4.1 Quantum numbers and atomic orbitals

Quantum Numbers and Atomic Orbitals

Quantum numbers and atomic orbitals describe the structure of the hydrogen atom at the quantum level. By solving the Schrödinger equation for the hydrogen atom, you get a set of discrete numbers that fully specify an electron's energy, angular momentum, and spin. These quantum numbers also determine the shapes and orientations of the regions where the electron is likely to be found (the orbitals).

The four quantum numbers (principal, angular momentum, magnetic, and spin) together define every possible state an electron can occupy. They're the foundation for understanding not just hydrogen, but atomic structure in general, including electron configurations, spectral lines, and chemical bonding.

Quantum Numbers for Electrons

Four Fundamental Quantum Numbers

The state of an electron in a hydrogen atom is fully described by four quantum numbers: the principal quantum number ($n$), the angular momentum quantum number ($l$), the magnetic quantum number ($m_l$), and the spin quantum number ($m_s$). All four arise directly from solving the Schrödinger equation for the Coulomb potential.

Together, they specify:

• Energy level of the electron
• Shape of its orbital
• Orientation of that orbital in space
• Spin direction (up or down)

Each electron in an atom must have a unique set of all four quantum numbers. This is the Pauli exclusion principle, and it's what prevents all electrons from collapsing into the lowest energy state.

Quantum Numbers and Atomic Structure

Quantum numbers do more than label states in hydrogen. They form the framework for understanding electron configuration in multi-electron atoms, which in turn explains periodic trends like atomic size and ionization energy.

They also govern which electronic transitions are allowed (through selection rules), which is why atoms emit and absorb light only at specific wavelengths. In chemistry, quantum numbers underpin ideas like orbital hybridization and molecular orbital theory.

Physical Significance of Quantum Numbers

Energy and Orbital Characteristics

Principal quantum number ($n$): This determines the electron's energy and the overall size of the orbital. It takes positive integer values: $n = 1, 2, 3, \ldots$ Higher $n$ means higher energy and a larger average distance from the nucleus. For hydrogen specifically, the energy depends only on $n$:

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

Angular momentum quantum number ($l$): This determines the shape of the orbital and is related to the magnitude of the electron's orbital angular momentum. For a given $n$, $l$ can take integer values from $0$ to $n - 1$. Each value of $l$ corresponds to a subshell label:

• $l = 0 \rightarrow s$
• $l = 1 \rightarrow p$
• $l = 2 \rightarrow d$
• $l = 3 \rightarrow f$

The orbital angular momentum magnitude is $L = \hbar\sqrt{l(l+1)}$.

Spatial Orientation and Spin

Magnetic quantum number ($m_l$): This specifies the orientation of the orbital in space. It takes integer values from $-l$ to $+l$, including zero, giving $2l + 1$ possible values for each $l$. Physically, $m_l$ determines the z-component of the orbital angular momentum: $L_z = m_l \hbar$. This is what causes orbitals to behave differently in an external magnetic field.

Spin quantum number ($m_s$): This represents the electron's intrinsic angular momentum, which has no classical analogue. It takes only two values: $+\frac{1}{2}$ (spin-up) or $-\frac{1}{2}$ (spin-down). Spin is responsible for the splitting of spectral lines in a magnetic field (the Zeeman effect) and is essential for understanding fine structure in atomic spectra.

Shapes and Orientations of Atomic Orbitals

Orbital shapes come from the angular part of the hydrogen wave function (the spherical harmonics). The radial part determines how the probability density falls off with distance, while the angular part gives the characteristic shapes you need to know.

s and p Orbitals

s orbitals ($l = 0$):

• Spherically symmetric, with no angular dependence
• No angular nodes (though higher-$n$ s orbitals have radial nodes)
• A single lobe centered on the nucleus
• Examples: $1s$, $2s$, $3s$

p orbitals ($l = 1$):

• Dumbbell-shaped with two lobes on opposite sides of the nucleus
• One nodal plane passes through the nucleus
• Three orientations exist ($m_l = -1, 0, +1$), conventionally labeled $p_x$, $p_y$, and $p_z$, each aligned along its respective axis
• First appear at $n = 2$

d Orbitals and Beyond

d orbitals ($l = 2$): These have more complex shapes and first appear at $n = 3$. There are five d orbitals ($m_l = -2, -1, 0, +1, +2$):

• $d_{xy}$, $d_{xz}$, $d_{yz}$: four lobes oriented between the coordinate axes
• $d_{x^2 - y^2}$: four lobes pointing along the x and y axes
• $d_{z^2}$: a unique shape with two lobes along the z-axis and a toroidal (doughnut-shaped) ring in the xy-plane

All orbital shapes are derived from the angular part of the wave function (spherical harmonics $Y_l^{m_l}(\theta, \phi)$). Higher angular momentum orbitals ($f$, $g$, etc.) exist but have increasingly complex geometries. Understanding d orbital shapes is particularly important for transition metal chemistry and crystal field theory.

Allowed Values for Quantum Numbers

Restrictions on Quantum Numbers

The quantum numbers aren't independent of each other. Each one constrains the next:

1. $n$ can be any positive integer: $n = 1, 2, 3, \ldots$

2. $l$ ranges from $0$ to $n - 1$ (so for $n = 3$, the allowed values are $l = 0, 1, 2$)

3. $m_l$ ranges from $-l$ to $+l$ in integer steps, giving $2l + 1$ values (so for $l = 2$, you get $m_l = -2, -1, 0, +1, +2$)

4. $m_s$ is always $+\frac{1}{2}$ or $-\frac{1}{2}$

These restrictions aren't arbitrary. They come directly from the boundary conditions and mathematical requirements of the Schrödinger equation's solutions (specifically, requiring the wave function to be single-valued and normalizable).

Counting States

You can use these rules to count the total number of states in any shell. For a given $n$, the total number of distinct quantum states is $2n^2$. For example, $n = 3$ gives $2(3)^2 = 18$ states. This counting is what underlies the structure of the periodic table: the rows get longer as $n$ increases because more quantum states become available.

Implications and Applications

• The allowed combinations determine every possible electron state in hydrogen
• For multi-electron atoms, the same quantum numbers apply, but energies also depend on $l$ (not just $n$) due to electron-electron interactions and shielding
• The aufbau principle uses these quantum numbers to build up electron configurations from lowest to highest energy
• Selection rules for allowed transitions (e.g., $\Delta l = \pm 1$) follow directly from the properties of these quantum numbers and the matrix elements of the dipole operator