# Technical difficulties in calculating hydrogen probabilities

• spaghetti3451
In summary, the question is how to differentiate the expression for the probability of finding an electron in a thin shell in a hydrogenic atom, in order to determine the radius at which the radial probability is maximum. There are two ways to approach this problem: by taking the probability for finding the electron in a thin shell and dividing it by the volume, or by calculating the mean radius. In both cases, the dR dependency drops out, and for s-wavefunctions, the dΩ integration is just a constant 4π and need not be considered.
spaghetti3451
I am trying to prove to myself that the most probable distance for a 1s electron in a H atom is the Bohr radius.

The probability of finding an electron (for any given state in a hydrogenic atom) in a spherical shell of thickness dr at a distance r from the nucleus is $\left|R_{nl}\right|^{2} r^{2} dr = \left|\psi_{nlm}\right|^{2} 4\pi r^{2} dr$.

Given this, I need to differentiate this expression wrt r to obtain the radius at which the radial probability is maximum. This is where I face the problem. How do I differentiate either of the above expressions when there's a dr in each of them?

There are several different ways to look at this problem.

1) Take probability for finding the electron in a thin shell

$$P_{nlm}(R,dR) = \int_\Omega d\Omega \int_R^{R+dR} dr\,r^2\,|\psi_{nlm}|^2$$

devide by the volume

$$V(R,dR) = \int_\Omega d\Omega \int_R^{R+dR} dr\,r^2$$

$$p_{nlm}(R) = \frac{P_{nlm}(R,dR)}{V(R,dR)}$$

and determine the maximum via

$$\frac{\partial p_{nlm}(R)}{\partial R} = 0$$

I think this is what you have in mind. You will find that (when deviding by V) the dR dependency drops out as expected.

2) Calculate the mean radius (or the more general case of the average value for rn)

$$\langle r^n \rangle = \int_\Omega d\Omega \int_0^\infty dr\,r^{2+n}\,|\psi_{nlm}|^2$$

Of course in both cases for nlm=n00, i.e. for s-wavefunctions, the dΩ integration is just a constant 4π and need not be considered.

## 1. What are technical difficulties in calculating hydrogen probabilities?

Technical difficulties in calculating hydrogen probabilities refer to challenges or obstacles that scientists face when trying to determine the likelihood of hydrogen atoms or molecules in a given system or reaction. This could include issues with data collection, analysis, or interpretation.

## 2. Why is it important to accurately calculate hydrogen probabilities?

Accurately calculating hydrogen probabilities is crucial for understanding various chemical and physical processes, such as the formation of molecules, reactions, and energy transfer. It also has practical applications in fields such as material science, renewable energy, and pharmaceutical research.

## 3. What factors can affect the accuracy of hydrogen probability calculations?

Some factors that can impact the accuracy of hydrogen probability calculations are experimental errors, limitations of computational methods, and incomplete data. The complexity and unpredictability of hydrogen bonds, which are crucial in determining probabilities, can also contribute to inaccuracies.

## 4. How do scientists overcome technical difficulties in calculating hydrogen probabilities?

Scientists use a combination of experimental techniques, such as spectroscopy, and computational methods, such as molecular dynamics simulations, to overcome technical difficulties in calculating hydrogen probabilities. They also constantly refine and improve these methods to increase accuracy.

## 5. Can technical difficulties in calculating hydrogen probabilities be completely eliminated?

While advancements in technology and methods have greatly improved the accuracy of hydrogen probability calculations, it is unlikely that all technical difficulties can be completely eliminated. This is due to the inherent complexity of hydrogen bonds and the dynamic nature of chemical systems. However, scientists continue to strive for more accurate and reliable methods to understand the behavior of hydrogen in various environments.

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