Standard derivation and probability density

In summary, you calculate the standard derivation for the mean distance of an electron from the nucleus. You are given the mean distance (<r>), and the probability density. You integrate by parts 4 times to get -3a^2. Putting this into the standard derivation leads to the square root of minus 21.a^2/4, which gives a complex value. You find the error in the calculation and correct it.
  • #1
the riddick25
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Homework Statement


the question asks you to calculate the standard derivation for the mean distance of an electron from the nucleus.
you are given the mean distance (<r>), and the probability density

Homework Equations


delta r = sqrt (<r^2> - <r>^2)
<r> = 3.a/2

The Attempt at a Solution


We had to calculate <r^2> by integrating a given value.
after integrating by parts 4 times i ended up with -3a^2
putting this into the standard derivation lead to the square root of minus 21.a^2/4
this obviously gives a complex value.
i was wondering if this can be correct, or if the negative sign has no meaning in the calculation.
i have read from some physics books but i can't find an answer
 
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  • #2
<r^2> can never be negative (otherwise, it would imply that measuring r would tend to yield an imaginary number!). Check your calculations again.
 
  • #3
i've checked my calculations and i still get a value of -3a^2

i suppose there could be a typo somewhere, but apart from that my answer seems correct for the integral i did.
you end up with;
4/a^3 X (24/(-2/a)^5) = -3a^2

thanks anyway :)
 
  • #4
What can I say? I mean, thousands of people, students and professors alike, have done this calculation without it being negative, and even using logic itself, <r^2> cannot be negative.

I mean, all I can say is to check your calculations again, there has to be an error somewhere. Either you used the wrong wave function, or you didn't multiply by the complex conjugate, or you didn't integrate correctly, or something...

I'm sure if you put your full calculation here someone will point out to you where you went wrong. That's all I can advise.
 
  • #5
thanks for your help, i can understand it better now

i have just found the error in the calulation (this bit wasnt actucally done by me), i guess i just glanced over it and forgot to correct it


thanks again :)
 

1. What is standard deviation?

Standard deviation is a measure of how much the data values in a dataset vary from the mean or average value. It is calculated by finding the square root of the variance, which is the average of the squared differences from the mean.

2. How is standard deviation different from variance?

Standard deviation and variance are both measures of variability in a dataset, but standard deviation is the square root of the variance. This means that standard deviation is in the same units as the data, while variance is in squared units.

3. How is probability density related to standard deviation?

Probability density is a concept used in statistics to describe the likelihood of a continuous variable falling within a certain range of values. Standard deviation is used to calculate the probability density in a normal distribution, as it measures the spread of data around the mean.

4. What does a high standard deviation indicate?

A high standard deviation indicates that the data values in a dataset are more spread out from the mean, while a low standard deviation indicates that the data values are closer to the mean. In other words, a high standard deviation suggests a higher level of variability in the data.

5. How is standard deviation used in hypothesis testing?

In hypothesis testing, standard deviation is used to calculate the standard error, which is used in determining the p-value and confidence intervals. It is also used to assess the significance of differences between group means in a sample.

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