Quantum Mechanics | Brilliant Math & Science Wiki (2024)

Quantum mechanics [QM] is a branch of physics which describes physical systems so that properties like the energy or angular momentum are discrete quantities that are multiples of a smallest unit or quantum. A famous physicist named Erwin Schrödinger made an example of quantum mechanics, often called Schrödinger's cat. It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics for every day objects. In this "paradox", Schrödinger states that if you would have put a cat in a box with a bunker with unstable gun powder that has a 50% chance of killing the cat in an hour, right before you open the box with the cat, the cat would be both dead and alive at the same time, and you will only know if it is dead or alive once you open the box.This is also called the many worlds theory. QM replaces classical particles with wave-like states, which results in physical predictions that violate everyday intuition but are nonetheless correct. Quantum-mechanical effects are typically only visible on atomic or subatomic scales since e.g. the spacing of energy levels is on par with the typical energies of an atom; in macroscopic (classical) systems these effects become undetectable. However, in certain cases quantum effects become important on larger scales, such as in condensed matter physics or cosmology.

The Standard Model of particle physics is a quantum theory of physics, although it is expressed as a more sophisticated development of quantum mechanics called quantum field theory which is consistent with relativity.


  • The Development of Quantum Mechanics
  • Brief Overview of the Formalism
  • Physical Systems and Phenomenology
  • References

The Development of Quantum Mechanics

Initial experimental evidence for the quantum nature of reality came from Planck's solution to the black-body radiation problem in 1900. According to physics as of the turn of the twentieth century, a black-body in thermal equilibrium with its environment should emit radiation with infinite power. Planck solved this problem by assuming that a black-body could only emit radiation with energy in discrete packets: \(E = h \nu\) where \(\nu\) was the frequency of the radiation and \(h \approx 6.626 \times 10^{-34} \text{ J}\cdot\text{s}\) was a constant now called the Planck constant.

Quantum Mechanics | Brilliant Math & Science Wiki (1) Power distribution of radiation from a black-body according to the Planck's theory at different temperatures compared to the classical prediction [2].

Five years later, Albert Einstein would make the next step in clarifying Planck’s claim in his examination of the photoelectric effect. In the photoelectric effect, shining light on metals causes them to emit electrons immediately provided that the light is of high enough frequency, without regard to the intensity of the light. Einstein's solution was the proposition that light came in "quantized packets of energy" that described a fundamental particle of light, the photon. However, the interference and diffraction of light observed over the previous century e.g. through Young's double-slit experiment supported the idea that light was a wave. Einstein unified the two ideas by suggesting that photons travel by the propagation of electromagnetic radiation (light waves), giving rise to the modern term wave-particle duality.

Quantum Mechanics | Brilliant Math & Science Wiki (2) In the photoelectric effect, low frequency light does not eject electrons from a metal such as Potassium. The higher the frequency of the light, the more energetic the emitted electron [3].

In 1924, Louis de Broglie furthered the development of QM by making another bold claim: all particles possess wave-particle duality, where the momentum and energy of a particle are given by two relations:

\[p= \hbar k = \frac{h}{\lambda}\] and \[E = \hbar \omega = \frac{{p}^{2}}{2m}\]

where \(k = \frac{2 \pi}{\lambda}\), \(\omega = 2 \pi \nu\), and the constant \(\hbar = \frac{h}{2\pi}\) is known as the reduced Planck constant.

de Broglie's hypothesis radically changed our understanding of matter. His "matter-waves" generalize Einstein's photon such that every particle we observe in the universe has the property of being both a particle and a wave. Interference and diffraction experiments performed with electrons have confirmed de Broglie's claim.

Lastly, experimental results from atomic physics also provided early evidence for quantum mechanics. The discrete spectral lines in the hydrogen emission spectrum showed that energy levels in atoms were quantized. This was initially explained by Bohr's atomic model in which the electrons were in circular orbits around the nucleus, although this theory was later shown to be inaccurate.

Throughout the remainder of the early twentieth century, a number of physicists and mathematicians like Dirac, Hilbert, von Neumann, Heisenberg, Schrödinger and others worked to solidify the theoretical underpinnings that explained these atomic and subatomic phenomena.

Brief Overview of the Formalism

QM describes physical systems by a wave-like expression called a wavefunction, \(\Psi\), which encodes the probability of measuring properties like position, spin, or momentum to be in some particular location or state. Since particles must be measured to be in some state, the total probability over all states must sum to one, and the wavefunction must evolve so that this total probability is always one. This constrains wavefunctions to evolve according to the Schrödinger equation.

Observable quantities in quantum mechanics no longer take definite values. Rather, they are governed by probability distributions specified by the wavefunction \(\Psi\). The expectation value of an observable, corresponding to the mean of this distribution, is its classically measured value. Correspondingly, the equations governing the evolution of the expectation value of an observable usually correspond to the classical equations of motion for that observable. The variance or standard deviation of the probability distribution describes the uncertainty of finding the physical system in any particular state. For certain pairs of variables like position and momentum, called conjugate variables, the standard deviations obey the Heisenberg uncertainty principle:\[\sigma_x \sigma_p \geq \frac{\hbar}{2}.\]

Quantum Mechanics | Brilliant Math & Science Wiki (3) Wavefunctions plotted in position and momentum space, showing that the two are conjugate variables as given by the Heisenberg uncertainty principle. Note the size of the errors \(\Delta x\) and \(\Delta p\) for each wavefunction [4].

Furthermore, observables correspond to operators which act on the wavefunction \(\Psi\). For instance, the momentum operator acting on the position wavefunction can be written as:\[\hat{p} \Psi (x) = -i\hbar \frac{\partial \Psi (x) }{\partial x}.\]Operators in QM are typically denoted with a "hat" to distinguish them from other functions or numbers.

The expectation value of momentum is then written:\[\langle \hat{p} \rangle = \int_{-\infty}^{\infty} dx\, \Psi^{\ast} (x) \hat{p} \Psi(x),\]where the asterisk denotes complex conjugation.

If the momentum of a state is measured, the state collapses into an eigenstate of momentum with eigenvalue equal to the measured momentum. For instance, the eigenstates of momentum are the plane waves \(\Psi(x) = e^{ikx / \hbar}\) with eigenvalue \(k\), because:\[\hat{p} \Psi(x) = -i\hbar \frac{\partial}{\partial x} e^{ikx / \hbar}= -i\hbar \frac{ik}{\hbar} e^{ikx /\hbar} = k \Psi (x).\]

Extending this formalism to operators corresponding to other observables is essentially identical. Observables in quantum mechanics correspond to Hermitian operators, so the Spectral Theorem from functional analysis says that any state \(\Psi\) can be written out in a basis of eigenstates of an observable.

There exist several other ways of formulating quantum mechanics, often used in advanced work. One common approach which generalizes more nicely to relativistic quantum field theory is Feynman's path integral formalism. This approach uses a generalization of Lagrangian mechanics where the relevant Lagrangian is integrated over all possible paths between two classical endpoints, so it is sometimes called the sum-over-paths formalism.

Physical Systems and Phenomenology

Although quantum mechanics is primarily applicable on the scale of atomic interactions, its effects and related phenomena are visible in a wide range of physical systems.

Quantum Mechanics | Brilliant Math & Science Wiki (4) An array of plots of the probability density for the electron in its lowest-energy states of the hydrogen atom [5].

One of the first successes of quantum mechanics was in explaining the emission spectrum of the hydrogen atom and similar atoms by correctly predicting the quantized energy levels of the electron. QM also correctly described how the electron could be bound to a nucleus without losing energy through radiation and inspiraling to the nucleus, and thus provided the first explanation for the stability of atoms.

Perhaps the most popularized example of a quantum effect due to its extreme defiance of classical intuition, still on the atomic scale, is quantum tunneling. Classically, a particle bound in a potential (i.e., stuck in a box) at low energy is trapped there, unable to escape. In QM, however, there is always a probability of measuring particles to have escaped the potential and thus "tunneled" through the walls of the box. As a result, there is a nonzero but inconceivably miniscule probability that if you were to attempt to walk through a wall, you would succeed! More practically, quantum tunneling is responsible for the \(\alpha\)-decay of radioactive nuclei explained via the Gamow model, since the alpha particles must use quantum tunneling to escape the strong nuclear binding force.

Quantum Mechanics | Brilliant Math & Science Wiki (5) Quantum tunneling gives particle wavefunctions a nonzero probability of penetrating a potential barrier [6].

Quantum Mechanics | Brilliant Math & Science Wiki (6) Velocity distribution of rubidium atoms throughout the process of forming a Bose-Einstein condensate, in which a large fraction of atoms occupy the lowest-energy quantum state [7].From the discovery of spin, the intrinsic angular momentum of particles, many new physical effects of quantum origin were understood. Particles are categorized into spins either in half-integer multiples of \(\hbar\), in which case they are called fermions, or integer multiples of \(\hbar\), in which case they are called bosons. Through quantum statistical mechanics and the Spin-Statistics Theorem, it was shown that populations of fermions and bosons behave differently from each other. In particular, no two fermions can occupy the same quantum state, a property called the Pauli exclusion principle in chemistry. The quantum statistical mechanics of particles has colossal influence throughout physics; for instance, it is responsible for the stability of neutron stars, the formation of "macroscopic" quantum states called Bose-Einstein condensates, and even the solid feeling of your desk.

A large body of work in quantum mechanics has gone towards understanding radiation. Classically, it was understood that atoms could both absorb and spontaneously emit light. Since the photon is a boson, Einstein was able to reproduce Planck's original distribution for the power spectrum of black-body radiation using the quantum statistics of bosons. Equally importantly, Einstein used his derivation to predict the stimulated emission of radiation from atoms, showing that the presence of spontaneously emitted photons could drive the further emission of photons at an exponentially growing pace. In other words, Einstein predicted the laser decades before its invention!

Quantum Mechanics | Brilliant Math & Science Wiki (7) A spontaneously emitted photon drives stimulated emission of more photons [8].

Quantum Mechanics | Brilliant Math & Science Wiki (8) Quantum entanglement causes correlations in spin measurements too large to be classically explained [9].

Spin has also been very important in quantum mechanics due to the discovery of quantum entanglement, which Einstein called "spooky action at a distance." It is possible to create two electrons so that their states are not independent; when the spins of these electrons are measured, there is a strong (anti-)correlation between the results, even if the electrons are separated by too large of a distance to influence each other! The fact that these correlations cannot be explained by some unseen variable hidden in the physical system is a theorem in quantum mechanics known as Bell's theorem. Entanglement, quantum teleportation and in general manipulation of superpositions of spin states has enormous implications in cryptography and is the basis of the nascent field of quantum computing.

Quantum mechanics provides an elaborate theoretical framework for describing the interactions between particles via perturbation theory and scattering theory. At high energies such as those present in the Large Hadron Collider (LHC) at CERN, this framework is upgraded to relativistic quantum field theory and the Standard Model of particle physics. It is the hope that the LHC or similar particle colliders will reveal physics beyond our current knowledge of quantum theory.

Quantum Mechanics | Brilliant Math & Science Wiki (9) The diagrammatic shorthand for the mathematics of particle scattering in quantum field theory, called Feynman diagrams [10].

Quantum Mechanics | Brilliant Math & Science Wiki (10) Artist's concept of a black hole and the radiation / accretion disk surrounding it [11]. Quantum effects can even be visible on enormous scales, like the size of the galaxy or even the universe. Quantum mechanics and quantum theories beyond the Standard Model such as string theory can have effects in cosmology and black hole physics. For instance, effects from string theory may alter how the early universe evolved, and leave a characteristic signature in the cosmic microwave background that permeates the universe. Black holes themselves are theorized to directly exhibit quantum effects through Hawking radiation, radiation which escapes black hole event horizons due to quantum vacuum fluctuations.


[1] Griffiths, David J. Introduction to Quantum Mechanics. Second Edition. Pearson: Upper Saddle River, NJ, 2006.

[2] Image from https://upload.wikimedia.org/wikipedia/commons/thumb/1/19/Black_body.svg/2000px-Black_body.svg.png under Creative Commons licensing for reuse and modification.

[3] Image from https://commons.wikimedia.org/w/index.php?curid=39529 under CC BY-SA 3.0.

[4] Image from https://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Quantum_mechanics_travelling_wavefunctions.svg/2000px-Quantum_mechanics_travelling_wavefunctions.svg.png under Creative Commons licensing for reuse and modification.

[5] Image from https://upload.wikimedia.org/wikipedia/commons/e/e7/Hydrogen_Density_Plots.png under Creative Commons licensing for reuse and modification.

[6] Image from https://upload.wikimedia.org/wikipedia/commons/1/1d/TunnelEffektKling1.png under Creative Commons licensing for reuse and modification.

[7] Image from https://upload.wikimedia.org/wikipedia/commons/a/af/Bose_Einstein_condensate.png under Creative Commons licensing for reuse and modification.

[8] Image from https://upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Stimulated_emission-2.svg/640px-Stimulated_emission-2.svg.png under Creative Commons licensing for reuse and modification.

[9] Image from https://upload.wikimedia.org/wikipedia/en/thumb/e/e2/Bell.svg/1195px-Bell.svg.png under Creative Commons licensing for reuse and modification.

[10] Image from https://en.wikipedia.org/wiki/Feynman_diagram#/media/File:Feynman-diagram-ee-scattering.png under Creative Commons licensing for reuse and modification.

[11] Image from https://www.flickr.com/photos/ghost_of_kuji/395419629/in/photostream/ under Creative Commons licensing for reuse and modification.

Quantum Mechanics | Brilliant Math & Science Wiki (2024)
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