Talk:Making sense of quantum mechanics/observational properties
Here we have a key issue in quantum mechanics. In my idea, observable = observational property. However the value of the observable is not necessarily the observed value (through a measurement). There are at least two reasons that could make them differ:
- experimental error in measurement
- intrinsic indeterminacy
The first reason is a common source of error. The second reason is peculiar to the nature of quantum objects and quantum measurements. If we visualize quantum objects as little needles and a quantum measurement as an interaction between two needles, we may get some insight: even if the position of a needle (say the geometrical center) is definite at a particular time, if the value were measured at that time, it could give a different position, because what is measured is the location of the point of interaction (=collision), which is generally not the geometrical center.
So we have an indeterminacy in position that is proportional to the length of the needle. For a velocity measurement, we have an indeterminacy proportional to the difference in translational velocity at both ends (due to the spinning motion). Arjen Dijksman 22:54, 18 September 2007 (UTC)
- In the language of linear algebra, a quantum observable is a linear operator( ie a linear map, or if you like, a matrix), on a possibly infinite-dimensional vector space, and an observed value is an eigenvalue of this linear operator. -Hillgentleman|Talk 23:20, 19 September 2007 (UTC)
- Yes. However we could notice some difference between an observable (a physical property) and its linear operator. The way Dirac introduces the linear operator in The Principles of Quantum Mechanics (1930) illustrates this. The chain between linear operator and observable is made of correspondences:
- at §7, the linear operators correspond to the dynamical variables,
- at §10: When we make an observation we measure some dynamical variable, and further: We call a real dynamical variable whose eigenstates form a complete set an observable.
- So, we have a correspondence between an operator (something that performs an operation on a state vector, which may be assimilated to an act of measurement) and an observable (a dynamical variable or property that is undetermined as long as the act of measurement has not given one of its possible outcomes).
- I personally have the impression that there is a flaw when we say that an observed value is an eigenvalue of the linear operator. Eigenvalues are well determined values of measurement outcomes, while observed values are subject to experimental error and intrinsic indeterminacy. For example, the measurement of the energy of system in a state with discrete energy levels (say a particle in a box) will nearly never yield exactly one of its eigenvalues. In this light, I feel uncomfortable when I read Dirac (§10): We can infer that, with the dynamical system in any state, any result of a measurement of a real dynamical variable is one of its eigenvalues [hmmm...]. Conversely, every eigenvalue is a possible result of a measurement of the dynamical variable for some state of the system [yes, no problem with that], since it is certainly the result if the state is an eigenstate belonging to this eigenvalue [No! it maybe the theoretical result, but not the ascertained result of the act of measurement]. Arjen Dijksman 09:50, 23 September 2007 (UTC)