Module 3 part A1

In previous discussions, we considered a single electron confined within a one-dimensional potential energy box, leading to the understanding that electrons can occupy discrete energy levels, known as energy states.
The probability density of these electrons defines the likelihood of their presence within the box relative to their energy levels.

However, real solid-state materials consist of billions of electrons, leading to complex interactions such as Coulomb interactions between electrons, nuclei, and between nucleuses themselves.
This results in a transition from individual energy states to broader energy bands, which describe ranges of energy that can be occupied by charge carriers, like electrons.

Allowed Bands: These are the ranges of energy where electrons can reside and where there are available charge carriers.
Forbidden Bands: These are the gaps where electrons cannot exist due to the lack of available energy states.
Knowledge of these energy bands in semiconductors is vital for understanding electrical conductivity, optical properties, and the functionality of solid-state devices.

When moving to a three-dimensional framework, the Schrodinger equation becomes more complex, especially when using spherical coordinates.
Solutions to this equation can be found using the method of separation of variables, yielding solutions for the radial, angular (theta), and azimuthal (phi) components of the wave function.

Each wave function has associated quantum numbers, crucial for identifying their properties:
- Principal Quantum Number (n): Indicates the energy level and takes integer values (1, 2, 3, ...).
- Orbital Quantum Number (l): Defines the shape of the orbital, ranging from 0 to n-1.
- Magnetic Quantum Number (m): Describes the orientation of the orbital, taking values from -l to +l.

The Pauli Exclusion Principle states that no two electrons can occupy the same quantum state within a system. This principle is essential for understanding electron configurations in atoms and molecules.
For example, in a one-electron hydrogen atom:
- The state is defined by quantum numbers n=1, l=0, and m=0, allowing for 2 electrons in the spin state (s = ±1/2).
- For hydrogen with n=2, the combinations increase, allowing for configurations of up to 8 electrons.

In helium, with its two electrons, the 1s orbital is completely filled, making it inert.
Neon, with 10 electrons, has fully filled 1s, 2s, and 2p orbitals, rendering it also inert.
An atom with 11 electrons (such as sodium) has a valence electron that occupies the higher energy state, making it reactive and enabling interactions with other atoms.

When analyzing the hydrogen atom, the potential function relates to the Coulomb interaction, with calculations yielding an energy of approximately 13.6 eV.
As the quantum number n increases, the energy states decrease in binding strength to the nucleus, becoming weaker for higher n values.
Probability densities can be calculated to illustrate where the electron is most likely to be found in relation to the nucleus, showcasing how these densities change with distance.

When two hydrogen atoms come close to each other, their probability functions can overlap, resulting in the splitting of energy levels according to the Pauli Principle.
The original single-electron energy states then split into doublets due to interactions, leading to a more complex energy landscape for atoms near each other.

As more atoms come together, the splitting of energy states becomes pronounced, generating a continuum of energy bands instead of distinct levels.
These bands represent the ranges within which electrons can exist, including allowed bands (1s, 2s, 2p) and forbidden bands, which are gaps between these allowed bands.
The phenomena observed in solid-state physics largely arise from these overlapping energy bands resulting from the interactions of a vast number of electrons.