6.3 Development of Quantum Theory

Light and matter behave as both waves and particles. This wave-particle duality is a central idea in quantum mechanics, explaining phenomena like the photoelectric effect and electron diffraction. It challenges our classical understanding of physics, where something was either a wave or a particle, never both.

Quantum mechanics describes electrons in atoms using wave functions called orbitals. These represent the probability of finding an electron in a specific region around the nucleus. Four quantum numbers define an electron's energy, orbital shape, orbital orientation, and spin.

Wave-Particle Duality and Quantum Mechanics

Wave-particle duality in physics

Both matter and electromagnetic radiation can behave as waves and as particles. This isn't intuitive, but multiple experiments confirm it.

Electromagnetic radiation as particles: Light travels as waves, but it also comes in discrete packets of energy called photons. The photoelectric effect provides strong evidence for this. When light of sufficient frequency hits a metal surface, it ejects electrons. Classical wave theory predicted that brighter light (higher amplitude) should always eject electrons regardless of frequency, but that's not what happens. Only light above a certain frequency works, no matter how bright it is. Einstein explained this by treating light as photons, each carrying energy:

$E = h\nu$

where $h$ is Planck's constant ($6.626 \times 10^{-34} \text{ J·s}$) and $\nu$ is the frequency of light. Each photon either has enough energy to knock an electron loose, or it doesn't.

Matter as waves: Electrons and other particles also exhibit wave-like behavior. In the double-slit experiment, electrons fired one at a time produce an interference pattern on a detector, something only waves should do. De Broglie proposed that any particle has an associated wavelength:

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

where $m$ is the particle's mass and $v$ is its velocity. For everyday objects, this wavelength is unimaginably tiny and undetectable. But for something as light as an electron, the wavelength is large enough to observe wave behavior like diffraction.

Max Planck introduced the concept of energy quantization before any of this, proposing that energy is emitted and absorbed in discrete amounts (quanta) rather than continuously. This laid the foundation for all of quantum theory.

Quantum model of atomic electrons

The quantum mechanical model describes electrons using mathematical wave functions ($\Psi$) called orbitals. An orbital is not a fixed path an electron follows. Instead, it represents a probability distribution, showing where you're most likely to find the electron. The probability at any point is proportional to $\Psi^2$ (the square of the wave function), which gives you the electron density or "electron cloud."

Types of orbitals:

• s orbitals are spherical in shape (1s, 2s, 3s, etc.)
• p orbitals have a dumbbell shape, oriented along the x, y, or z axes (2p, 3p, etc.)
• d orbitals have more complex, clover-like shapes (starting at 3d)
• f orbitals are even more complex (starting at 4f)

Nodes are regions where the probability of finding an electron is zero. The number of nodes increases with the energy of the orbital. For example, the 1s orbital has zero nodes, while the 2s orbital has one spherical node. The total number of nodes for any orbital equals $n - 1$.

Niels Bohr proposed an earlier model with quantized electron energy levels (fixed orbits), which correctly predicted hydrogen's emission spectrum. The full quantum mechanical model built on Bohr's idea of quantized energy but replaced fixed orbits with probability-based orbitals.

Quantum numbers for electron states

Every electron in an atom is described by a unique set of four quantum numbers. Think of them as an electron's "address" within the atom.

• Principal quantum number ($n$): Indicates the energy level and shell. Takes positive integer values (1, 2, 3, ...). Higher $n$ means higher energy and a larger average distance from the nucleus.
• Angular momentum quantum number ($l$): Determines the subshell and shape of the orbital. Takes integer values from 0 to $n - 1$. Each value corresponds to a subshell type:
  • $l = 0$ → s subshell
  • $l = 1$ → p subshell
  • $l = 2$ → d subshell
  • $l = 3$ → f subshell
• Magnetic quantum number ($m_l$): Specifies the orientation of the orbital in space. Takes integer values from $-l$ to $+l$. This is what determines how many orbitals exist within a subshell (s has 1, p has 3, d has 5, f has 7).
• Spin quantum number ($m_s$): Describes the electron's intrinsic spin. Only two possible values: $+\frac{1}{2}$ (spin up) or $-\frac{1}{2}$ (spin down).

The Pauli exclusion principle states that no two electrons in the same atom can share all four quantum numbers. Since each orbital is defined by a specific combination of $n$, $l$, and $m_l$, this means each orbital holds a maximum of two electrons, and those two must have opposite spins.

Fundamental concepts in quantum mechanics

• The Heisenberg uncertainty principle states that you cannot simultaneously know both the exact position and exact momentum of a particle. The more precisely you measure one, the less precisely you can know the other. This is why we describe electrons with probability clouds rather than exact paths.
• The Schrödinger equation is the fundamental equation of quantum mechanics. Solving it for an atom gives you the wave functions (orbitals) and their associated energies. For this course, you won't need to solve it, but you should know it's the mathematical basis for the orbital model.
• Wave function collapse refers to what happens when you measure a quantum system. Before measurement, a particle can exist in a superposition of multiple states simultaneously. Once observed, it "collapses" into one definite state. The Copenhagen interpretation is the most widely taught framework for understanding this: quantum mechanics is fundamentally probabilistic, and a particle doesn't have a definite state until it's measured.