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Lesson Directive // Quantum MechanicsREF_CORE

The Ultraviolet Catastrophe

EE==hhff

Hover over a variable in the formula above, or see glossary below:

EE
Photon Energy
Joules (J) or eV
hh
Planck's Constant
J·s
ff
Frequency
Hertz (Hz)

In 1900, Max Planck solved a crisis in physics. Classical theory predicted that hot objects should radiate infinite energy at high frequencies (the "ultraviolet catastrophe"). Planck proposed that energy comes in discrete packets, or **quanta**, of size E = hf.

INSIGHT: Energy is quantised — it exists in discrete chunks, not a continuous flow.

Wave-Particle Duality

In 1905, Einstein used Planck's idea to explain the photoelectric effect: light behaves as particles (photons) when it hits a metal and ejects electrons. Yet light also exhibits wave interference. Matter itself — electrons, protons — also displays wave-like diffraction. This is wave-particle duality.

INSIGHT: Light and matter are both waves AND particles, depending on how you observe them.

Heisenberg's Uncertainty Principle

Werner Heisenberg showed that it is fundamentally impossible to simultaneously know both the exact position x and exact momentum p of a particle: \Delta x \cdot \Delta p \geq \frac{\hbar}{2}. This is not a limitation of instruments — it is a fundamental property of the universe.

INSIGHT: The more precisely you know position, the less precisely you can know momentum — and vice versa.
Detailed Theory & ReferencesEXT_DOC

Quantum Mechanics: Foundations

Quantum mechanics governs the behaviour of matter and energy at atomic and subatomic scales. It replaces deterministic classical mechanics with a probabilistic description.

Planck's Quantum Hypothesis (1900)

Max Planck resolved the ultraviolet catastrophe by postulating that electromagnetic oscillators emit or absorb energy only in discrete amounts: En=nhf(n=0,1,2,…)E_n = nhf \quad (n = 0, 1, 2, \ldots)

where h≈6.626×10−34h \approx 6.626 \times 10^{-34} J·s is Planck's constant.

The Photoelectric Effect (Einstein, 1905)

Einstein proposed that light consists of discrete quanta (photons), each carrying energy E=hfE = hf. Above a threshold frequency f0f_0, electron kinetic energy scales linearly with frequency: Kmax⁡=hf−ϕK_{\max} = hf - \phi where ϕ\phi is the work function. Einstein won the 1921 Nobel Prize for this work.

de Broglie Wavelength

All matter has an associated wavelength (de Broglie, 1924): λ=hp=hmv\lambda = \frac{h}{p} = \frac{h}{mv}

Confirmed by the Davisson-Germer electron diffraction experiment (1927).

The Heisenberg Uncertainty Principle

Δx⋅Δp≥ℏ2\Delta x \cdot \Delta p \geq \frac{\hbar}{2}

where ℏ=h/(2π)\hbar = h / (2\pi) is the reduced Planck constant. Similarly: ΔE⋅Δt≥ℏ/2\Delta E \cdot \Delta t \geq \hbar / 2.

The Schrödinger Equation

iℏ∂Ψ∂t=H^Ψi\hbar \frac{\partial \Psi}{\partial t} = \hat{H} \Psi

The wavefunction Ψ\Psi gives the probability amplitude for finding the particle in a given state. The probability density is ∣Ψ∣2|\Psi|^2.

References

AI NOTICE

AI Assistance Disclaimer: This module uses AI-assisted educational models and interactive visual representations to help explain scientific and mathematical concepts. For formal research or academic evaluation, please verify formulas and data against standard primary reference materials.