Quantum state space

Last updated July 01, 2024

In physics, a quantum state space is an abstract space in which different "positions" represent not literal locations, but rather quantum states of some physical system. It is the quantum analog of the phase space of classical mechanics.

Relative to Hilbert space

In quantum mechanics a state space is a separable complex Hilbert space. The dimension of this Hilbert space depends on the system we choose to describe.^[1]^[2] The different states that could come out of any particular measurement form an orthonormal basis, so any state vector in the state space can be written as a linear combination of these basis vectors. Having an nonzero component along multiple dimensions is called a superposition. In the formalism of quantum mechanics these state vectors are often written using Dirac's compact bra–ket notation.^[3]^: 165

Examples

The spin state of a silver atom in the Stern–Gerlach experiment can be represented in a two state space. The spin can be aligned with a measuring apparatus (arbitrarily called 'up') or oppositely ('down').^[4] In Dirac's notation these two states can be written as $|u\rangle ,|d\rangle$ . The space of a two spin system has four states, $|uu\rangle ,|ud\rangle ,|du\rangle ,|dd\rangle$ .

The spin state is a discrete degree of freedom; quantum state spaces can have continuous degrees of freedom. For example, a particle in one space dimension has one degree of freedom ranging from $-\infty$ to $\infty$ . In Dirac notation, the states in this space might be written as $|q\rangle$ or $|\psi \rangle$ .^[5]^: 302

Relative to 3D space

Even in the early days of quantum mechanics, the state space (or configurations as they were called at first) was understood to be essential for understanding simple quantum-mechanical problems. In 1929, Nevill Mott showed that "tendency to picture the wave as existing in ordinary three dimensional space, whereas we are really dealing with wave functions in multispace" makes analysis of simple interaction problems more difficult.^[6] Mott analyzes $\alpha$ -particle emission in a cloud chamber. The emission process is isotropic, a spherical wave in quantum mechanics, but the tracks observed are linear.

Alpha particle track in a cloud chamber AlphaTrackRutherfordScattering3.jpg — Alpha particle track in a cloud chamber

As Mott says, "it is a little difficult to picture how it is that an outgoing spherical wave can produce a straight track; we think intuitively that it should ionise atoms at random throughout space". This issue became known at the Mott problem. Mott then derives the straight track by considering correlations between the positions of the source and two representative atoms, showing that consecutive ionization results from just that state in which all three positions are co-linear. ^[7]

Relative to classical phase space

Classical mechanics for multiple objects describes their motion in terms of a list or vector of every object's coordinates and velocity. As the objects move, the values in the vector change; the set of all possible values is called a phase space.^[8]^: 88 In quantum mechanics a state space is similar, however in the state space two vectors which are scalar multiples of each other represent the same state. Furthermore, the character of values in the quantum state differ from the classical values: in the quantum case the values can only be measured statistically (by repetition over many examples) and thus do not have well defined values at every instant of time. ^[5]^: 294

Related Research Articles

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 mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces, and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.

<span class="mw-page-title-main">Pauli exclusion principle</span> Quantum mechanics rule: identical fermions cannot occupy the same quantum state simultaneously

In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins cannot simultaneously occupy the same quantum state within a system that obeys the laws of quantum mechanics. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940.

Quantum mechanics is a fundamental theory in physics that describes the behavior of nature at and below the scale of atoms. It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science.

The Schrödinger equation is a partial differential equation that governs the wave function of a 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 mechanics, wave function collapse, also called reduction of the state vector, occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an observation, and is the essence of a measurement in quantum mechanics, which connects the wave function with classical observables such as position and momentum. Collapse is one of the two processes by which quantum systems evolve in time; the other is the continuous evolution governed by the Schrödinger equation.

Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system.

<span class="mw-page-title-main">Wave function</span> Mathematical description of the quantum state of a system

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.

Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum jumps supplanted the Bohr model's electron orbits. It did so by interpreting the physical properties of particles as matrices that evolve in time. It is equivalent to the Schrödinger wave formulation of quantum mechanics, as manifest in Dirac's bra–ket notation.

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.

In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the context of quantum field theory (QFT), the equations determine the dynamics of quantum fields. The solutions to the equations, universally denoted as $ψ$ or $Ψ$ , are referred to as "wave functions" in the context of RQM, and "fields" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a wave equation or are generated from a Lagrangian density and the field-theoretic Euler–Lagrange equations.

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:

<span class="mw-page-title-main">Spin-1/2</span> Type of matter

In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of 1/2. The spin number describes how many symmetrical facets a particle has in one full rotation; a spin of 1/2 means that the particle must be rotated by two full turns before it has the same configuration as when it started.

First quantization is a procedure for converting equations of classical particle equations into quantum wave equations. The companion concept of second quantization converts classical field equations in to quantum field equations.

In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light c, and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. Non-relativistic quantum mechanics refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. Relativistic quantum mechanics (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them also work with special relativity.

Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory.

In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. A weather map, with the surface temperature described by assigning a number to each point on the map, is an example of a scalar field. A surface wind map, assigning an arrow to each point on a map that describes the wind speed and direction at that point, is an example of a vector field, i.e. a 1-dimensional (rank-1) tensor field. Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics. For instance, the electric field is another rank-1 tensor field, while electrodynamics can be formulated in terms of two interacting vector fields at each point in spacetime, or as a single-rank 2-tensor field.

In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a quantum-mechanical prediction for the system represented by the state. Knowledge of the quantum state, and the quantum mechanical rules for the system's evolution in time, exhausts all that can be known about a quantum system.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. In application, understanding symmetries can also provide insights on the eigenstates that can be expected. For example, the existence of degenerate states can be inferred by the presence of non commuting symmetry operators or that the non degenerate states are also eigenvectors of symmetry operators.

References

↑ McIntyre, David (2012). Quantum Mechanics: A Paradigms Approach (1st ed.). Pearson. ISBN 978-0321765796.
↑ Bengtsson, Ingemar; Życzkowski, Karol (2017). Geometry of Quantum States (2nd ed.). Cambridge University Press. ISBN 978-1139207010.
↑ Schiff, Leonard (1949). Quantum mechanics. McGraw-Hill.
↑ Susskind, Leonard; Friedman, Art; Susskind, Leonard (2014). Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics]. The theoretical minimum / Leonard Susskind and George Hrabovsky. New York, NY: Basic Books. ISBN 978-0-465-06290-4.
1 2 Messiah, Albert (1966). Quantum Mechanics. North Holland, John Wiley & Sons. ISBN 0486409244.
↑ "The wave mechanics of ∝-Ray tracks". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 126 (800): 79–84. 1929-12-02. doi: 10.1098/rspa.1929.0205 . ISSN 0950-1207.
↑ Figari, Rodolfo; Teta, Alessandro (2013). "Emergence of classical trajectories in quantum systems: the cloud chamber problem in the analysis of Mott (1929)". Archive for History of Exact Sciences. 67 (2): 215–234. arXiv: 1209.2665 . doi:10.1007/s00407-012-0111-z. ISSN 0003-9519. S2CID 253891627.
↑ Susskind, Leonard; Hrabovsky, George; Susskind, Leonard (2014). The theoretical minimum: what you need to know to start doing physics. The theoretical minimum / Leonard Susskind and George Hrabovsky (Paperback 1. publ ed.). New York: Basic Books. ISBN 978-0-465-07568-3.