Editing Probability amplitude (section)

== Physical overview ==
Neglecting some technical complexities, the problem of [[quantum measurement]] is the behaviour of a quantum state, for which the value of the [[observable]] {{mvar|Q}} to be measured is [[uncertainty principle|uncertain]]. Such a state is thought to be a [[quantum superposition|coherent superposition]] of the observable's ''[[eigenstate]]s'', states on which the value of the observable is uniquely defined, for different possible values of the observable.

When a measurement of {{mvar|Q}} is made, the system (under the [[Copenhagen interpretation]]) [[state vector reduction|''jumps'' to one of the eigenstates]], returning the eigenvalue belonging to that eigenstate. The system may always be described by a [[linear combination]] or [[Quantum superposition|superposition]] of these eigenstates with unequal [[weight function|"weights"]]. Intuitively it is clear that eigenstates with heavier "weights" are more "likely" to be produced. Indeed, which of the above eigenstates the system jumps to is given by a probabilistic law: the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical weight squared. These numerical weights are called probability amplitudes, and this relationship used to calculate probabilities from given pure quantum states (such as wave functions) is called the [[Born rule]].

Clearly, the sum of the probabilities, which equals the sum of the [[Square (algebra)#In complex numbers|absolute squares]] of the probability amplitudes, must equal 1. This is the [[#Normalization|normalization]] requirement.

If the system is known to be in some eigenstate of {{mvar|Q}} (e.g. after an observation of the corresponding eigenvalue of {{mvar|Q}}) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of {{mvar|Q}} (so long as no other important forces act between the measurements). In other words, the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. If the set of eigenstates to which the system can jump upon measurement of {{mvar|Q}} is the same as the set of eigenstates for measurement of {{mvar|R}}, then subsequent measurements of either {{mvar|Q}} or {{mvar|R}} always produce the same values with probability of 1, no matter the order in which they are applied. The probability amplitudes are unaffected by either measurement, and the observables are said to [[commutator|commute]].

By contrast, if the eigenstates of {{mvar|Q}} and {{mvar|R}} are different, then measurement of {{mvar|R}} produces a jump to a state that is not an eigenstate of {{mvar|Q}}. Therefore, if the system is known to be in some eigenstate of {{mvar|Q}} (all probability amplitudes zero except for one eigenstate), then when {{mvar|R}} is observed the probability amplitudes are changed. A second, subsequent observation of {{mvar|Q}} no longer certainly produces the eigenvalue corresponding to the starting state. In other words, the probability amplitudes for the second measurement of {{mvar|Q}} depend on whether it comes before or after a measurement of {{mvar|R}}, and the two observables [[commutator|do not commute]].