3.1 Hydrogen atom and quantum numbers

Quantum Mechanical Model of Hydrogen

The hydrogen atom is the simplest atomic system, with just one proton and one electron. Solving the Schrödinger equation for this system gives us atomic orbitals: mathematical functions that describe where the electron is likely to be found. This solution also explains why atoms emit light at only specific wavelengths, producing discrete spectral lines rather than a continuous spectrum.

Schrödinger Equation and Atomic Orbitals

The Schrödinger equation treats the electron not as a particle orbiting the nucleus, but as a wave described by a wave function ($\psi$). The square of the wave function, $|\psi|^2$, gives the probability density for finding the electron at a given point in space. The regions where this probability is significant are what we call atomic orbitals.

Solving the Schrödinger equation for hydrogen reveals that the electron can only occupy specific, quantized energy states. The allowed energies depend on the principal quantum number $n$:

$E_n = -\frac{13.6 \text{ eV}}{n^2}$

where $n$ is a positive integer (1, 2, 3, ...). The negative sign means the electron is bound to the nucleus. At $n = 1$ (the ground state), the energy is $-13.6$ eV. At $n = 2$, it's $-3.4$ eV. As $n \to \infty$, the energy approaches zero, which corresponds to a free electron.

Implications and Applications

When an electron transitions from a higher energy level to a lower one, it emits a photon whose energy equals the difference between the two levels. This is why hydrogen's emission spectrum consists of sharp lines at specific wavelengths rather than a continuous glow. The Balmer series (visible light) corresponds to transitions down to $n = 2$, while the Lyman series (ultraviolet) corresponds to transitions down to $n = 1$.

The Heisenberg uncertainty principle places a fundamental limit on how precisely you can simultaneously know an electron's position and momentum. You cannot trace the electron's path around the nucleus the way you'd track a planet. Instead, the wave function gives you a probability distribution, and that's the best description nature allows.

The hydrogen atom solution serves as the foundation for understanding more complex atoms. While multi-electron atoms require additional considerations (electron-electron repulsion, shielding), the orbital framework and quantum number system developed for hydrogen carries over directly.

Quantum Numbers and Atomic Structure

Four Quantum Numbers

Each electron in an atom is described by four quantum numbers. Together, they specify the electron's energy, the shape of its orbital, the orbital's orientation, and the electron's spin.

Principal quantum number ($n$) determines the main energy level and the average distance of the electron from the nucleus. It takes positive integer values: 1, 2, 3, ... Larger $n$ values mean higher energy and a larger orbital.

Angular momentum quantum number ($l$) describes the shape of the orbital. For a given $n$, $l$ can range from 0 to $n - 1$. Each value of $l$ corresponds to a letter designation:

• $l = 0$ → s orbital
• $l = 1$ → p orbital
• $l = 2$ → d orbital
• $l = 3$ → f orbital

These letter names come from old spectroscopic terminology: sharp, principal, diffuse, and fundamental.

Magnetic quantum number ($m_l$) specifies the orientation of the orbital in space. It takes integer values from $-l$ to $+l$, including 0. For a p orbital ($l = 1$), the three allowed values are $m_l = -1, 0, +1$, giving three distinct spatial orientations.

Spin quantum number ($m_s$) describes the electron's intrinsic angular momentum. It has only two possible values: $+\frac{1}{2}$ (spin up) and $-\frac{1}{2}$ (spin down). Spin is not a classical rotation but an intrinsic quantum property with no direct macroscopic analogy.

Uniquely Identifying Electrons

The Pauli exclusion principle states that no two electrons in the same atom can share the same set of all four quantum numbers. This means each orbital (defined by $n$, $l$, and $m_l$) can hold at most two electrons, and those two must have opposite spins. This principle is what forces electrons into progressively higher energy levels and ultimately explains the structure of the periodic table.

Energy Levels and Orbitals in Hydrogen

Energy Levels and Subshells

In hydrogen, energy depends only on $n$. The ground state ($n = 1$) is the lowest energy state, while $n = 2, 3, 4, ...$ are excited states. A key feature of hydrogen specifically: all subshells within the same shell have the same energy. The 2s and 2p subshells are degenerate (equal in energy) in hydrogen. This degeneracy breaks down in multi-electron atoms due to electron-electron interactions.

Each shell contains $n$ subshells. Here's how that plays out:

• $n = 1$: one subshell (1s)
• $n = 2$: two subshells (2s, 2p)
• $n = 3$: three subshells (3s, 3p, 3d)
• $n = 4$: four subshells (4s, 4p, 4d, 4f)

Orbitals and Electron Capacity

Within each subshell, the number of orbitals is given by $2l + 1$:

• s subshell ($l = 0$): 1 orbital
• p subshell ($l = 1$): 3 orbitals
• d subshell ($l = 2$): 5 orbitals
• f subshell ($l = 3$): 7 orbitals

Since each orbital holds a maximum of two electrons (one spin up, one spin down), the total electron capacity of a shell is $2n^2$. The first shell holds 2 electrons, the second holds 8, the third holds 18, and so on.

Shapes and Orientations of Atomic Orbitals

The angular momentum quantum number $l$ determines the shape of an orbital, while the magnetic quantum number $m_l$ determines its orientation in space.

s, p, and d Orbitals

s orbitals ($l = 0$) are spherically symmetric. They look the same from every direction, with a single orientation ($m_l = 0$). The probability density is highest near the nucleus and decreases with distance. Higher-$n$ s orbitals (2s, 3s, ...) are larger and contain radial nodes, which are spherical surfaces where the probability density drops to zero.

p orbitals ($l = 1$) have a dumbbell shape with two lobes on opposite sides of the nucleus, separated by a nodal plane where the probability of finding the electron is zero. The three p orbitals ($p_x$, $p_y$, $p_z$) are aligned along the three Cartesian axes, corresponding to $m_l = -1, 0, +1$.

d orbitals ($l = 2$) have five orientations and more complex shapes. Four of them ($d_{xy}$, $d_{xz}$, $d_{yz}$, $d_{x^2-y^2}$) have a cloverleaf pattern with four lobes. The fifth ($d_{z^2}$) has a dumbbell shape along the z-axis with a torus (donut) in the xy-plane. These orbitals are particularly important for describing the chemistry of transition metals.

f Orbitals and Implications for Bonding

f orbitals ($l = 3$) have seven orientations and highly intricate shapes that are difficult to visualize. They become relevant in the electronic structure of lanthanides and actinides, where electrons fill the 4f and 5f subshells.

The shapes and orientations of orbitals directly influence chemical bonding. Bonds form through the overlap of orbitals on adjacent atoms, and the spatial distribution of those orbitals determines the geometry and directionality of bonds. This is why, for example, carbon's sp$^3$ hybridized orbitals produce tetrahedral geometry, and why p orbitals can overlap side-by-side to form pi bonds. Understanding orbital shapes is the first step toward predicting molecular geometry and reactivity.