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Quantum Mechanics: An Introduction
Modern Physics
Quantum mechanics describes matter and light at the smallest scales, where energy comes in packets, particles behave like waves, and outcomes are governed by probabilities.
What quantum mechanics is
Quantum mechanics grew from the discovery that energy is not continuous at small scales but comes in discrete packets called quanta. Max Planck found in 1900 that light is emitted in packets of energy E = hf, and Einstein showed that light itself arrives as particles called photons. Louis de Broglie then proposed that matter also has a wavelength, lambda = h/p, and experiments confirmed that electrons diffract like waves. A quantum particle is described by a wavefunction, whose square gives the probability of finding the particle in a given place. Measurement does not reveal a pre-existing definite position; it gives one outcome with a probability set by the wavefunction.
The core ideas
Three ideas sit at the heart of the theory. Quantisation means that bound systems, such as electrons in atoms, can only have certain energies. Wave-particle duality means that light and matter show wave behaviour in some experiments and particle behaviour in others, depending on how they are measured. The uncertainty principle states that position and momentum cannot both be known with arbitrary precision: sharpening one blurs the other. None of these is a failure of our instruments; they are built into how nature works at atomic scales, and the theory's predictions have been tested to extraordinary accuracy.
Where you see it
Quantum mechanics explains why atoms are stable, why the periodic table has its shape, and why materials conduct, insulate or glow as they do. It underlies lasers, transistors, LEDs, MRI scanners and solar cells: the whole of modern electronics is applied quantum mechanics. Quantum tunnelling lets particles pass through barriers they classically could not cross, which is essential in flash memory and in the fusion reactions that power the Sun. The theory is strange, but it is also the most precisely tested theory in physics, and it works every time it is put to the test.
📝 Worked example: energy of a photon
A photon has frequency f = 5.0 x 10^14 Hz. Use E = h f, with Planck constant h = 6.63 x 10^-34 J s. Substitute: E = (6.63 x 10^-34) x (5.0 x 10^14). E = 3.315 x 10^-19 J. So one photon carries about 3.3 x 10^-19 J. That is a tiny amount of energy, which is why a bright light contains an enormous number of photons.
Common mistakes
- Saying a particle is in two places at once in the everyday sense: superposition is a statement about the quantum state before measurement, not a claim that a ball is visibly in two places.
- Treating uncertainty as a measurement problem: the uncertainty principle is a property of nature, not just a limit of our instruments.
- Thinking quantisation applies to everything: energy is quantised in atoms, but many quantities, such as the energy of a free particle, can take a continuous range of values.
Practice questions
- What is the energy of a photon of frequency 6.0 x 10^14 Hz?
Show answer
E = (6.63 x 10^-34) x (6.0 x 10^14) = 3.98 x 10^-19 J.
- An electron has momentum 1.0 x 10^-24 kg m/s. What is its de Broglie wavelength?
Show answer
lambda = h/p = (6.63 x 10^-34)/(1.0 x 10^-24) = 6.63 x 10^-10 m, about 0.66 nm.
- Can the position and momentum of an electron both be known exactly?
Show answer
No. The uncertainty principle forbids knowing both exactly at the same time.
Frequently asked questions
Does quantum mechanics mean nothing is predictable?
No. Quantum mechanics gives probabilities for outcomes; the pattern only becomes predictable when many particles are involved.
Why do we not notice quantum effects in everyday life?
Because Planck's constant is extremely small, so quantum effects are negligible for large objects and only show up at atomic scales.