H Atom Energy Levels

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Any electron associated with an atom has a wavefunction that describes its position around the nucleus as well as an energy. This energy consists of two components: kinetic and potential energies. The kinetic energy is a consequence of the electron having mass and moving at a certain speed. The potential energy is a result of the electrostatic, or Coulombic, force that attracts oppositely charged particles (i.e. the electron and the proton nucleus). The hydrogen atom energy levels can be obtained through solving the Schrödinger Equation.

Introduction

Using the power series method, the solution of the hydrogen atom Schrödinger Equation leads to quantized energy levels

\[ E_n=-\dfrac{m_ee^4}{8\epsilon{_0}^2h^2n^2}=-\dfrac{m_ee^4}{32\pi{^2}\epsilon{_0^2}\hbar{^2}n^2}\;\;n=1,2,..\]

This energy can be written in terms of the Bohr radius, \( a_0\):

\[ E_n=-\dfrac{e^2}{8\pi{^2}\epsilon{_0}a_0n^2}\;\; n=1,2,...\]

The energy is negative due to the attractive nature of the Coulombic interaction. This is alternatively visualized as an atom whose electron has been moved infinitely far away. The potential energy of the electron is defined as zero as there is no interaction at infinite distance. As the electron approaches the nucleus, the amount of potential energy decreases as the electron gets nearer to the nucleus.

One interesting feature of the energy levels are that they are not spaced evenly, but rather the spacing diminishes exponentially as \( E_n\propto 1/{n^2} \). The limit of these energies as n approaches infinity is the dissociation energy, the point at which the electron escapes the Coulombic pull of the nucleus.

Another important aspect to note is that the energy is solely dependent on the principal quantum number n, hence the 3s, 3p and 3d orbitals all have the same energy. This is not the case for multi-electron atoms, where the s, p, or d orbitals in a given n level have different energies.

References

McQuarrie, Donald A. Quantum Chemistry. 2nd ed. United States Of America: University Science Books, 2008. 321-24.

Problems

1. How much energy is required to excite the hydrogen electron from its ground state to the first excited state?

2. According to quantum mechanics, what is the dissociation energy of an H atom?