# 1.6H: The ground state of Hydrogen

Hydrogen is the simplest atoms, which only contains an electron and a proton. The ground state of hydrogen is the lowest allowed energy level and has zero angular momentum. However, it is the most stable state in which a single electron occupied the 1s atomic orbital.

## Introduction

The Third Law of Thermodynamics states that a system at absolute zero temperature exists in its ground state. Therefore, its entropy is determined by the degeneracy of the ground state.The ground state is the lowest energy state and the energy of the ground sate is called zero-point energy. The diameter of a hydrogen atom in its ground state is about 1 x 10-8cm. In order to provide the ground states of the hydrogen atom, we need to solve the Schrödinger equation.

## The wavefunction of the ground state of hydrogen

A hydrogen atom’s ground state wavefunction is a spherically symmetric distribution in the nucleus, in which the largest at the center and reduces exponentially at larger distance. The function is known as the 1s atomic orbital. Hydrogen has more than one ground state exists, in which is said to be degenerate.

The ground state wavefunction is ψ1s(r)=(1/π1/2a3/2)e-r/a

Since the probability density is1s(r)|2, therefore, ρ1s(r)=|ψ1s(r)|2=(1/πa3)e-2r/a

Next, we need to consider that ρ1s is in spherical coordinates dV=r2sin(φ)dr dθ dφ, and then multiplied by r2

Since ψ1s is spherically symmetric, we have to integrate over θ and φ to get the radial probability density P1s(r)=(4/a3)r2e-2r/a

Energy level of the ground state of hydrogen

In order to find out the energy of the particle, we used the equation E=h2n2/8mL2, where h is the Planck constant, n is the energy state, m is the mass of the particle, and L is the width.The ground state of hydrogen corresponds to energy level n(the principle quantum number)=1, thus, l(angular momentum quantum number)=0, ml(magnetic quantum number)=0. An electron in the ground state for hydrogen has energy -13.6eV, in which is also called the Rydberg constant.

The quantum numbers:

Principle quantum number n=1,2,3,...

Angular momentum quantum number l=0,1,2,...,n-1

magnetic quantum number ml=-l,,,,l

## References

1. Housecroft, Catherine E., and Alan G. Sharpe. Inorganic Chemistry. 3rd ed. Harlow: Pearson Education, 2008. Print.
2. Kuhn, Hans, Horst-Dieter Försterling, and David H. Waldeck. Principles of Physical Chemistry. 2nded. Hoboken, NJ: John Wiley, 2009. Print.
3. Sobolewski, Andrzej L., and Wolfgang Domcke. "Photophysics of Malonaldehyde: An ab Initio Study." The Journal of Physical Chemistry A103.23 (1999): 4494-504. Print.