Quantum Mechanics

Wave-particle duality, the Schrödinger equation, and the uncertainty principle - the rules that govern the subatomic world.

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Photon Energy Calculator

Formula
E = hf = hc/λ
λ Wavelength
nm
Photon Energy
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1 Wave-Particle Duality

Light and matter exhibit both wave and particle properties. Light behaves as a wave (interference, diffraction) but also as particles called photons (photoelectric effect). Electrons behave as particles (tracks in detectors) but also as waves (electron diffraction). This duality is fundamental - not a limitation of our measurement, but the true nature of reality.

The double-slit experiment is the clearest demonstration of duality. When electrons are fired one at a time through two slits, they still build up an interference pattern on a detector screen - as if each electron passes through both slits simultaneously. The moment you try to detect which slit an electron used, the interference pattern disappears and you get two bands, as if the electron were a classical particle. Observation itself collapses the quantum superposition.

e⁻ source slit 1 slit 2 interference pattern barrier

Double-slit experiment: electrons fired one at a time still create an interference pattern, revealing their wave nature.

E = hf = hc/λ
h = 6.626 × 10⁻³⁴ J·s (Planck's constant)f = frequency (Hz), λ = wavelength (m)c = 3 × 10⁸ m/s (speed of light)
De Broglie wavelength: λ = h/p = h/(mv). Every moving object has a wavelength, but for macroscopic objects it's undetectably small. A baseball at 40 m/s has λ ≈ 10⁻³⁴ m - far smaller than an atom.
Worked Example 1

Photon Energy of Green Light

Problem: Green light has a wavelength of 550 nm. Calculate the energy of one photon in both Joules and electron volts.
Apply E = hc/λ
λ = 550 nm = 550 × 10⁻⁹ m
E = (6.626 × 10⁻³⁴)(3 × 10⁸) / (550 × 10⁻⁹)
E = 1.988 × 10⁻²⁵ / 5.5 × 10⁻⁷
E = 3.61 × 10⁻¹⁹ J = 2.25 eV
Answer: 3.61 × 10⁻¹⁹ J (2.25 eV). Visible light photons range from ~1.65 eV (red, 750 nm) to ~3.1 eV (violet, 400 nm). UV photons carry even more energy, which is why they cause sunburn.

2 The Uncertainty Principle

Heisenberg's uncertainty principle states that certain pairs of properties cannot both be known precisely at the same time. The more precisely you know a particle's position, the less precisely you can know its momentum, and vice versa.

Δx · Δp ≥ ħ/2
Δx = uncertainty in position (m)Δp = uncertainty in momentum (kg·m/s)ħ = h/(2π) = 1.055 × 10⁻³⁴ J·s

This isn't about measurement error - it's a fundamental property of nature. A particle genuinely does not have a precise position and momentum simultaneously.

Worked Example 2

Uncertainty Principle - Electron in an Atom

Problem: An electron is confined to a region of Δx = 1.0 × 10⁻¹⁰ m (roughly the size of an atom). What is the minimum uncertainty in its velocity?
Apply Heisenberg's principle
Δp ≥ ħ/(2Δx) = (1.055 × 10⁻³⁴)/(2 × 1.0 × 10⁻¹⁰)
Δp ≥ 5.28 × 10⁻²⁵ kg·m/s
Δv = Δp/m = 5.28 × 10⁻²⁵ / 9.109 × 10⁻³¹
Δv ≥ 5.79 × 10⁵ m/s ≈ 580 km/s
Answer: At least 580 km/s uncertainty. That's ~0.2% the speed of light! Electrons in atoms are inherently "fuzzy" - they don't orbit like planets but exist as probability clouds.

3 The Schrödinger Equation

The Schrödinger equation is the fundamental equation of quantum mechanics - it describes how the quantum state (wave function ψ) of a system evolves over time.

iħ ∂ψ/∂t = Ĥψ
ψ = wave function (complex-valued)Ĥ = Hamiltonian operator (total energy)|ψ|² = probability density
Time-dependent

Describes how ψ evolves: iħ ∂ψ/∂t = Ĥψ. Used for non-stationary states.

Time-independent

For stationary states: Ĥψ = Eψ. Solving this gives allowed energy levels (eigenvalues).

Worked Example 3

Particle in a Box - Energy Levels

Problem: An electron is confined to a 1D box of width L = 0.5 nm. Calculate the first three energy levels.
Energy levels for particle in a box
E_n = n²π²ħ² / (2mL²) = n²h² / (8mL²)
E₁ = (1)²(6.626×10⁻³⁴)² / (8 × 9.109×10⁻³¹ × (0.5×10⁻⁹)²)
E₁ = 2.41 × 10⁻¹⁹ J = 1.50 eV
E₂ = 4 × 1.50 = 6.02 eV
E₃ = 9 × 1.50 = 13.5 eV
Answer: E₁ = 1.50 eV, E₂ = 6.02 eV, E₃ = 13.5 eV. Energy levels scale as n² - the gaps between levels grow with n. This quantization of energy is what makes quantum mechanics "quantum."

4 Quantum Tunneling

In classical physics, a ball can't pass through a wall. In quantum mechanics, a particle has a non-zero probability of passing through a potential energy barrier that it classically shouldn't be able to cross. The probability decreases exponentially with barrier width and height.

T ≈ e^(−2κL)
T = tunneling probabilityκ = √(2m(V₀−E)/ħ²)L = barrier width, V₀ = barrier height
Real applications: Tunnel diodes, flash memory, scanning tunneling microscopes (STM), and nuclear fusion in the Sun all rely on quantum tunneling. Without it, the Sun wouldn't shine - protons don't have enough energy to classically overcome their mutual repulsion.

5 Quantum Spin & Pauli Exclusion

Quantum spin is an intrinsic angular momentum that particles possess - it's not actual spinning but a fundamental quantum property. Electrons have spin ±½. The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state simultaneously.

Electron spin
s = ½, m_s = +½ or −½
Photon spin
s = 1 (boson, no exclusion)
Proton/Neutron
s = ½ (fermion, exclusion applies)
Higgs boson
s = 0

The Pauli exclusion principle is why the periodic table exists - electrons must fill different orbitals rather than all collapsing to the lowest energy state. It's also why matter is solid rather than collapsing.

6 Quantum Entanglement

When two particles become entangled, measuring the state of one instantly determines the state of the other, regardless of the distance between them. Einstein called this "spooky action at a distance." It's been experimentally verified and is the basis of quantum computing and quantum cryptography.

EPR Paradox

Einstein, Podolsky, and Rosen argued entanglement implied quantum mechanics was incomplete. Bell's theorem (1964) showed local hidden variables can't explain it - quantum mechanics is genuinely non-local.

Applications

Quantum teleportation, quantum key distribution (QKD), quantum computing (qubits). China's Micius satellite demonstrated entanglement over 1,200 km in 2017.

Worked Example 4

De Broglie Wavelength of an Electron

Problem: An electron is accelerated through a potential difference of 100 V. What is its de Broglie wavelength? Would it show diffraction through a crystal lattice (spacing ~0.3 nm)?
Find velocity, then wavelength
KE = eV = 1.602×10⁻¹⁹ × 100 = 1.602×10⁻¹⁷ J
v = √(2KE/m) = √(2 × 1.602×10⁻¹⁷ / 9.109×10⁻³¹)
v = 5.93 × 10⁶ m/s
λ = h/(mv) = 6.626×10⁻³⁴ / (9.109×10⁻³¹ × 5.93×10⁶)
λ = 0.123 nm = 1.23 Å
Answer: 0.123 nm - comparable to crystal lattice spacing (~0.3 nm), so yes, electron diffraction occurs! This was first demonstrated by Davisson and Germer in 1927, confirming de Broglie's hypothesis and earning them the Nobel Prize.

De Broglie Wavelength Calculator

Formula
λ = h / (mv)
m Mass (kg)
v Velocity (m/s)
De Broglie Wavelength
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Heisenberg Uncertainty Calculator

Formula
Δp ≥ ħ / (2·Δx)
Δx Position uncertainty (m)
m Particle mass (kg)
Minimum velocity uncertainty (Δv)
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Related Topics

Relativity
Nuclear Physics
Astrophysics