Each shell of an atom corresponds with an energy level. Solving the following equations will reveal these energy levels. Schrodinger equation This governs the (quantum-dynamical) dynamics of fundamental particles, such as electrons. This equation is for an electron within an atom*. Solutions

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With corresponding (non-relativistic), energies

It’s a bit complicated mathematically, but the most important part is that only certain sets of integers (, , ) correspond to physically meaningful states (specifically, finite wave functions). The number is the “principal” quantum number, on which the total energy (mostly) depends. The number is the “azimuthal” quantum number, roughly describing the total contribution to energy from angular motion (analogous to classical circular orbits). The number is the “magnetic” quantum number, indicating how much the angular motion is about the z-axis in the given set of coordinates (called “magnetic”, because this motion of the electrically charged electron produces a magnetic field pointing along the z-axis.).

These are the rules:

- Any positive integer can be used.
- Any non-negative integer less that.
- Any integer between (inclusive) and can be used

The first shell and the lowest energy correspond to. This allows only one value of () and one for () according to the rules above. There is therefore only one wavefunction, or “orbital”, in the first shell: The lonely 1s orbital.

Wait, there’s more! An intrinsic spin is also available to the electron.

The value of, which can be either “up” or “down”, is a variable that can take one of two possible values. There is only one orbital in the first Shell. However, there are two distinct states. Each state corresponds to an electron with “up” or “down” spin.

The Pauli exclusion principal is the final piece.

This rule states that two electrons cannot be in the same physical status. This rule is based on a deeper truth in quantum mechanics: electrons obey Fermi statistics and are identical particles. A valid multi-electron wavefunction switches sign but retains its magnitude if you swap two electrons. Let’s say you want to write down a wavefunction where two electrons, electron 1, and electron 2, are in the exact same state.

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It is obvious that if you swap and a wavefunction will be exactly the same, which is contrary to Fermi statistics. It is impossible for two electrons to be in the exact same physical state.

We can only hold two electrons if we go back to the first shell.

*It is impossible to solve for one particle in a system with many interacting particles. However, it provides a useful first approximation that gives insight into the qualitative behavior atoms.