# Probability in Quantum Mechanics

• Japa
In summary: Superposition is when a particle or wave is in more than one state at the same time. The uncertainty principle is the principle that we can't measure both the position and momentum of a particle with absolute precision. The observer effect is the fact that the behavior of a system changes when it is observed, this is most famously illustrated by the double slit experiment.
Japa
I'm a biology student, but I take some interest in physics and have read 2 of Stephen Hawking's divulgation books about modern physics and often stop to read something about it in wikipedia. Wich has revealed to obviously not be enough to understand some characteristics of Quantum Mechanics.

Even knowing the answers are on the article I'm reading on wipedia, I'm not able to understand it well, probably because they are written by physicists and therefore to some point for physicists. So I searched for some physics forum where maybe I could get some more friendly explanations. :P Can anyone help me?

The main matter that annoys me about Quantum Mechanics and to many other people it seems, is the apparent randomicity or probability it brings, making the determinable universe I always liked to think of impossible. But I'm not able to understand where this probability comes from.

I can understand that through the uncertainty principle, precise measures for both of a pair of characteristics for a particle or wave become unreachable for us, but that doesn't means the system itself doesn't have definite values, does it?

I can understand the uncertainty principle through the observer effect, observing changing the observed, but it seems that's not quite the nature of the principle. I read somewhere you could get a measure observing some particle entangled to another, getting information on it without disturbing it.
However, does a particle get entangled for position or momentum too, I've always seen examples for spin? Is there a pair for uncertainty also for spin?
But if measures can be made undirectly, where does the uncertainty arise from? Can't one measure both position and momentum for it precisely undirectly like that, without uncertainty?

Initially well defined states for wavefunctions start getting less precise over time, why is that? From wikipedia, it seems it is possible to measure else position or momentum quite exactly without getting the other measure. Does the lack of a measure in position influence the state of momentum over time?

Wavefunctions describe reality through probability values over time. But why does reality HAVE to coincide with the wavefunctions, why can't they be merely an aproximation of our reality due to unknown variables?

In wikipedia it says John Bell has theorized creating a quite reasonable theorem saying that if there were unknown variables, they should equally affect all particles, creating a correlation and therefore being detectable. But couldn't the variables have different states for different particles, creating different tendencies? Or maybe the same state on different particles being affected by similar but non equal conditions on the system?

My main question would be why does quantum mechanics have to suppose a probabilistic universe. And why do physicists seek for so unnatural explanations to justify it in the different interpratations of quantum mechanics. Why can't quantum mechanics be just the model to describe a system and the wave function collapse be origined on its true nature revealed.

I manage to see ways out of it in my personal view, but I realize they're due to my limited knowledge on it.

Sorry for the large amount of (likely stupid) doubts I've brought. And also for any english mistakes/incoherences.

I'll be really thankful if anyone can answer any of those questions.

Last edited:
Puzzled biology student said:
I'm a biology student, but I take some interest in physics and have read 2 of Stephen Hawking's divulgation books about modern physics and often stop to read something about it in wikipedia. Wich has revealed to obviously not be enough to understand some characteristics of Quantum Mechanics.

Even knowing the answers are on the article I'm reading on wipedia, I'm not able to understand it well, probably because they are written by physicists and therefore to some point for physicists. So I searched for some physics forum where maybe I could get some more friendly explanations. :P Can anyone help me?

The main matter that annoys me about Quantum Mechanics and to many other people it seems, is the apparent randomicity or probability it brings, making the determinable universe I always liked to think of impossible. But I'm not able to understand where this probability comes from.

My main question would be why does quantum mechanics have to suppose a probabilistic universe. And why do physicists seek for so unnatural explanations to justify it in the different interpratations of quantum mechanics. Why can't quantum mechanics be just the model to describe a system and the wave function collapse be origined on its true nature revealed.

The following should clear up your question and hopefully answer some others.

http://physicsweb.org/articles/world/15/9/1

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/DoubleSlit/DoubleSlit.html"

Firstly you need to understand superposition, the above account of the two slit and the explanation should clear up a few things.

Wavefunctions describe reality through probability values over time. But why does reality HAVE to coincide with the wavefunctions, why can't they be merely an aproximation of our reality due to unknown variables?

Consider the principles of superposition and decoherence, that lie at the heart of the http://plato.stanford.edu/entries/qm-copenhagen/" this should clear up some of the issues you have about probability and also reveal why matter in essence appears undeterministic.

A wave function is exactly what you say, a mathematical expression of something we cannot directly see, so it is not meant to be interpreted as a "snapshot" of reality, more as an approximation of what happens experimentally.

Bohr thought of the atom as real. Atoms are neither heuristic nor logical constructions. A couple of times he emphasized this directly using arguments from experiments in a very similar way to Ian Hacking and Nancy Cartwright much later. What he did not believe was that the quantum mechanical formalism was true in the sense that it gave us a literal (‘pictorial’) rather than a symbolic representation of the quantum world. It makes much sense to characterize Bohr in modern terms as an entity realist who opposes theory realism (Folse 1987). It is because of the imaginary quantities in quantum mechanics (where the commutation rule for canonically conjugate variable, p and q, introduces Planck's constant into the formalism by pq - qp = ih/2π) that quantum mechanics does not give us a ‘pictorial’ representation of the world. Neither does the theory of relativity, Bohr argued, provide us with a literal representation, since the velocity of light is introduced with a factor of i in the definition of the fourth coordinate in a four-dimensional manifold (CC, p. 86 and p. 105). Instead these theories can only be used symbolically to predict observations under well-defined conditions. Thus Bohr was an antirealist or an instrumentalist when it comes to theories.

The true nature of light waves are still a mystery, although we can infer it's properties from various experiments which attempt to measure where the wave isn't so as to leave the particle un-decohered. And of course the principals of superposition are aptly demonstrated by Feynmans two slit experiment, involving single photons.

Another interesting point is how do we measure say a photon? We need to use something to detect it, by doing so though we not only destroy its coherence but also impart energy into the system, thus we can know it's position but not it's momentum as the detection has changed the particle/wave in question.

I'm sure others are better able to answer some of your other questions, but hopefully the above links should resolve some of your issues.

Also a very simple explanation of Bell's theorem, courtesy of Dr Chinese.

http://drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm

http://drchinese.com/David/Bell_Theorem_Easy_Math.htm

A discussion of the EPR experiment and Bell's theorem.

Once you have all these under your belt, it would also be a good idea to do a search on Particle wave duality etc, some of the discussions here should give you a good picture of the often confusing world of quantum mechanics.

Some wryly humorous quotes from the Father of the Copenhagen Interpretation: Niels Bohr.

There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.

If anybody says he can think about quantum physics without getting giddy, that only shows he has not understood the first thing about them.

If quantum mechanics hasn't profoundly shocked you, you haven't understood it yet.

We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that it is not crazy enough.

Stop telling God what to do with his dice.

* A response to Einstein's assertion that "God doesn't play dice"

No, no, you're not thinking; you're just being logical.

How wonderful that we have met with a paradox. Now we have some hope of making progress.

Niels Bohr

Last edited by a moderator:
Japa said:
I'm a biology student, but I take some interest in physics and have read 2 of Stephen Hawking's divulgation books about modern physics and often stop to read something about it in wikipedia.

I would suggest S.Gasiorowicz “Quantum Physics” instead. In addition, consider receiving degree(s) in physics; it seems, it is very difficult to do something in biology without it.

Japa said:
The main matter that annoys me about Quantum Mechanics and to many other people it seems, is the apparent randomicity or probability it brings, making the determinable universe I always liked to think of impossible. But I'm not able to understand where this probability comes from.

In physics, the unjustified statements are not required to be confirmed experimentally by the obvious reason (it is impossible). Your statement is correct, therefore, it should be demonstrated unambiguously. It is not performed yet.

Japa said:
I'm a biology student, but I take some interest in physics and have read 2 of Stephen Hawking's divulgation books about modern physics and often stop to read something about it in wikipedia. Wich has revealed to obviously not be enough to understand some characteristics of Quantum Mechanics.

You're in good company. That's now already 80 years that nobody understands these issues perfectly !

You have to know that at its barest level, quantum mechanics is a formal theory (that means, mathematics and some intuition you'll need to make the relationship between the mathematical inputs and outputs, and "things in the lab") for which you GIVE some kind of "initial state" (however, not as with classical mechanics ; rather, a specification of what kind of "result of measurement you are sure of), and which will spit out a probability distribution of what you are going to measure.

You can stop there - many people do. The price to pay is that you have no clue of what's "really going on". This attitude is called the "shut up and calculate" approach.

Or you can try to "understand" the stuff. And then you're in trouble, because that's what people have been trying to do for more than 80 years now, without conclusive success. That doesn't mean that no progress has been made on the question! But one thing is sure: there is no consensus, and the reason is that no story is entirely satisfactory from all PoV.

You enter now the murky waters of "the interpretation of quantum theory", and "the measurement problem".

The main matter that annoys me about Quantum Mechanics and to many other people it seems, is the apparent randomicity or probability it brings, making the determinable universe I always liked to think of impossible. But I'm not able to understand where this probability comes from.

Actually, it is not the "randomness" in itself that is giving problems. In fact, even though *in principle* classical mechanics is deterministic, the fact that it depends upon the real number system implies that *in practice* it is also stochastic, given that we can never ever specify initial conditions "correctly" (that is, by assigning one and unique real number to each coordinate). We will always have to specify an approximate initial condition, and chaotic systems then lead automatically to totally stochastic behaviour after some time.

What is so disturbing in quantum theory is that fact that the randomness is not "ignorance randomness". It is not (as Einstein thought it had to be) so that it is our ignorance of a certain reality which makes that the outcomes can only be known stochastically (just as in the case of classical chaotic dynamics, only, with a fixed boundary). It seems to be some kind of "intrinsic randomness" which "pops up out of nothing".

I can understand the uncertainty principle through the observer effect, observing changing the observed, but it seems that's not quite the nature of the principle. I read somewhere you could get a measure observing some particle entangled to another, getting information on it without disturbing it.
However, does a particle get entangled for position or momentum too, I've always seen examples for spin? Is there a pair for uncertainty also for spin?

There is no difference in principle between the spin uncertainty principle (the 3 components of angular momentum are a bit like the position and momentum in this respect) and the position/momentum uncertainty principle. But spin is simply easier, both experimentally and theoretically (finite-dimensional spaces etc...)

But if measures can be made undirectly, where does the uncertainty arise from? Can't one measure both position and momentum for it precisely undirectly like that, without uncertainty?

There's always something that will bite you !

Wavefunctions describe reality through probability values over time. But why does reality HAVE to coincide with the wavefunctions, why can't they be merely an aproximation of our reality due to unknown variables?

That's what Einstein also thought, that the stochastic character of quantum theory is like any other stochastic description, namely an "ignorance description". In other words, there's something missing in our description of reality, and that something determines what's actually going on. Quantum theory describes us the statistics of that something, but without giving us the details. Such an approach is called a "hidden variable" interpretation.
It doesn't work. Well, it does work, but not as expected.

There are 3 steps in this "discovery".

The first one was a failure. von Neuman thought he had the proof, once and for all, that there is no "ignorance description" possible. He thought he had a mathematical proof of the impossibility of rendering the stochastic behaviour of quantum theory by a set of hidden variables. It turned out that he made a mistake in his proof.

The second step was Bohm. Bohm devised a hidden variable theory, Bohmian mechanics, which is stochastically equivalent to quantum theory. (as such, his theory was a counter example to von Neumann's erroneous "theorem"). Bohmian mechanics is an interesting theory in its own right. It merits to be studied.

However, Bohmian mechanics has a problem. It uses "action at a distance" (as in Newtonian gravity) in a very peculiar way: particles on Alpha Centauri can influence particles in my lab in just as much a way as particles nearby. This is a bit strange - but ok. However, what is really annoying with Bohmian mechanics is that the way that this "action at a distance" works, means that it cannot be a relativistically correct theory.
(it can reproduce relativistically correct statistical predictions if this is put in by hand, but the inner workings do not respect the basic postulates of relativity).

There's a lot of semantic wars over this, but the short statement is that Bohmian mechanics is not "local" (which is a necessary condition for a theory to be in agreement with the spirit of relativity).

Now, this was encouraging. Maybe it would be possible to formulate a theory, similar to Bohmian mechanics, which IS local. Bell set himself out to try to find this. Unfortunately, he discovered himself that this is impossible. That is the content of Bell's theorem: no LOCAL hidden variable theory (in the spirit of Bohmian mechanics or other) can ever reproduce all the stochastical predictions of quantum theory.

This is von Neumann's theorem, but with an extra assumption, namely, that the hidden variable theory respects relativity in its inner workings.

There.

That's where we are.

Some people say that, if this is the case, then we should just kick relativity in the butt. They work on refinements of Bohmian mechanics. Others say that this is so ridiculous, that the statistical predictions of quantum theory must be wrong. Unfortunately, many experiments have been done that suggest that these predictions are correct. I am careful in the wordings, because there are some people who try to pick apart all of these experiments, and suggest that not everything is 100% sure (the detectors could work differently than expected etc...). Let's just say that the margin for this is small and decreasing.

Other people try to make sense of quantum mechanics as it is. I could go into that, but I would seriously suggest that, if you're interested in this, that you FIRST learn quite well the formal side of quantum theory (and keep yourself happy with the "shut up and calculate" approach), because to understand the reasons and subtleties of each of these attempts, you'll need in any case to be fluent with the formalism.

But just know that your question is legitimate, and that there simply is no easy answer you might have skimmed over.

## 1. What is probability in quantum mechanics?

In quantum mechanics, probability refers to the likelihood of a particular outcome or measurement occurring in a quantum system. It is represented by a complex number known as the probability amplitude, which is squared to give the actual probability of the outcome.

## 2. How is probability used in quantum mechanics?

Probability is used to describe the behavior of subatomic particles and their interactions in a quantum system. It allows us to make predictions about the outcomes of experiments and measurements on these particles.

## 3. What is the role of uncertainty in quantum mechanics?

Uncertainty is a fundamental aspect of quantum mechanics and is related to the probabilistic nature of the theory. According to the Heisenberg uncertainty principle, it is impossible to know both the position and momentum of a particle with absolute certainty.

## 4. How is probability different in classical mechanics and quantum mechanics?

In classical mechanics, probability can be described by a single value, whereas in quantum mechanics it is represented by a probability distribution. Additionally, in classical mechanics, probabilities are determined by the initial conditions of a system, while in quantum mechanics they are determined by the wave function of the system.

## 5. How does the concept of superposition relate to probability in quantum mechanics?

In quantum mechanics, superposition refers to the ability of a quantum system to exist in multiple states at the same time. This is related to probability in that the probability of a particular outcome is determined by the superposition of all possible states of the system.

• Quantum Physics
Replies
12
Views
643
• Quantum Physics
Replies
17
Views
1K
• Quantum Physics
Replies
6
Views
352
• Quantum Physics
Replies
18
Views
2K
• Quantum Physics
Replies
12
Views
1K
• Quantum Physics
Replies
10
Views
1K
• Quantum Physics
Replies
7
Views
1K
• Quantum Physics
Replies
6
Views
1K
• Quantum Physics
Replies
11
Views
1K
• Quantum Physics
Replies
3
Views
841