The Uncertainty Principle

[thylacine]

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ok. An observable is modeled by a self adjoint operator on the Hilbert Space. It seems like there would be a lot of self adjoint operators on the Hilbert Space. But there are only a few observables. Are all the self adjoint operators on the Hilbert Space some kind of observables? If not, what do they make of the ones that are not? And how do they decide which ones are and which ones aren't? If all of the self adjoint operators are observables, why are there so few self adjoint operators?

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I think this discussion could generate some semantic problems. The word "observable" means some kind of observable property of the particle or system. But it also means a self-adjoint operator on the underlying Hilbert space. (This fact alone demonstrates that they are generally believed to be in one-to-one correspondence.) To try to be clear, I will call the physical properties "real observables" and the operators either "observables" or just "self-adjoint operators."

A self-adjoint operator determines a mapping that takes a state x to a probability distribution on the real line. So your question amounts to this: given such a mapping, can we cook up an experiment whose outcome has the right distribution for each state x? I think it is generally believed that this is true. So yes, every self-adjoint operator corresponds to a real observable. In practice, maybe one can cook up examples of self-adjoint operators whose corresponding real observables are presently unknown.

But maybe not. For instance, suppose one is not concerned with position, momentum, etc., but simply wants to measure spin. The appropriate Hilbert space for this is C^2. (C is the complex numbers.) The self-adjoint operators are simply 2x2 matrices A with complex entries such that A is equal to its own conjugate transpose. There are three "primitive" real observables: the spin in the x, y, and z directions. Their matrices are given by

S_x = [0 1; 1 0],
S_y = [0 -i; i 0], and
S_z = [1 0; 0 -1].

(There is actually a constant of h/4pi in front of each of these, where h is Planck's constant.) We can generate other (real) observables by taking real linear combinations of these operators. For example, S_x + S_y would correspond to spin the direction of the vector (1,1,0). (Or, perhaps more accurately, this measures sqrt{2} times the spin in that direction.) In this way, we can generate a set of observables which is a 3-dimensional vector space over the reals.

Think, however, about how to construct an arbitrary self-adjoint operator on C^2. To build such a matrix, one selects any two real numbers for the main diagonal, then any complex number for the upper right entry. The lower left entry is then determined. So the entire space of observables is a 4-dimensional vector space over the reals. It follows that not all observables can be realized as real linear combinations of the spin matrices.

The problem is resolved by noting that we are not restricted to taking linear combinations. We may also take functions of these matrices. For example, we could measure 2^{x-spin}. This is a real observable. Its corresponding self-adjoint operator is

2^[0 1; 1 0] = [5/4 3/4; 3/4 5/4].

Notice that this operator is not a real linear combination of the spin matrices. In general, every self-adjoint operator on C^2 can be written as a real-valued function of a linear combination of the spin matrices. Which function and which linear combination to use are not hard to compute. In this way, every self-adjoint operator can be associated to a real observable. I do not know if something similar can be done for more complicated systems.

Edit: By "more complicated systems," I mean more complicated Hilbert spaces. For example, once you want to measure position and momentum, you must work in an infinite dimensional Hilbert space. The nature of self-adjoint operators in infinite dimensions is very different from finite dimensions.

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I don't follow. Why not just add I=[1,0;0,1] to your list?

[jason1990]

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I don't follow. Why not just add I=[1,0;0,1] to your list?

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Aha... Why didn't I think of that? So, for example,

2^S_x = (3/4)S_x + (5/4)I.

Both observables amount to measuring spin in the x-direction, which is 1 or -1, and mapping it to 2 or 1/2 respectively. This actually looks like an interesting way to derive a lot of matrix identities.

Does this answer my other question:

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Could we have obtained the physical interpretation of C directly from the interpretations of S_z and B? [B = 2^S_x]

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Given the above, we can write

C = S_z + 2^S_x
= S_z + (3/4)S_x + (5/4)I
= (5/4)(0.6 S_x + 0.8 S_z) + (5/4)I.

From here it is clear that C corresponds to measuring v-spin, which is 1 or -1, and mapping it to 5/2 or 0 respectively.

[jason1990]

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This actually looks like an interesting way to derive a lot of matrix identities.

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So, in general,

f(S_x) = [(f(1) - f(-1))/2]*S_x + [(f(1) + f(-1))/2]*I.

I thought this was interesting, but maybe it is trivial. I have to think on it some more.

[jason1990]

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I don't follow. Why not just add I=[1,0;0,1] to your list?

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Okay, here is one more comment on this. PairTheBoard wants a physical interpretation. Given an operator A, he wants to know the real observable quantity being measured. Without I on the list, the linear combinations look like

aS_x + bS_y + cS_z.

In this case, the quantity being measured is spin in a certain direction. With I on the list, one has for example

aS_x + bI.

The experiment you would perform that corresponds to this operator is one in which you first measure the x-spin, and then you apply the function ax + b to your measurement. So if you add I to the list and want to talk about physical interpretations, you must still introduce the idea of a function of a measured quantity, and not a measured quantity itself. Maybe there is no real distinction between the two, but my non-physicist's intuition makes me feel like there is.

Granted, even aS_x is a*(x-spin). But we can change units so that this is x-spin itself. We cannot do any unit changing trick to make a*(x-spin) + b anything other than a function of x-spin.

[PairTheBoard]

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ok. An observable is modeled by a self adjoint operator on the Hilbert Space. It seems like there would be a lot of self adjoint operators on the Hilbert Space. But there are only a few observables. Are all the self adjoint operators on the Hilbert Space some kind of observables? If not, what do they make of the ones that are not? And how do they decide which ones are and which ones aren't? If all of the self adjoint operators are observables, why are there so few self adjoint operators?

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I think this discussion could generate some semantic problems. The word "observable" means some kind of observable property of the particle or system. But it also means a self-adjoint operator on the underlying Hilbert space. (This fact alone demonstrates that they are generally believed to be in one-to-one correspondence.) To try to be clear, I will call the physical properties "real observables" and the operators either "observables" or just "self-adjoint operators."

A self-adjoint operator determines a mapping that takes a state x to a probability distribution on the real line. So your question amounts to this: given such a mapping, can we cook up an experiment whose outcome has the right distribution for each state x? I think it is generally believed that this is true. So yes, every self-adjoint operator corresponds to a real observable. In practice, maybe one can cook up examples of self-adjoint operators whose corresponding real observables are presently unknown.

But maybe not. For instance, suppose one is not concerned with position, momentum, etc., but simply wants to measure spin. The appropriate Hilbert space for this is C^2. (C is the complex numbers.) The self-adjoint operators are simply 2x2 matrices A with complex entries such that A is equal to its own conjugate transpose. There are three "primitive" real observables: the spin in the x, y, and z directions. Their matrices are given by

S_x = [0 1; 1 0],
S_y = [0 -i; i 0], and
S_z = [1 0; 0 -1].

(There is actually a constant of h/4pi in front of each of these, where h is Planck's constant.) We can generate other (real) observables by taking real linear combinations of these operators. For example, S_x + S_y would correspond to spin the direction of the vector (1,1,0). (Or, perhaps more accurately, this measures sqrt{2} times the spin in that direction.) In this way, we can generate a set of observables which is a 3-dimensional vector space over the reals.

Think, however, about how to construct an arbitrary self-adjoint operator on C^2. To build such a matrix, one selects any two real numbers for the main diagonal, then any complex number for the upper right entry. The lower left entry is then determined. So the entire space of observables is a 4-dimensional vector space over the reals. It follows that not all observables can be realized as real linear combinations of the spin matrices.

The problem is resolved by noting that we are not restricted to taking linear combinations. We may also take functions of these matrices. For example, we could measure 2^{x-spin}. This is a real observable. Its corresponding self-adjoint operator is

2^[0 1; 1 0] = [5/4 3/4; 3/4 5/4].

Notice that this operator is not a real linear combination of the spin matrices. In general, every self-adjoint operator on C^2 can be written as a real-valued function of a linear combination of the spin matrices. Which function and which linear combination to use are not hard to compute. In this way, every self-adjoint operator can be associated to a real observable. I do not know if something similar can be done for more complicated systems.

Edit: By "more complicated systems," I mean more complicated Hilbert spaces. For example, once you want to measure position and momentum, you must work in an infinite dimensional Hilbert space. The nature of self-adjoint operators in infinite dimensions is very different from finite dimensions.

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wow. That is great stuff. I always wondered what they were talking about with those "Operators". So it looks like they construct the Hilbert Space according to the Primitive Physical Observables they know about. I was thinking they had some general apriori Hilbert Space and then discovered the Primitive Physical Observables they could apply to it. But of course. How would they know what that Hilbert Space was to begin with.

PairTheBoard

[m_the0ry]

Even though the uncertainty principle is most often applied when comparing position and speed, it applies to any of the conjugate pair variables.

It also says frequency and time measurements have a minimum uncertainty, mass and velocity, etc. etc...

[Metric]

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isn't it not just h but h/4pi or h-bar/2? or are you guys just simplifying it?

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Yeah, people always simplify this kind of stuff. The factor of 2 isn't the important thing (I often don't remember factors of 2 myself) -- the important thing is the concept that "simultaneously measuring two non-commuting observables" is exactly analogous mathematically to "simultaneously diagonalizing non-commuting matrices." This was one of the most profound realizations in 20th century physics.