# Taking the average of a formula

1. May 7, 2015

### SU403RUNFAST

1. The problem statement, all variables and given/known data
I have the total energy of a hydrogen atom E and need to take the average. The uncertainty in deltax=r so the uncertainty in deltap=hbar/2r The average values of x^2 and p^2 can be identified with the squares of the corresponding uncertainties, and the constant value of E is by definition the same as its average (what does this mean, <E^2>=(E)^2?), take the average of E and use it to estimate the minimum value of E and the minimizing value of deltap, and the corresponding value of r.

2. Relevant equations
<x^2>=(deltax)^2
<p^2>=(deltap)^2
<E>.=________________
E=K+U
K=(p^2)/2m
U=-(ke^2)/r

3. The attempt at a solution
I do not know how to take the average, i know that when i take the average of <E> the potential energy term U has an r at the bottom, and that <1/r> is not the same as 1/<r>, so I believe i must replace the r first. i know that deltar=(hbar)/(2deltap) but Im not sure what taking the average is doing to the equation. Is it changing the variables in any way? I originally took the derivative of E with respect to r and set it equal to zero, giving me r=(hbar^2)/(me^2), this is bohr's radius=.5 angstroms, and E=-13.6 eV. The kinetic energy can be rewritten as K=(hbar^2)/(2mr^2). I tried latexing the preview showed all the writing though sorry.

2. May 8, 2015

### Simon Bridge

Average of what over what?

Is this the way the problem statement was written when you got it?
What is the context? i.e. what part of physics are you currently supposed to be learning?

Taking some guesses:
$\renewcommand{\ex}[1]{\langle #1 \rangle}$
Usually: $\Delta A^2 = \ex{A^2}-\ex{A}^2$ but you seem to have a situation where $\ex p = 0$ so $\Delta p^2 = \ex{p^2}$. Same with x. Is this hydrogen atom a particle in a box?

Note: the classical equations are the averages of the quantum ones.
i.e. $$\ex{E_K} = \frac{\ex{p^2}}{2m}$$

Last edited: May 8, 2015
3. May 8, 2015

### SU403RUNFAST

E is the total energy of the hydrogen atom in my equations above (E=K+U) , i need to take the average of E. I know the first term is (deltaP^2)/2m, but the second term (U) has 1/r in it. The average of 1/r is not the same as 1 over the average of r. Do i replace r using uncertainty principle? deltxdeltap=hbar/2, where deltax=r. How do i take the average of the equation E.

4. May 8, 2015

### SU403RUNFAST

taking the average of the second part of E=K+U, i dont know how to. U=ke^2/r, what is the average of this part

5. May 8, 2015

### Simon Bridge

I cannot advise you properly because I don't have enough information.
i.e. could you take <U> as the value of the potential energy when the kinetic energy is <EK>?
It seems likely since you know E but I don't really know because you have not told me what you are doing this for.
Check your notes for similar calculations. Good luck.