# Modified Coulomb model for hydrogen, minimising error

1. Aug 13, 2013

### bobred

1. The problem statement, all variables and given/known data
The ground energy can be approximated as

$E_{1}^{(1)}\approx-\frac{4b^2}{a_0^2}E_R$

Find the largest value of b that would be consistent with the ground-state energy of a hydrogen atom that agrees with the predictions of the Coulomb model to one part in a thousand

2. Relevant equations
$E_R=13.6$eV
$a_0$ is the Bohr radius

3. The attempt at a solution
I can transpose to make b the subject but not sure about the one part in a thousand.

2. Aug 15, 2013

### Basic_Physics

It refers to the difference between the two values as a fraction of the value. The values being the ground state energy of the hydrogen atom as per the two formulas. The difference should then be no more than 1000th or 0.001 of the expected value.

For example for a value of 10.0 cm +/- 0.5 cm the uncertainty 1 in 20 of the value.

3. Aug 21, 2013

### bobred

Hi
So $13.6 \pm0.0136$ $\dfrac{0.0136}{13.6}$ would be one part in a thousand.

But how do I use that to find the maximum value of b so that $E_1$ agrees with $E_R$ to one part in a thousand?

4. Aug 21, 2013

### Basic_Physics

You have to find a value for b so that the ground-state energy of the atom according to the approximate formula, and the value from the Coulomb model do not differ by more than 1/1000. That is the difference between the two values, Δ, has to be such that

Δ/13.6 ≤ 0.001

5. Aug 21, 2013

### bobred

Of course, thanks.