2.1 Basic principles of quantum mechanics

Fundamental Concepts of Quantum Mechanics

Quantum mechanics describes how matter and energy behave at the atomic and molecular scale. It replaced classical physics for these systems because classical models couldn't explain key experimental results, like the photoelectric effect or atomic emission spectra. For photochemistry, quantum mechanics is essential: it tells you why molecules absorb specific wavelengths of light, how excited states form, and what drives photochemical reactions.

Two foundational ideas run through everything here: energy is quantized (it comes in discrete packets, not continuous amounts), and particles behave as both waves and particles depending on how you observe them. The Schrödinger equation ties these ideas together mathematically.

Wave-Particle Duality of Matter

Wave-particle duality means that light and matter can exhibit both wave-like and particle-like properties, depending on the experiment. A beam of electrons, for example, produces an interference pattern in a double-slit experiment (wave behavior), yet those same electrons hit a detector at discrete points (particle behavior). This has been demonstrated not just for electrons but even for large molecules like buckminsterfullerene ($C_{60}$).

The de Broglie relation connects a particle's momentum to its wavelength:

$\lambda = \frac{h}{p}$

where $h$ is Planck's constant ($6.626 \times 10^{-34} \text{ J·s}$) and $p$ is the particle's momentum. For macroscopic objects, $\lambda$ is negligibly small, which is why you don't see wave behavior in everyday life. But for electrons and other light particles, the wavelength is significant and directly affects how they interact with matter.

This duality is what forced physicists to abandon purely classical descriptions. The photoelectric effect, for instance, can only be explained by treating light as discrete photons (particles), while diffraction and interference require a wave description.

Heisenberg Uncertainty Principle

The uncertainty principle sets a fundamental limit on how precisely you can simultaneously know certain pairs of properties. For position and momentum:

$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$

where $\hbar = h / 2\pi$. This isn't about measurement error or imperfect instruments. It's a built-in feature of quantum systems. The more precisely you pin down a particle's position, the less precisely you can know its momentum, and vice versa.

There's also an energy-time form:

$\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$

This version matters a lot in photochemistry. A short-lived excited state (small $\Delta t$) has a large uncertainty in energy (large $\Delta E$), which directly broadens spectral lines. So when you see a broad absorption or emission peak, part of that width comes from the uncertainty principle limiting how well-defined the energy of a transient state can be.

The uncertainty principle also explains why we describe electrons in orbitals (probability clouds) rather than fixed orbits. You simply can't assign an electron a precise trajectory.

Quantum Mechanical Systems and Energy

Applications of the Schrödinger Equation

The time-independent Schrödinger equation is the central equation for finding the allowed energy states of a quantum system:

$\hat{H}\Psi = E\Psi$

Here, $\hat{H}$ is the Hamiltonian operator (which encodes the total energy of the system), $\Psi$ is the wavefunction, and $E$ is the energy of that state. Solving this equation for a given system gives you both the allowed energies and the corresponding wavefunctions.

Particle in a box is the simplest model and a good starting point. Imagine a particle confined to a one-dimensional box of length $L$ with infinitely high walls. Solving the Schrödinger equation with the boundary condition that $\Psi = 0$ at the walls gives:

• Energy levels: $E_n = \frac{n^2 h^2}{8mL^2}$, where $n = 1, 2, 3, \ldots$
• Wavefunctions: $\Psi_n(x) = \sqrt{\frac{2}{L}} \sin\!\left(\frac{n\pi x}{L}\right)$

Notice that $n$ can only be a positive integer, so energy is automatically quantized. The number of nodes (zero-crossings) in the wavefunction increases with $n$, and higher $n$ means higher energy.

The quantum harmonic oscillator models molecular vibrations. A diatomic molecule's bond stretching, near equilibrium, behaves approximately like a harmonic oscillator with energy levels:

$E_n = \left(n + \frac{1}{2}\right)\hbar\omega$

The $\frac{1}{2}\hbar\omega$ term is the zero-point energy, meaning a molecule always retains some vibrational energy even at absolute zero. This has real consequences for photochemistry, since it affects the baseline energy from which electronic transitions occur.

For any wavefunction, the probability density $|\Psi|^2$ tells you the likelihood of finding the particle at a given location. This is how quantum mechanics replaces the classical idea of a definite trajectory with a probability distribution.

Quantization of Energy

Quantum mechanics explains why atoms and molecules have discrete energy levels rather than a continuous range. This is what produces the sharp lines in atomic emission spectra and the structured bands in molecular spectra.

Atomic orbitals are described by four quantum numbers:

• $n$ (principal): determines the energy level and size of the orbital
• $l$ (angular momentum): determines the orbital shape ($l = 0$ is s, $l = 1$ is p, $l = 2$ is d, $l = 3$ is f)
• $m_l$ (magnetic): determines the orbital's orientation in space
• $m_s$ (spin): specifies the electron's spin ($+\frac{1}{2}$ or $-\frac{1}{2}$)

Molecular orbitals form when atomic orbitals on different atoms combine. The LCAO (linear combination of atomic orbitals) approach produces bonding orbitals (lower energy, electron density between nuclei) and antibonding orbitals (higher energy, node between nuclei). The energy gap between these orbitals determines what wavelengths of light a molecule can absorb.

Spectroscopic transitions occur when a photon's energy matches the gap between two allowed energy levels. Not all transitions are equally probable; selection rules, derived from quantum mechanical considerations of symmetry and angular momentum, dictate which transitions are allowed and which are forbidden. For example, in atomic spectra the rule $\Delta l = \pm 1$ governs which orbital transitions can occur.

These electronic transitions are at the heart of photochemistry. When a molecule absorbs a photon and moves to an excited electronic state, the resulting energy can drive chemical reactions. Photosynthesis and the chemistry of vision both depend on specific electronic excitations that quantum mechanics predicts and explains.