Quantum Mechanics Questions: Superposition, Bohm Mechanics

[StevieTNZ]

Hey there,

I am very interested in Quantum Mechanics and its philosophical implications. Here are a few questions:

1. What is the accepted definition of superposition? If we used an object, for example, and its position, a superposition would say it is in two or more places at once? Would those 'position values' be real, as in the object actually existing in two different places, or would the wavefunction permit no object existing at any position - then the wavefunction would merely describe potential positions an object could be at?

2. I see Bohm Mechanics has been up for discussion a lot on this forum. I've seen a lot of articles about non-local realism and inequalities testing such a notion. Has Bohm Mechanics actually be falisfied (e.g. by the Before-Before experiment)? I saw an article on Nature entitled 'An Experimental Test of Non-Local Realism' [Sept. 2007] but was the defined realism in that article the same kind of realism Bohm Mechanics is implying?

Thanks for any answers on either question! :)

 

[Fredrik]

1. What is the accepted definition of superposition?

The least mathematical answer I can give you is that if A and B are two possible results of a measurement, and the system has been prepared in a state such that the theory predicts that the probabilities P(A) and P(B) of getting the results A and B respectively, satisfy neither (P(A)=0 and P(B)=1) nor (P(A)=1 and P(B)=0), then the system is said to be in a superposition of the states associated with A and B.

What I mean by "the state associated with X" is the state the system is in after going through a preparation procedure that guarantees that the result will be X, i.e. a preparation procedure such that P(X)=1.

If we used an object, for example, and its position, a superposition would say it is in two or more places at once? Would those 'position values' be real, as in the object actually existing in two different places,...

It's important to understand that the theory doesn't answer such questions. The so-called "interpretations" of quantum mechanics are attempts to answer them. Different interpretations may give you different answers. Note that interpretations aren't theories. They are simply additional assumptions tacked onto the theory that don't change the theory's predictions. They are strictly speaking not a part of science.

There are many-worlds interpretations in which the particle can be said to be in both places. Some would say that "objective collapse" interpretations also describe the particle as being in two places, and they're probably right about that, but I doubt that any of those interpretations is even consistent.

I think the most useful way to think about QM is as a set of rules that tells us how to calculate probabilities of possible results of experiments. Don't think of a wavefunction as a representation of what "actually happens". Think of it as a probability measure, i.e. a function you use to assign probabilities to possibilities.

...then the wavefunction would merely describe potential positions an object could be at?

In quantum mechanics, the mathematical representation of "either here or there" is different than "superposition of here and there", and they lead to different predictions. So we can say with certainty that it's not the case that "superposition of here and there" really means "either here or there".

 

[StevieTNZ]

In quantum mechanics, the mathematical representation of "either here or there" is different than "superposition of here and there", and they lead to different predictions. So we can say with certainty that it's not the case that "superposition of here and there" really means "either here or there".

Just to make sure I understand what you've posted in the quote section;
A superposition doesn't mean that the particle exists in those probabilities - merely it only show us what could happen but which don't exist until measurement?

E.g. superposition of points A and B - the particle doesn't exist at either point A or B.

 

[Fredrik]

Just to make sure I understand what you've posted in the quote section;
A superposition doesn't mean that the particle exists in those probabilities - merely it only show us what could happen but which don't exist until measurement?

E.g. superposition of points A and B - the particle doesn't exist at either point A or B.

That's right, because in terms of state operators (density matrices) "either at A or at B" would be expressed as

$$\frac{1}{2}|A\rangle\langle A|+\frac{1}{2}|B\rangle\langle B|$$

while a superposition of A and B would be something like

$$\frac{1}{2}\left(|A\rangle+|B\rangle\right)\left(\langle A|+\langle B|\right)\right)=\frac{1}{2}\left(|A\rangle\langle A|+|A\rangle\langle B|+|B\rangle\langle A|+|B\rangle\langle B|\right)$$

Never mind if you don't understand what these expressions mean. The point is that the two situations "either A or B" and "superposition of A and B" are represented by different state operators, and lead to different predictions about results of experiments.