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In theoretical physics, quantum geometry is the set of mathematical concepts that generalize geometry to describe physical phenomena at distance scales comparable to the Planck length. At such distances, quantum mechanics has a profound effect on physical phenomena.
Each theory of quantum gravity uses the term "quantum geometry" in a slightly different fashion. String theory, a leading candidate for a quantum theory of gravity, uses it to describe exotic phenomena such as T-duality and other geometric dualities, mirror symmetry, topology-changing transitions[ clarification needed ], minimal possible distance scale, and other effects that challenge intuition. More technically, quantum geometry refers to the shape of a spacetime manifold as experienced by D-branes, which includes quantum corrections to the metric tensor, such as the worldsheet instantons. For example, the quantum volume of a cycle is computed from the mass of a brane wrapped on this cycle.
In an alternative approach to quantum gravity called loop quantum gravity (LQG), the phrase "quantum geometry" usually refers to the formalism within LQG where the observables that capture the information about the geometry are well-defined operators on a Hilbert space. In particular, certain physical observables, such as the area, have a discrete spectrum. LQG is non-commutative. [1]
It is possible (but considered unlikely) that this strictly quantized understanding of geometry is consistent with the quantum picture of geometry arising from string theory.
Another approach, which tries to reconstruct the geometry of space-time from "first principles" is Discrete Lorentzian quantum gravity.
Differential forms are used to express quantum states, using the wedge product: [2]
where the position vector is
the differential volume element is
and x1, x2, x3 are an arbitrary set of coordinates, the upper indices indicate contravariance, lower indices indicate covariance, so explicitly the quantum state in differential form is:
The overlap integral is given by:
in differential form this is
The probability of finding the particle in some region of space R is given by the integral over that region:
provided the wave function is normalized. When R is all of 3d position space, the integral must be 1 if the particle exists.
Differential forms are an approach for describing the geometry of curves and surfaces in a coordinate independent way. In quantum mechanics, idealized situations occur in rectangular Cartesian coordinates, such as the potential well, particle in a box, quantum harmonic oscillator, and more realistic approximations in spherical polar coordinates such as electrons in atoms and molecules. For generality, a formalism which can be used in any coordinate system is useful.
Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread.
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. Due to the larger relative mass of a nucleus compared to an electron, the coordinates of the nuclei in a system are approximated as fixed, while the coordinates of the electrons are dynamic. The approach is named after Max Born and his 23-year-old graduate student J. Robert Oppenheimer, the latter of whom proposed it in 1927 during a period of intense ferment in the development of quantum mechanics.
In quantum physics, a wave function is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters ψ and Ψ. Wave functions are complex-valued. For example, a wave function might assign a complex number to each point in a region of space. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density of measuring a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called normalization. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.
An operator is a function over a space of physical states onto another space of states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square of the modulus of this quantity represents a probability density.
The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons. Only a small subset of all possible fermionic wave functions can be written as a single Slater determinant, but those form an important and useful subset because of their simplicity.
In physics, the S-matrix or scattering matrix is a matrix which relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. In quantum mechanics, it means the particle is in a region of uniform potential, usually set to zero in the region of interest since the potential can be arbitrarily set to zero at any point in space.
In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.
In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is: where ħ is the reduced Planck constant, i the imaginary unit, x is the spatial coordinate, and a partial derivative is used instead of a total derivative since the wave function is also a function of time. The "hat" indicates an operator. The "application" of the operator on a differentiable wave function is as follows:
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle.
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.
In differential geometry, a field of mathematics, a Courant algebroid is a vector bundle together with an inner product and a compatible bracket more general than that of a Lie algebroid.
In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring. It is a fundamental concept in all areas of quantum physics.
A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with two neutrons, depending on the isotope, held together by the strong force. Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be used to estimate the ground state energy and wavefunction of the atom. Historically, the first such helium spectrum calculation was done by Albrecht Unsöld in 1927. Its success was considered to be one of the earliest signs of validity of Schrödinger's wave mechanics.
Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
