Quantum States
Is you is or is you ain't my baby? Probably. Probably not.
----------------------------------------------------------------------------------
------------------------------------------------
What is meant by a state? Compare this with what we might mean by a problem, and by extension, to problem-solving skills.
A state is the kind of thing you recognize as an ordinary circumstance of life. A condition of something. The way things are around you at the moment characterizes a state.
You can define a state of something as the values of a set of properties that identify the thing effectively -- usually relative to your skills setting. For instance, if you were playing tennis or golf, the state of the ball would be given in terms of its position, its velocity, and its spin. These are the properties that define the path of the ball in space-time. Other properties of the ball -- like its weight (mass) -- are presumed to remain constant during play and are not considered relevant -- until they change significantly, at which time you toss the ball in favor of a new one.
It's one thing to talk about attributes of the object in question (the ball), but quite another to say that the values can in principle be given for any particular moment of play. I don't mean they have to be measured, but rather that they could be measured if we cared to exert the effort. The idea is that they have values that could be determined as accurately as needed for that moment if we had the equipment. The equipment makes it possible to read (observe perceive, recognize, diagnose, measure) the properties. The method of measurement lets us see the properties, and the type of equipment determines the type of information we can get in the observation. In classical physics we can "see" the properties accurately and jointly. Often the unaided eyes are sufficient.
This was the state of affairs when Heisenberg came along with his quantum theory to change the notion entirely. Because of the quantum nature of the light source and its finite wave length, the answer could only be known to within the range of the wave length. And that put into question its real existence! (If you can't see it, how can you say it's there?)
More significantly, still, Heisenberg propounded the principle of indeterminacy, that conjugate pairs of state properties, like position and velocity (momentum), couldn't be measured precisely at the same time, that in fact the more precisely you measured one of them, the more uncertainty you produced in the other one!
----------------------------------------------
State variables are the defining properties of the state. They are the physical properties deemed to be observables, like position, velocity (or momentum), mass, energy, and so on. In classical physics, such conjugate variables are considered capable of being observed simultaneously and can define a state completely.
With quantum mechanics, however, all that has changed, and so have the specifics of a state. A quantum state is therefore a far cry from a classical state. The idea of a state is now in a real state! No longer are state variables simultaneously measurable with unlimited accuracy. They can't be observed jointly and accurately.
Furthermore, quantum states are expressed in terms of probabilities -- you now have to deal with probability "states." The state of the (atomic) particle in question at any moment is a distribution of probabilities, each one giving the likelihood of the particle having a specific property (like a position). The particle could be in any one of an infinite number of real states, each with a certain probability. But the particle can't be said to have a position until the position is actually measured (observed).
What's more, the variables aren't evaluated merely as real quantities, such as one might expect for the position and velocity of a tennis ball, say. You have to contend with properties characterized by complex numbers -- numbers with imaginary components!
E. H. Walker interprets the imaginary nature of the variables to mean that the particle has a potential for being in any particular state. Also, to put icing on the cake and defy logic, states can be put together according to the principle of superposition to form new, seemingly self contradictory states. That is, quantum states satisfy the superposition principle -- they can be added together and given a probabilistic interpretation.
To illustrate, suppose you're told I'm either at the department store or the theater. (The state variable here is position and there are two possible values.) Suppose, too, there is a 30% likelihood that I'm at the store and a 70% chance that I'm at the theater. In the classical view I am definitely at one place or the other, although the likelihood is greater that I would be at the theater. But certainly I can't be at both at the same time
In the quantum update, however, I would not be at one place or the other, exclusively. Rather, since QM admits superposition (the adding of state vectors), the two states can be combined into one. So it turns out that there is a state having a 30% probability that I would be at the store and a 70% probability that I would be at the theater, which means I could have been at the store and at the theater at the same time(!), just before I was observed to be at the theater, say, should an observation have been made. (The observation itself collapses the quantum nature and determines where I am! And of course I wouldn't have been as I am finally observed to be, since I would only have been in some potential state!)
The idea is inconceivable to most of us but is nonetheless accepted by most quantum physicists as being a feature of the real underworld. So an electron (proton, quark, or any of the rest of these underworld creatures) could be (each with a certain probability) here and there and there and there ... at the same time before the situation is resolved in perception!
We have a world in which the observer is a factor determining the state of a system. In my view of the matter, the observer becomes a component of the system being observed. Observation creates a behavioral environment.
At work here, to repeat, is the principle of superposition. To try to clarify this concept, say you walk from your abode (point A) to the next corner of the block (point B), then you turn 90 degrees and walk from that corner to the diagonal corner of your block (point C).
In vector language the two walks can be added together to generate an equivalent walk from point A to point C. This is an example of the superposition principle applied to the components of a vector in two-space. Walking is here an operator that transforms vector A into vector B, and vector B into vector C, or A into C.
In this example the operator is said to commute, because you can just as easily walk from C to B and from B to A. That is, you can reverse the operations. However, if the path from A to B is a steep downhill walk, you just might not be able to walk back from B to A. In that event, the operator would not commute. In quantum physics the operators are mostly of the non-commutative variety, and this is what causes most of the conceptual difficulties.
----------------------------------------------
In the Heisenberg Matrix Theory -- as compared to the (equivalent) Schroedinger Wave Theory -- eigenvectors and eigenvalues become important ingredients. To compare them with ordinary numbers and ordinary variables, suppose you have an ordinary function that maps any real number into a simple multiple of that number. For instance, let the function be:
f(x) = 2x.
This expression can also be written in the simple mapping form:
f:x ® 2x.
You can see that the function is linear (in x). It uses the real multiple 2 to transform the real independent variable, x, into the real dependent variable, 2x. If x is 5, then:
f(x) = f(5) = 2 times 5.
In the same way, a vector that is transformed into some real multiple of itself by an operator is called an eigenvector, and the multiplier in the transformation is called the eigenvalue. That's all there is to it.
Note that, like f(x) = 2x, this operator is also linear. The term, operator, replaces the word, function, and may be seen as a generalization of 'function,' when function is understood in numerical terms. And 'vector' replaces 'variable.' So this linear operator transforms a vector (function) into some real multiple of itself.
The strange thing about quantum mechanics is that the measurement of a quantum state -- a superposition of different probability states -- gets transformed by the measurement (operation, observation, reading) into an eigenvector having the appropriate eigenvalue. For example, if the operation measures the velocity (momentum) of an electron, the operator is the momentum operator that takes the state of the electron to the special eigenvector with its eigenvalue, which is one of the possible values of momentum for the electron. To use a simple and direct phrase by John Polkinghorne in The Quantum World:
[T]he possible results of measuring an observable are just the set of eigenvalues of the corresponding operator.
As it happens, the principles of quantum mechanics can be expressed conveniently in terms of the theory of vector spaces. For one thing, vector spaces can accommodate quantum states having any number of state variables, because you can construct vector spaces of any number of dimensions. Secondly, vectors are superposible, as we see here, so they can readily express the superposition of quantum states. And thirdly, vector spaces can be written in equation form as the linear vector sum of any number of dimension (or unit) vectors, as follows:
V = a1I + a2J + a3K + ... anL.
In this equation, I, J, K,... L are the unit dimension vectors over the vector space, and the coefficients a1 through an provide the real scalar magnitudes of the vectors in the space. This is the standard definition of an n-dimensional vector space.
But there's a difference between this interpretation and the vector spaces used in quantum mechanics. That difference creates a puzzle wrapped in a riddle, especially for physicists in the classical tradition, because the coefficients a1 through an are no longer simply real numbers. On the contrary, they are complex numbers, which is to say they have an imaginary component! As Walker says, the imaginary numbers identify a sort of potential for being -- rather than actually being -- in one state or another when not observed. And so quantum mechanics uses imaginary vector spaces. How's that for knockin' the skin off an orange!
----------------------------------------------
A curious thing about the imaginary (or complex) number is that it is used in various other ways to solve problems. For one thing, as you know, it is used to expand the solution space for quadratic equations. It is also used in engineering problems as a holding place and partition for variable pairs. And it is touted seriously as a potential mechanism for expanding the solution space of logical (statement) equations.
-------------------------------------