Concepts from Quantum Mechanics

Quantum Mechanics

is the theory of the mechanics governing extremely small systems, comparable to the size of atoms. It was developed at the start of the 20th century to describe phenomena that classical mechanics was unable to explain. These phenomena included the theory of blackbodies, the photoelectric effect, and the wave-like nature of particles.

Quantization

is the phenomenon in which certain quantities in quantum mechanics can only take certain discrete values (unlike in classical mechanics, in which quantities can vary continuously). For example, in the Bohr model of the atom, electrons can only have discrete values of energy, and are said to occupy energy levels. The of energy was discovered by Max Planck, who, in his studies on blackbodies, proposed that electromagnetic energy could only be emitted in discrete, packets. Today, we call these packets photons, which are individual particles of light. Mathematically, Planck discovered that the energy of a photon is directly proportional to its frequency, with the constant of proportionality being Planck’s constant. Planck’s constant appears in many equations in quantum mechanics; it is represented by h, and it also has a form called the reduced Planck’s constant, which equals Planck’s constant over 2 π and is represented by ħ, pronounced “h-bar.”

Superposition

is the principle by which a quantum system can exist in a state that is a simultaneous combination of two or more discrete states. For example, consider a system in which a single particle can be in either position A or position B. In quantum mechanics, there is also a state that combines the state where the particle is in position A and the state where the particle is in position B. According to the Copenhagen interpretation of quantum mechanics, the particle in the state is in neither position A nor B. However, when the system is observed (such as by an experimenter attempting to take a measurement of the particle’s position), the state collapses, and the experimenter will observe the particle to be in either position A or position B. The exact location of the particle does not exist until an external observer intervenes. The apparent absurdity of this real quantum phenomenon is demonstrated in the Schrödinger’s cat thought experiment, in which a cat in a box exists in a quantum superposition of being alive and dead, until the experimenter opens the box to discover either a living or a dead cat.

Wave-particle duality

is the idea that quantum systems can display both -like and -like properties. For example, while light was classically thought to be a —as demonstrated by the diffraction pattern produced in Thomas Young’s double-slit experiment—some phenomena, such as the Planck model of blackbodies mentioned above, can only be explained by light existing as discrete photons. In the photoelectric effect, light shined onto a metal causes the metal to emit electrons. However, the electrons are only emitted when the energy of the light is above a minimum threshold. The threshold is dependent on the type of metal being studied and is called the work function. The existence of this minimum threshold could only be explained by viewing light as photons, an explanation that helped Albert Einstein win the Nobel Prize for Physics in 1921. While those studies showed that waves could act like , the duality was shown to be two-sided when electrons—classically thought to be —were shown to also act like . In the Davisson–Germer experiment, electrons were diffracted off a nickel crystal. The diffraction pattern was similar to one predicted by Bragg’s law, which governs the -like diffraction of X-rays. The electrons were acting like with a (called the de Broglie wavelength) equal to Planck’s constant divided by momentum. Later, electrons were shown to also produce the same diffraction pattern as light in a double-slit experiment setup.

Heinsenberg’s uncertainty principle

roughly states that the value of certain pairs of properties cannot be known simultaneously with arbitrary precision. For example, if an experimenter attempts to very precisely measure the momentum of a particle, they will not be able to precisely measure its position, while if they instead attempt to precisely measure its momentum, they will not be able to precisely measure its position. It is most commonly applied to position and momentum, but is true for any canonically conjugate pair of variables, such as energy and time. More precisely, the uncertainty principle states that the product of the standard deviation of position and momentum must be greater than Planck’s constant divided by 4 π. The uncertainty principle can be rigorously proved using the Fourier transform. Quantum states that have the lowest possible uncertainty are called coherent states. microscope is a thought experiment that provides an argument for the uncertainty principle using classical optics.

Spin

is a quantity possessed by elementary particles in quantum mechanics. It is a form of intrinsic angular momentum. Despite the name, spin does not refer to a particle literally spinning; it is a purely quantum mechanical phenomenon with no classical analogue. can take on integer or half-integer values, and it can be positive or negative, corresponding to whether the angular momentum vector points “up” or “down”; for example, the electron has a of ±½, while photons have a of ±1. Particles with half-integer are called fermions, and particles with integer are called bosons. The quantization of angular momentum was demonstrated in the Stern–Gerlach experiment, in which particles traveling through a magnetic field were alternately deflected up or down depending on the direction of their .

Quantum tunneling

is a phenomenon in which a quantum particle passes through a potential energy barrier that it could not pass through classically. For example, in classical mechanics, if a particle does not have enough energy to get over a hill, it must always roll back down the hill. In quantum mechanics, however, there is a small probability it will get over the hill. Quantum tunneling is a consequence of the wave-like behavior of particles, and it corresponds to an exponentially decaying wavefunction. George Gamow used tunneling to mathematically explain how an alpha particle escapes from a nucleus despite not having enough energy to do so classically. It also forms the basis for how scanning tunneling microscopes (STMs) work. In STMs, a small voltage is applied, which allows electrons to tunnel through a vacuum. By measuring the resultant current, it is possible to map an atomic surface. In stars, tunneling is used to help overcome the Coulomb barrier to allow nuclear fusion to occur. Tunneling also allows Esaki diodes to display negative resistances.

Entanglement

occurs when the state of each particle in a group cannot be described independently of the other particles. This can occur even when the particles are separated by large distances, an apparent violation of the principle of locality. was described by Einstein as “spooky action at a distance” and was first introduced in the paper “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”, by Einstein, Podolsky and Rosen (EPR). They introduced the EPR paradox, in which two entangled particles A and B are physically separated by an extremely large distance, say many light-years apart. The spin of A is then measured, so the spin of B is now instantaneously determined, even though no information could possibly have traveled between A and B (special relativity limits the speed of causality to be no greater than the speed of light). Bell’s theorem states that no theory with hidden variables is able to explain entanglement. particles also violate Bell’s inequalities. The quantum eraser experiment can be explained using . In this experiment, a variant of the classic double-slit experiment, two photons are created, and measuring anything about the first photon affects the second photon, which can destroy an interference pattern created by the second photon.

Pauli exclusion principle

states that no two identical fermions can occupy the same quantum state at the same time. It is a result of the fact that fermions’ half-integer spin means they have antisymmetric wavefunctions, and thus the wavefunction of two identical fermions in the same quantum state cancels out to zero. A consequence of the exclusion principle is that the two electrons in the same atomic orbital must have opposite values for spin. The was first formulated by Wolfgang _____for electrons, before the proof of the spin-statistics theorem allowed it to be extended to all fermions. Electron degeneracy pressure in white dwarfs is a consequence of the exclusion principle, since it means electrons packed into a very small volume must occupy states with higher kinetic energy, resulting in a large pressure.

Bose-Einstein condensate (BEC)

is a state of matter formed when a low-density gas of bosons is cooled to temperatures very close to absolute zero. A forms when a very high proportion of the particles simultaneously occupy the lowest-energy quantum state (or ground state). While the Pauli exclusion principle prevents fermions from occupying the same quantum state, there is no such restriction for bosons. are sometimes called the “fifth state of matter.” The first was observed when Carl Wiemann and Eric Cornell used lasers to cool rubidium-87 atoms. This won them and Wolfgang Ketterle the 2001 Nobel Prize in Physics.

Wavefunction

is a mathematical description of a quantum state in an isolated system, and a way of formalizing many of the ideas discussed above. It is usually represented using a uppercase or lowercase Greek letter psi (Ψ or ψ respectively). In the Copenhagen interpretation, the wavefunction is viewed as a “probability amplitude,” meaning the square of the wavefunction gives the probability density of measuring a particle in a particular place. For example, consider a one-dimensional particle in a box (also called the infinite square well). The for the particle in a box is represented by a sine wave, with the period of the sine wave depending on the energy level of the particle. The crests and troughs of the wave represent positions where the particle has a high probability of being located (were an observer to measure the system and collapse the wavefunction), while at the nodes of the wave, the particle has a zero probability of being at that position. To ensure the total probability measure is one, the spatial integral of the wavefunction times its complex complement must equal one. This condition is called normalization. An atomic orbital can be represented as a wavefunction, representing the possible positions an electron can be in.

Operators

are mathematical functions that act on wavefunctions to yield values of physical quantities called observables. For example, if you know a particle’s wavefunction and want to calculate its momentum, you apply the momentum operator to the wavefunction, and the result equals the particle’s momentum (the observable of interest) multiplied by its wavefunction. The corresponding to total energy in quantum mechanics has a special name, the Hamiltonian. Using the techniques of linear algebra, in quantum mechanics can be represented by matrices, while wavefunctions can be represented by column vectors. The observables are thus the eigenvalues of the (while the wavefunction is the eigenvector).

The commutator

is a function that measures the degree to which the two operators obey the commutative property. It is represented by a pair of square brackets; the commutator of operators A and B is written [A, B] and is equal to A B − B A. If the of two operators is zero, the operators are said to commute, and their corresponding observables can be simultaneously measured. The between the position and momentum operators is not zero, instead equaling i ħ, and thus position and momentum cannot be simultaneously measured (i.e., the uncertainty principle). The ability of an operator to commute with the Hamiltonian has a special property; if an operator commutes with the Hamiltonian, its corresponding observable is a conserved quantity. The is analogous to the Poisson bracket from classical mechanics.

The Schrödinger equation

is a partial differential equation governing how the wavefunction changes in time and space. It has both time-dependent and time-independent forms. The time-independent form states that the Hamiltonian operator applied to the wavefunction equals energy times the wavefunction, while the time-dependent form states that the Hamiltonian applied to the wavefunction equals i ħ times the time derivative of the wavefunction. The Schrödinger equation can be seen as a quantum generalization of Newton’s second law (using Ehrenfest’s theorem). The has exact analytical solutions only for relatively simple systems, such as the particle in a box, the quantum harmonic oscillator, and the hydrogen atom. For more complicated systems, techniques like perturbation theory, the variational principle, and the WKB approximation are used to obtain approximate solutions to the Schrödinger equation.