Quantum Mechanics and Quantum Chemistry (v1.1)
March 30, 2022 Note: Much of this essay is a synthesis and summarization of lectures by Dr. Erica W. Carlson of Purdue University.
The Macro and Micro Worlds
We are all familiar with Newtonian physics. Laws like "a body at rest continues to be at rest and a body in motion continues to be in motion in a straight line unless a force is applied" feel intuitively obvious. They match our everyday experiences with macro-objects and the macro world we operate in. Einstein’s laws of special and general relativity, which generalize and extend Newtonian physics, are amazing at first glance, but they also apply to macro-objects.
However, in the world of the exceedingly small—like the atom or smaller—laws are vastly different and governed by quantum mechanics. Quantum mechanics can technically be applied to much larger objects, but quantum effects rapidly decrease as the mass of objects increases, becoming virtually unmeasurable in the macro-world.
No one so far has explained why the strange world of quantum mechanics behaves the way it does; scientists continue to work on a foundational explanation. However, its rules have been verified by numerous experiments, and no violations are known to date. This blog introduces key rules of quantum mechanics, focusing on how they underpin chemistry, thermodynamics, particle physics, blackbody radiation, quantum computers, electrical conduction, superconductivity, color vision, fluorescent lighting, magnets, and tunneling.
Wave-Particle Duality and Function Collapse
The first strange rule is that a particle behaves as both a wave and a particle.
The Double-Slit Example: Shooting electrons at two closely spaced slits in a barrier results in a wave interference pattern on a recording wall behind it. It is as if the single electron went through both slits simultaneously.
However, observation or measurement of the position, momentum, or energy of the particle collapses the wave function. It becomes a localized particle and stays so for some time. If we place a detector to spy on which slit the electron went through, the wave behavior disappears, and it behaves strictly as a particle—hitting the wall like a tennis ball. This is wave-particle duality. Even light behaves alternately as a photon particle or a light wave. Until observed, a system exists in a superposition quantum state, meaning it can hold multiple positions or momenta at the same time.
To illustrate how strange this is, Erwin Schrödinger proposed a famous thought experiment: Schrödinger's Cat. Suppose we have a sealed box containing a radioactive atom, a Geiger counter, a hammer, a vial of poison gas, and a cat. If the Geiger counter detects a single atomic decay, the hammer drops, breaks the vial, and the gas kills the cat. According to quantum mechanics, until the box is opened, the atom is in a superposition of both decayed and undecayed states. Consequently, the cat is simultaneously both dead and alive until an observer opens the box, forcing the system to collapse into one definitive state.
Similar superposition effects are observed in superconductivity, where electrical current can flow in both directions simultaneously.
The Heisenberg Uncertainty Principle
The second foundational rule is that you cannot simultaneously know a particle's exact position and momentum with absolute certainty. Mathematically, this is expressed as:
σx × σp ≥ hbar/2
Where:
σx is the uncertainty in position.
σp is the uncertainty in momentum (mass * velocity).
h is Planck's constant. hbar is h / 2 × π, The value of h is 6.626 * 10-34J Hz.
Because Planck's constant is an exceedingly small number, these quantum effects completely vanish into the background as the mass of an object grows large. This probabilistic nature is fundamental and intrinsic to the universe; it is not a limitation of our current measurement technology or data gathering techniques.
Bounded Waves and Quantization
Waves that are bounded—like a vibrating clothesline anchored on both ends, air in a flute, or a drumhead vibrating in two dimensions—form standing waves with discrete harmonics. You cannot have "1.5" stationary nodes on a guitar string; you must have integer values (1, 2, 3, etc.).
For electrons bound to an atom, their position, speed, angular momentum, and energy are similarly quantized into discrete integer values. An electron in an atom forms a three-dimensional standing wave around the nucleus, described by the Schrödinger Wave Equation.
Why does the Schrödinger Equation use imaginary numbers (i)?
The inclusion of the imaginary unit i (sqrt-1) simply condenses a system of two real differential equations into a more compact form. Complex numbers should be thought of as a two-dimensional vector space over the real numbers rather than something mystical. They are exceptionally good at simplifying equations involving spatial symmetry. Ultimately, the imaginary factor corresponds mathematically to a 90-degree geometric rotation.
How Quantum Mechanics Builds the Periodic Table
The nucleus of an atom consists of positively charged protons and neutral neutrons. Because the nucleus is thousands of times heavier than an electron, the nucleus remains stationary while the electrons exist around it.
The electron does not collapse directly into the proton (despite their intense electrical attraction) because of the Uncertainty Principle. If an electron were perfectly stationary on top of a proton, its position uncertainty (σx) would be zero, requiring its momentum (σp) to become infinitely large, causing it to fly away. It finds a stable energy balance by forming a standing wave close to the nucleus.
The electron populates lowest-energy states before filling higher-energy states. If you attempt to combine multiple stationary states of different energies for a single electron, the state becomes unstable; the electron will emit electromagnetic radiation like an antenna and sink back down to a stable, single stationary state.
Electron Configurations Explained by Wave Shapes
The s Orbitals (Columns 1–2): The lowest energy state is the 1s orbital. This is a stable, spherical standing wave—analogous to a 3D drumhead where the center bounces up and down while the edges remain stationary. Because electron spin is quantized (either "up" or "down"), up to two electrons can occupy this state. This accommodates Hydrogen (1 electron) and Helium (2 electrons).
The Pauli Exclusion Principle states that no two electrons can occupy the exact same quantum state simultaneously. Because electrons are fermions (particles with half-integer spin), they obey Fermi-Dirac statistics, which mathematically enforces this exclusion.
The next energy level is 2s (Lithium and Beryllium), which resembles a drumhead with an additional internal, concentric stationary ring. This pattern repeats down the first two columns (3s, 4s, 5s, etc.) with increasing nodes.
The p Orbitals (Columns 13–18): After Beryllium comes a block of six elements (Boron through Neon). These correspond to radial harmonics called 2p orbitals. Imagine a drumhead divided by a stationary line right through the center, where the left and right halves vibrate out of phase. In three dimensions, this wave can be oriented along three distinct axes: p_x, p_y, and p_z. With two spin states per orientation, we get exactly 3 × 2 = 6 electron slots, creating the right-hand block of the periodic table.
The d Orbitals (Transition Metals): In the center of the table lies a block of 10 elements (the transition metals, rows 3d, 4d, 5d, etc.). The geometry of these standing waves has 5 distinct geometric orientations (z2, x2-y2, xy, yz, zx). This resembles a drumhead with two intersecting stationary lines forming quadrants. Five orientations multiplied by two spin states yields exactly 10 elements per row.
The f Orbitals (Lanthanides and Actinides): The rare earth elements comprise a block of 14 columns (4f and 5f states). Their underlying geometry is even more complex, featuring 7 distinct spatial orientations (resembling a drumhead divided symmetrically by three intersecting lines). At this scale, the massive number of electrons begin interacting and repelling one another significantly, creating unique architectural twists.
Where is the Orbital Angular Momentum Hiding?
If electrons exist as static standing waves, how can they possess orbital angular momentum? The answer lies within the complex numbers of the wave function.
If you calculate a mathematical superposition of two orientations using the imaginary unit—for example, 2p_x + i(2p_y) (where 2p_y is rotated 90 degrees relative to 2p_x)—the resulting probability wave transforms into a spinning donut. This physical rotation generates a current, which reveals the quantized angular momentum:
s orbitals: l = 0
p orbitals: l = 1\hbar
d orbitals: l = 2\hbar
f orbitals: l = 3\hbar
The Quantum Origins of Magnetism and Chemical Bonds
Both the inherent spin of an electron and its orbital motion around a nucleus generate tiny magnetic fields. If an atomic orbital is fully occupied, the paired electrons possess opposite spins, causing their magnetic fields to cancel each other out perfectly. Net magnetic fields occur only in atoms with partially filled outermost orbitals (such as Iron, Nickel, or Cobalt in the d-block, or Neodymium in the f-block). For a material to become a permanent magnet, these individual atomic fields must align along a macro-axis—a cooperative behavior dictated by Pauli's exclusion principle and electron repulsion.
Chemical bonds are formed exclusively by these outermost, high-energy valence electrons. When atoms approach each other, their atomic orbitals merge and distort into completely new, hybridized standing wave shapes wrapped around the combined nuclei.
Carbon’s Versatility: In a diamond, carbon atoms hybridize into rigid, symmetric 3D tetrahedra bonded to four neighboring carbons, producing an incredibly hard structure. In graphene (and pencil graphite), carbon atoms bond into flat, 2D hexagonal sheets. The fourth valence electron remains oriented perpendicularly, delocalizing and floating freely across the entire sheet. This allows the sheets to slide past one another effortlessly, acting as a superb lubricant.
Noble Gases: Helium has a completed 1s shell. Because its wave states are fully occupied and highly stable, it has no energetic incentive to hybridize or react with other elements.
In structural chemistry, these interactions manifest primarily as strong, direct sigma bonds or overlapping pi bonds. By combining versatile carbon, hydrogen, and oxygen atoms via these quantum mechanics rules, we get organic chemistry, biochemistry, and ultimately, human life.
Deep Quantum Mysteries: Tunneling and Entanglement
Two of the most non-intuitive features of quantum mechanics are tunneling and entanglement.
Quantum Tunneling
Because particles act as waves, their probability functions don't drop to zero instantly when they hit an energy barrier; the wave decays exponentially through it. If the barrier is thin enough, the wave function extends through to the other side, meaning there is a non-zero probability that an electron will simply appear on the far side of an otherwise impassable wall. This phenomenon is actively harnessed in macro-technology, including modern solid-state flash memory and Tunneling Field-Effect Transistors (TFETs).
Quantum Entanglement
When two particles become entangled through a localized interaction, their wave functions become inextricably linked. If they are subsequently separated by vast distances—even across the universe—measuring a property (like spin) of one particle instantly reveals the correlated state of its entangled companion.
This instantaneous collapse across vast distances initially caused a massive paradox. Albert Einstein famously rejected this implication, calling it "spooky action at a distance" because it appeared to violate his theory of Special Relativity, which asserts that no signal or information can travel faster than the speed of light (c).
Physicists have since reconciled this tension via the No-Communication Theorem:
While the state collapse of entangled particles is indeed instantaneous, no usable information or energy is transmitted through the collapse itself.
When you measure your local particle, the outcome is completely random (e.g., a 50/50 chance of spin-up). You instantly know the distant particle is spin-down, but you cannot maliciously force your particle into a specific state to "send" a deliberate message.
To decode or utilize the quantum correlation to perform a task (like quantum cryptography or teleportation protocols), the two observers must still communicate using standard, classical channels (like fiber-optic lines or radio waves). Because these classical channels are strictly limited by the speed of light, relativity remains unviolated, and causality is preserved.
Superposition and entanglement serve as the absolute bedrock of quantum computing. As classical silicon microchips reach their physical limits, quantum computing may be the definitive pathway forward for complex computational simulation—until our technology eventually runs directly into the fundamental barriers of the Planck scale and the speed of light.
NEXT: Quantum Mechanics Applications and Light
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