# Statistics and Heisenberg

• I
• Erribert
In summary, the conversation delves into the application of statistics in quantum mechanics, particularly in relation to the Uncertainty Principle. The conversation touches on the definition of population and how it relates to the HUP, as well as the role of statistics in predicting future measurements. The expert also clarifies the misconception that statistics can be applied to a singular individual or event.
Erribert
to begin with I am a biophysicist so my question is very naive.

It is my understanding that the Uncertainty Principle deals with a single event (particle). It is also my understanding that quantum physics contains a lot of statistics (probability).

The question is: in the area of probability, a population is required and the equation contains a denominator of n-1. From this I assume that statistics of this kind cannot apply to a “singularity”; that is, a singular thing. For example statistics does not apply to a singular individual.

If the Heisenberg Uncertainty Principle applies to a singular “particle”, what kind of statistics is used, if any?

Erribert said:
The question is: in the area of probability, a population is required and the equation contains a denominator of n-1.
Doing statistics does require defining a population, but there is no requirement that there be an equation involving a denominator of n-1. Perhaps you are thinking about a very specific application of statistics - the estimation of population variance? Perhaps when you mention "n" you are thinking about "sample size" and not "population" size. Populations in statistics need not be finite populations.

The statistical interpretation of the Heisenberg Uncertainty principle does involve assuming we perform independent repeated experiments. The population defined by the experiments is more vague than a specific finite population such as "the population of citizens of the USA". Nevertheless, it is conceptually a population.

All predictions of quantum mechanics are probabilistic or statistical.

Quantum mechanics can be interpreted as applying to individual preparations, in the sense that statistical results are built up from many individual results.

vanhees71
Erribert said:
to begin with I am a biophysicist so my question is very naive.

It is my understanding that the Uncertainty Principle deals with a single event (particle). It is also my understanding that quantum physics contains a lot of statistics (probability).

The question is: in the area of probability, a population is required and the equation contains a denominator of n-1. From this I assume that statistics of this kind cannot apply to a “singularity”; that is, a singular thing. For example statistics does not apply to a singular individual.

If the Heisenberg Uncertainty Principle applies to a singular “particle”, what kind of statistics is used, if any?

That appears to be an erroneous understanding of the HUP here. Let's start from the very beginning.

Say you have two observables, A and B, which have the HUP relation:

ΔA ΔB = ħ/2

Now, what are ΔA and ΔB?

Statistically, these are equivalent to the standard deviation, i.e. we define each one as:

ΔA = √(<A2> - <A>2)

and similarly for ΔB. I will assume that you know what those angled-brackets are, since you appear to know statistics.

So already, from this, applying the HUP to ONE single measurement of A and B makes no sense, because there is no statistical spread in just one measurement! The HUP tells you the spread in measurement of observable A and B over many, many measurements.

But does that mean that it is completely irrelevant in a single measurement? No. As I've stated in my Insight article, once you know the HUP relation, then it tells you how well you are able to predict the NEXT measurement. So if ΔA has a huge spread, then your ability to predict what the next measurement of A will yield will not be very good.

This is nothing more than standard statistics. The major difference here from classical system is that now, two different observables are somehow entertwined. The more you are better able to predict the outcome of one observable, the less accurate you are to predict the outcome of the other observable. This is the QM part.

Zz.

Bandersnatch and vanhees71
Thanks

Stephen Tashi said:
Doing statistics does require defining a population, but there is no requirement that there be an equation involving a denominator of n-1. Perhaps you are thinking about a very specific application of statistics - the estimation of population variance? Perhaps when you mention "n" you are thinking about "sample size" and not "population" size. Populations in statistics need not be finite populations.

The statistical interpretation of the Heisenberg Uncertainty principle does involve assuming we perform independent repeated experiments. The population defined by the experiments is more vague than a specific finite population such as "the population of citizens of the USA". Nevertheless, it is conceptually a population.

Thanks. My point was that statistics cannot be applied to a population. The statistics of car crashes have no bearing on a single individual driving a car. To believe that it does means that individual does not know much about statistics :-). “There are lies, damn lies, and statistics”.

My graduate advisor told me one day: “If you have to use statistics, then the study is not important”. We all know that statistics are manipulative. Why develop a science that is based on it?

Okay, so quantum wave collapse in a multiverse... That is mysticism. Don’t get me wrong, I love what the mystics come up with. It is most entertaining.

In my opinion,
Cheers

Erribert said:
quantum wave collapse in a multiverse... That is mysticism

This is completely irrelevant to the topic of this thread. Please stay on topic.

Erribert said:
My point was that statistics cannot be applied to a population.
huh? Statistics are EXACTLY what can be applied to a population. They are in fact ABOUT populations, not about individuals. Either you have this backwards or I'm really misunderstanding you.

russ_watters
PeterDonis said:
This is completely irrelevant to the topic of this thread. Please stay on topic.

Actually I believe it was right on target. I was alluding to the replacement of reality with descriptive equations of such. I apologize if I wasn’t clear.

Still, I don’t understand how probably can apply to a single particle. In wave collapse, a billion probabilities are condensed into a singular reality. Would this not take probability out of the description of this reality?

I am trying to ask the same question, just in a different way.

I apologize for the confusion.

Cheers

phinds said:
huh? Statistics are EXACTLY what can be applied to a population. They are in fact ABOUT populations, not about individuals. Either you have this backwards or I'm really misunderstanding you.

I apologize, my bad. I meant that statistics cannot be applied to a population of one.

My goal is to try to understand whether the Heisenberg Uncertainty Principle is applied to a single particle. I am certainly no expert in that. I do have a good understanding of statistics, but I have not (yet) gone through the derivation of the HUP. Perhaps I should have done that before asking. I was being lazy by approaching the forum prematurely.

Thank you for the catch.

Erribert said:
I was alluding to the replacement of reality with descriptive equations of such. I apologize if I wasn’t clear.

You certainly weren't. All theories of physics involve "replacement of reality with descriptive equations". That's how we build physical models that make predictions. If that's "mysticism", then all physics is mysticism.

Erribert said:
My goal is to try to understand whether the Heisenberg Uncertainty Principle is applied to a single particle.

PeterDonis said:

Thank you. I just had a phone conversation with an old friend who recently retired from particle physics. He explained to me what I wished to know. I can now drop this thread.

All the best to you in your research.

PeterDonis said:
You certainly weren't. All theories of physics involve "replacement of reality with descriptive equations". That's how we build physical models that make predictions. If that's "mysticism", then all physics is mysticism.

I am an experimental scientist. I believe in the scientific method that Newton used to refine his theories. So, no, I do not consider that mystical. I misspoke. I should have used the word theoretical.

Throughout my long life, theories in physics have constantly changed through experimentation. That is my style of research. It’s the constant change that makes science interesting to me. “Today’s science will be tomorrow’s mythology”.

I do respect your point of view. I was told to temper my language. I do not like negative confrontation, and will do my best to keep my replies to those posts that are constructive.

Every forum has its rules. I need to abide by such rules.

Best

Erribert said:
I misspoke. I should have used the word theoretical.

Yes, "theoretical" would be fine.

Erribert said:
The question is: in the area of probability, a population is required and the equation contains a denominator of n-1.
In statistics, one should distinguish population variance from sample variance, as explained e.g. in
http://www.macroption.com/population-sample-variance-standard-deviation/
One of them has ##n## in the denominator, while the other has ##n-1## in the denominator. In quantum mechanics one uses the variance with ##n##, so technically there is no problem with ##n=1##. But in practice one usually uses a large ##n##, so the difference between two variances is negligible.

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Demystifier said:
In statistics, one should distinguish population variance from sample variance, as explained e.g. in
http://www.macroption.com/population-sample-variance-standard-deviation/
One of them has ##n## in the denominator, while the other has ##n-1## in the denominator.
Some people define "sample variance" to have an n in the denominator. (e.g. http://mathworld.wolfram.com/SampleVariance.html )

However, since there is a difference between "n" as sample size and "n" as population size, the distinction you mention between "sample variance" and "population variance" still exists.

Erribert said:
If the Heisenberg Uncertainty Principle applies to a singular “particle”, what kind of statistics is used, if any?
The Heisenberg uncertainty principle does not apply to a single particle as such. It applies to many repeated measurements on the same object, where the object may be a particle or something else.

Since you are a biophysicist, let me take an analogy from biology. Instead of a particle, let us consider a chicken. One chicken. You never measure all properties of the chicken at once. You measure one property of the chicken, e.g. it's weight. Each time you measure it's weight, you may get a slightly different number (perhaps because your weight scale is not perfect). So if you measure it ##n## times, you get ##n## weights ##x_1##, ..., ##x_n##. From this you can compute the mean and the variance by formulas you probably already know. The variance is computed with ##n## (not ##n-1##) in the denominator.

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## 1. What is the relationship between statistics and Heisenberg's uncertainty principle?

The relationship between statistics and Heisenberg's uncertainty principle lies in the fact that both concepts deal with uncertainty and unpredictability. Statistics is a branch of mathematics that deals with the collection, analysis, and interpretation of data, while Heisenberg's uncertainty principle is a fundamental principle in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a particle. This principle is often applied in statistical analysis of quantum systems.

## 2. How does Heisenberg's uncertainty principle impact statistical experiments?

Heisenberg's uncertainty principle has a significant impact on statistical experiments because it introduces a fundamental limit to the precision with which certain physical quantities can be measured. This means that in statistical experiments involving quantum systems, there will always be a degree of uncertainty and unpredictability in the results, making it challenging to obtain exact and precise measurements.

## 3. Can statistics be used to explain Heisenberg's uncertainty principle?

No, statistics cannot fully explain Heisenberg's uncertainty principle. While statistics can provide a mathematical framework for analyzing and interpreting data related to quantum systems, it cannot fully explain the underlying principles of quantum mechanics. Heisenberg's uncertainty principle is a fundamental principle of quantum mechanics that arises from the wave-particle duality of matter, which cannot be fully explained by statistics.

## 4. How do scientists use statistics in relation to Heisenberg's uncertainty principle?

Scientists use statistics in various ways in relation to Heisenberg's uncertainty principle. One common application is in the analysis of quantum systems, where statistical methods are used to interpret the data and make predictions about the behavior of particles. Additionally, statistics is also used to study the effects of Heisenberg's uncertainty principle on different physical systems and to better understand the limitations it imposes on measurement precision.

## 5. Are there any real-world applications of statistics and Heisenberg's uncertainty principle?

Yes, there are several real-world applications of statistics and Heisenberg's uncertainty principle. One example is in the development of technologies such as quantum computing, which relies on the principles of quantum mechanics and statistical analysis to process information. Additionally, the principles of Heisenberg's uncertainty principle are also applied in fields such as cryptography, where it is used to ensure the security of data transmission and communication.

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