10.4: Multi-State Model

Our model of a physical object, justified by the brief discussion of quantum mechanics in the previous two sections, is as follows. The object has a wave function \(\psi\) which, in principle, characterizes its behavior over time. This wave function may be difficult or impossible to calculate, and it can change, perhaps in unpredictable ways, when the object interacts with its environment.

The object has a finite (or perhaps countable infinite) number of “stationary states” that are easier to calculate (although for complicated objects finding them may still be impossible). Each of the stationary states has its own wave function \(\psi_j\) where \(j\) is an index over the stationary states. If the actual wave function is one of these stationary states (i.e., if this state is “occupied”) then the object stays in that state indefinitely (or until it interacts with its environment). Each stationary state has its own energy \(e_j\) and possibly its own values of other physical quantities of interest.

The object’s wave function can be expressed as a linear combination of the stationary states, in the form

\(\psi = \displaystyle \sum_{j} a_j \psi_j \tag{10.10}\)

where the \(a_j\) are complex numbers called expansion coefficients. If the object occupies one of the stationary states then all \(a_j\) are 0 except one of them. Without loss of generality the expansion coefficients can be defined so that the sum of their magnitudes squared is one:

\(1 = \displaystyle \sum_{j} |a_j|^2 \tag{10.11}\)

Measurement of an object’s property, such as its energy, involves an interaction with the object’s environment, and a change in the environment (if for no other reason than to record the answer). It is a consequence of quantum mechanics that if the object is in one of its stationary states and its energy is measured, the result of the measurement is simply the energy of that state, and the state does not change (i.e., the expansion coefficients, all of which are 0 except one, are not changed by the measurement). On the other hand, if the object is not in one of the stationary states, then the result of the measurement is the energy of one of the stationary states, and the object immediately assumes that stationary state. Thus after each measurement the object ends up in a stationary state. Which state? The probability that state \(j\) is the one selected is \(|a_j |^2\). Thus the expected value of the energy measured by an experiment is

\(\displaystyle \sum_{j} e_j |a_j|^2 \tag{10.12}\)

where \(e_j\) is the energy associated with the stationary state \(j\). Measurement in quantum mechanics is thus not like measurement of everyday objects, where it is assumed that the energy or other physical properties can be measured with arbitrary accuracy, and that such measurements need not perturb the object. The nature of quantum measurement is one more of those aspects of quantum mechanics that must be accepted even though it may not conform to intuition developed in everyday life.