes’ corresponding to the two outcomes, no grain corresponding to an intermediate wavelength is ever perceived as exposed, nor are two grains ever perceived as exposed by the observer.  