# Uncertainty in expectation value.

• swain1
In summary, when finding the uncertainty in position for the expectation value, it is necessary to also find the value of <r^2>. This can be done by integrating r^2 p(r) dr instead of rp(r)dr. There is a general form of integration between 0 and infinity for functions of r, such as the gaussian integral int[exp(-ax^2)] a>0. However, this integral cannot be evaluated directly and requires some substitutions. The solution is usually remembered as sqrt(pi/a), and for x^2n exp(-ax^2), it is found by differentiating both sides with respect to a. The calculated uncertainty for the ground state of hydrogen is sqrt(3)a/2, which may
swain1
When trying to work out the uncerainty in position of the expectation value I have read that you have to find <r^2> as well as <r>^2. I have worked out the value of 3a/2 for <r> but what do I have to do to find <r^2>. Do I just sqare the whole function before I integrate?
Also as I am integrating I found, in a book a table that had a general form of integrations between 0 and infinity. I used it without giving it much thought but how would you evaluate this? Thanks

Yes to find <r^2> you integrate r^2 p(r) dr instead of rp(r)dr. Or for any function of r, it's just int[f(r)p(r)dr].

I'm guessing the table was for the gaussian interal int[exp(-ax^2)] a>0? The integral can't be evaluated directly, you have to do some weird substitutions, if you want to know how to do it I can show you. It's easiest to remember the the soln is sqrt(pi/a), from 0 to infinity is half this, and for x^2n exp(-ax^2), differentiate both sides wrt a.

Ok thanks for that. I have done the inegral and worked out the uncertainty. The value I have got is sqrt(3)a/2. It seems like an awful lot to me. Anyway, do you know if this is right. I am doing this for the ground state of hydrogen. If I have got the correct value. Why is it so large?

Not entirely sure, what exactly was the integral? And what is a? I haven't done any quantum, not sure what this is about exactly.

$$\Delta r = \sqrt{<r^2> - <r>^2}$$

## 1. What is the concept of "uncertainty" in expectation value?

Uncertainty in expectation value refers to the variability or unpredictability in the outcome of a measurement or experiment. It is the difference between the actual value of a physical quantity and the average value that is expected based on a given probability distribution.

## 2. How is uncertainty in expectation value calculated?

Uncertainty in expectation value is calculated using the standard deviation of a probability distribution. This involves taking the square root of the variance, which is the average of the squared differences between each value and the mean. The larger the standard deviation, the greater the uncertainty in the expectation value.

## 3. How does uncertainty in expectation value impact scientific measurements and experiments?

Uncertainty in expectation value can have a significant impact on the accuracy and reliability of scientific measurements and experiments. It reflects the limitations and potential errors in the measurement process, and can affect the interpretation and validity of results.

## 4. Can uncertainty in expectation value be reduced or eliminated?

No, uncertainty in expectation value cannot be completely eliminated. It is an inherent aspect of measurement and is affected by factors such as instrument precision, sample size, and random fluctuations. However, it can be minimized by using more precise instruments and increasing the sample size.

## 5. How do scientists account for uncertainty in expectation value in their research?

Scientists use various statistical methods and techniques to account for uncertainty in expectation value in their research. This includes calculating and reporting the standard deviation or error of measurements, conducting repeated experiments, and performing statistical analyses to determine the confidence level or probability of error in their results.

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