Quantum state

Quantum mechanics

$\Delta x \, \Delta p \ge \frac{\hbar}{2}$

Fundamental concepts
Quantum state · Wave function Superposition · Entanglement Measurement · Uncertainty Exclusion · Duality Decoherence · Ehrenfest theorem · Tunneling

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In quantum physics, a quantum state is a mathematical object that fully describes a quantum system. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Quantum mechanics (QM or quantum theory) is a physical science dealing with the behavior of Matter and Energy on the scale of Atoms The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system Quantum superposition is the fundamental law of Quantum mechanics. Quantum entanglement is a quantum mechanical Phenomenon in which the Quantum states of two or more objects are linked together so that one object The framework of Quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and In Quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior The Ehrenfest theorem, named after Paul Ehrenfest, relates the time Derivative of the expectation value for a quantum mechanical operator In Quantum mechanics, quantum tunnelling is a nanoscopic phenomenon in which a particle violates the principles of Classical mechanics by penetrating a Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Quantum states can be statistically mixed, corresponding to an experiment involving a random change of the parameters. A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. States obtained in this way are called mixed states, as opposed to pure states which cannot be described as a mixture of others. When performing a certain measurement on a quantum state, the result is in general described by a probability distribution, and the form that this distribution takes is completely determined by the quantum state and the observable describing the measurement. The framework of Quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical However, unlike in classical mechanics, the result of a measurement on even a pure quantum state is only determined probabilistically. This reflects a core difference between classical and quantum physics.

Mathematically, a pure quantum state is typically represented by a vector in a Hilbert space. This article assumes some familiarity with Analytic geometry and the concept of a limit. In physics, bra-ket notation is often used to denote such vectors. Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical Linear combinations (superpositions) of vectors can describe interference phenomena. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics Quantum superposition is the fundamental law of Quantum mechanics. In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern Mixed quantum states are described by density matrices.

In a more general mathematical context, quantum states can be understood as positive normalized linear functionals on a C* algebra; see GNS construction. This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. In Functional analysis, given a C*-algebra A, the Gelfand-Naimark-Segal construction establishes a correspondence between cyclic *-representations of

1 Conceptual description
2 Formalism in quantum physics
3 Mathematical formulation
4 Notes
5 See also
6 Further reading

Conceptual description

The state of a physical system

The state of a physical system is a complete description of the parameters of the experiment. To understand this rather abstract notion, it is useful to first explore it in an example from classical mechanics.

Consider an experiment with a (non-quantum) particle of mass $m = 1$ which moves freely, and without friction, in one spatial direction.

We start the experiment at time $t = 0$ by pushing the particle with some speed into some direction. Doing this, we determine the initial position $q$ and the initial momentum^[1] $p$ of the particle. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product These initial conditions are what characterizes the state $σ$ of the system, formally denoted as $σ = (p, q)$ . We say that we prepare the state of the system by fixing its initial conditions.

At a later time $t > 0$ , we conduct measurements on the particle. The measurements we can perform on this simple system are essentially its position $Q (t)$ at time $t$ , its momentum $P (t)$ , and combinations of these. Here $P (t)$ and $Q (t)$ refer to the measurable quantities (observables) of the system as such, not the specific results they produce in a certain run of the experiment.

However, knowing the state $σ$ of the system, we can compute the value of the observables in the specific state, i. e. , the results that our measurements will produce, depending on $p$ and $q$ . We denote these values as $\langle P(t) \rangle _\sigma$ and $\langle Q(t) \rangle _\sigma$ . In our simple example, it is well known that the particle moves with constant velocity; therefore,

$\langle P(t) \rangle _\sigma = p, \quad \langle Q(t) \rangle _\sigma = pt+q.$

Now suppose that we start the particle with a random initial position and momentum. (For argument's sake, we may suppose that the particle is pushed away at $t = 0$ by some apparatus which is controlled by a random number generator. A random number generator (often abbreviated as RNG is a computational or physical device designed to generate a sequence of Numbers or symbols that lack any ) The state $σ$ of the system is now not described by two numbers $p$ and $q$ , but rather by two probability distributions. The observables $P (t)$ and $Q (t)$ will produce random results now; they become random variables, and their values in a single measurement cannot be predicted. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way However, if we repeat the experiment sufficiently often, always preparing the same state $σ$ , we can predict the expectation value of the observables (their statistical mean) in the state $σ$ . The expectation value of $P (t)$ is again denoted by $\langle P(t) \rangle _\sigma$ , etc.

These "statistical" states of the system are called mixed states, as opposed to the pure states $σ = (p, q)$ discussed further above. Abstractly, mixed states arise as a statistical mixture of pure states. In Statistics, a mixture density is a Probability density function which is a Convex combination of other probability density functions

Quantum states

Probability densities for the electron of a hydrogen atom in different quantum states

In quantum systems, the conceptual distinction between observables and states persists just as described above. The state $σ$ of the system is fixed by the way the physicist prepares his experiment (e. g. , how he adjusts his particle source). As above, there is a distinction between pure states and mixed states, the latter being statistical mixtures of the former. However, some important differences arise in comparison with classical mechanics.

In quantum theory, even pure states show statistical behaviour. Regardless of how carefully we prepare the state $ρ$ of the system, measurement results are not repeatable in general, and we must understand the expectation value $\langle A \rangle _\sigma$ of an observable $A$ as a statistical mean. It is this mean that is predicted by physical theories.

For any fixed observable $A$ , it is generally possible to prepare a pure state $σ A$ such that $A$ has a fixed value in this state: If we repeat the experiment several times, each time measuring $A$ , we will always obtain the same measurement result, without any random behaviour. Such pure states $σ A$ are called eigenstates of $A$ .

However, it is impossible to prepare a simultaneous eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement $Q (t)$ and the momentum measurement $P (t)$ (at the same time $t$ ) produce "sharp" results; at least one of them will exhibit random behaviour. ^[2] This is the content of the Heisenberg uncertainty relation. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain

Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system generally changes its state. More precisely: After measuring an observable $A$ , the system will be in an eigenstate of $A$ ; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure $A$ twice in the same run of the experiment, the measurements being directly consecutive in time, then they will produce the same results. This has some strange consequences however:

Consider two observables, $A$ and $B$ , where $A$ corresponds to a measurement earlier in time than $B$ . ^[3] Suppose that the system is in an eigenstate of $B$ . If we measure only $B$ , we will not notice statistical behaviour. If we measure first $A$ and then $B$ in the same run of the experiment, the system will transfer to an eigenstate of $A$ after the first measurement, and we will generally notice that the results of $B$ are statistical. Thus, quantum mechanical measurements influence one another, and it is important in which order they are performed.

Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called entangled states, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see entanglement. These entangled states lead to experimentally testable properties (Bell's theorem) that allow to distinguish between quantum theory and alternative classical (non-quantum) models. Bell's theorem is a theorem that shows that the predictions of Quantum mechanics (QM are not intuitive and touches upon fundamental philosophical issues that relate to modern

Schrödinger picture vs. Heisenberg picture

In the discussion above, we have taken the observables $P (t)$ , $Q (t)$ to be dependent on time, while the state $σ$ was fixed once at the beginning of the experiment. This approach is called the Heisenberg picture. In Physics, the Heisenberg picture is that formulation of Quantum mechanics where the operators (observables and others are time-dependent and the state vectors One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger picture. In Quantum mechanics, a state function is a linear combination (a superposition of eigenstates. Conceptually (and mathematically), both approaches are equivalent; choosing one of them is a matter of convention.

Both viewpoints are used in quantum theory. While non-relativistic quantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In quantum field theory (QFT the forces between particles are mediated by other particles

Formalism in quantum physics

See also: Mathematical formulation of quantum mechanics

Pure states as rays in a Hilbert space

Quantum physics is most commonly formulated in terms of linear algebra, as follows. The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. Linear algebra is the branch of Mathematics concerned with Any given system is identified with some Hilbert space, such that each vector in the Hilbert space (apart from the origin) corresponds to a pure quantum state. In addition, two vectors that differ only by a nonzero complex scalar correspond to the same state (in other words, each pure state is a ray in the Hilbert space). In Mathematics, scalar multiplication is one of the basic operations defining a Vector space in Linear algebra (or more generally a module in

Alternatively, many authors choose to only consider normalized vectors (vectors of norm 1) as corresponding to quantum states. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In this case, the set of all pure states corresponds to the unit sphere of a Hilbert space, with the proviso that two normalized vectors correspond to the same state if they differ only by a complex scalar of absolute value 1 (called a phase factor). In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed In Quantum Mechanics, a phase factor is a complex scalar number of Absolute value 1 that multiplies a bra or ket.

Bra-ket notation

Main article: Bra-ket notation

Calculations in quantum mechanics make frequent use of linear operators, inner products, dual spaces, and Hermitian conjugation. Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Mathematics, specifically in Functional analysis, each Linear operator on a Hilbert space has a corresponding adjoint operator. In order to make such calculations more straightforward, and to obviate the need (in some contexts) to fully understand the underlying linear algebra, Paul Dirac invented a notation to describe quantum states, known as bra-ket notation. Although the details of this are beyond the scope of this article (see the article Bra-ket notation), some consequences of this are:

The variable name used to denote a vector (which corresponds to a pure quantum state) is chosen to be of the form $|\psi\rangle$ (where the " $ψ$ " can be replaced by any other symbols, letters, numbers, or even words). Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical This can be contrasted with the usual mathematical notation, where vectors are usually bold, lower-case letters, or letters with arrows on top.
Instead of vector, the term ket is used synonymously.
Each ket $|\psi\rangle$ is uniquely associated with a so-called bra, denoted $\langle\psi|$ , which is also said to correspond to the same physical quantum state. Technically, the bra is an element of the dual space, and related to the ket by the Riesz representation theorem. There are several well-known theorems in Functional analysis known as the Riesz representation theorem.
Inner products (also called brackets) are written so as to look like a bra and ket next to each other: $\lang \psi_1|\psi_2\rang$ . In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. (Note that the phrase "bra-ket" is supposed to resemble "bracket". )

Spin, Many-body states

Main article: Mathematical formulation of quantum mechanics#Spin

It is important to note that in quantum mechanics besides, e. The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. g. , the usual position variable $\mathbf r$ , a discrete variable m exists, corresponding to the value of the z-component of the spin vector. This is some kind of intrinsic angular momentum, which does, however, not appear at all in classical mechanics and is in fact a legacy from Dirac's relativistic generalization of the theory. As a consequence, the quantum state of a system of N particles is described by a function with four variables per particle, e. g. $|\psi (\mathbf r_1,m_1;\dots ;\mathbf r_N,m_N)\rangle$ . Here, the variables m_ν assume values from the set { $- S ν, - S ν + 1,. . . , + S ν - 1, + S ν$ }, where $S ν$ (in units of Planck's reduced constant $\hbar$ ), is either a non-negative integer (0,1,2. . . ; bosons), or semi-integer (1/2,3/2,5/2,. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein . . ; fermions). In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. Moreover, in the case of identical particles, the above N-particle function must either be symmetrized (in the bosonic case) or anti-symmetrized (in the fermionic case) w. r. t. the particle numbers.

Electrons are fermions with S=1/2, photons (quanta of light) are bosons with S=1.

Apart from the symmetrization or anti-symmetrization, N-particle states can thus simply be obtained by tensor products of one-particle states, to which we return herewith. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector

Basis states of one-particle systems

As with any vector space, if a basis is chosen for the Hilbert space of a system, then any ket can be expanded as a linear combination of those basis elements. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics Symbolically, given basis kets $|k_i\rang$ , any ket $|\psi\rang$ can be written

$| \psi \rang = \sum_i c_i |k_i \rangle$

where c_i are complex numbers. In physical terms, this is described by saying that $|\psi\rang$ has been expressed as a quantum superposition of the states $|k_i\rang$ . If the basis kets are chosen to be orthonormal (as is often the case), then $c_i=\lang k_i|\psi\rang$ . In Mathematics, an orthonormal basis of an Inner product space V (i

One property worth noting is that the normalized states $|\psi\rang$ are characterized by

∑	\| c_i \| ² = 1
i

Expansions of this sort play an important role in measurement in quantum mechanics. In particular, If the $|k_i\rang$ are eigenstates (with eigenvalues $k i$ ) of an observable, and that observable is measured on the normalized state $|\psi\rang$ , then the probability that the result of the measurement is k_i is |c_i|². In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes (The normalization condition above mandates that the total sum of probabilities is equal to one. )

A particularly important example is the position basis, which is the basis consisting of eigenstates of the observable which corresponds to measuring position. If these eigenstates are nondegenerate (for example, if the system is a single, spinless particle), then any ket $|\psi\rang$ is associated with a complex-valued function of three-dimensional space:

$\psi(\mathbf{r}) \equiv \lang \mathbf{r} | \psi \rang$ . In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin

This function is called the wavefunction corresponding to $|\psi\rang$ . A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system

Superposition of pure states

Main article: Quantum superposition

One aspect of quantum states, mentioned above, is that superpositions of them can be formed. Quantum superposition is the fundamental law of Quantum mechanics. In Physics and Systems theory, the superposition principle, also known as superposition property, states that for all Linear systems If $|\alpha\rangle$ and $|\beta\rangle$ are two kets corresponding to quantum states, the ket

$c_\alpha|\alpha\rang+c_\beta|\beta\rang$

is a different quantum state (possibly not normalized). Note that which quantum state it is depends on both the amplitudes and phases (arguments) of $c α$ and $c β$ . In other words, for example, even though $|\psi\rang$ and $e^{i\theta}|\psi\rang$ (for real θ) correspond to the same physical quantum state, they are not interchangeable, since for example $|\phi\rang+|\psi\rang$ and $|\phi\rang+e^{i\theta}|\psi\rang$ do not (in general) correspond to the same physical state. However, $|\phi\rang+|\psi\rang$ and $e^{i\theta}(|\phi\rang+|\psi\rang)$ do correspond to the same physical state. This is sometimes described by saying that "global" phase factors are unphysical, but "relative" phase factors are physical and important.

One example of a quantum interference phenomenon that arises from superposition is the double-slit experiment. The photon state is a superposition of two different states, one of which corresponds to the photon having passed through the left slit, and the other corresponding to passage through the right slit. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena The relative phase of those two states has a value which depends on the distance from each of the two slits. Depending on what that phase is, the interference is constructive at some locations and destructive in others, creating the interference pattern.

Another example of the importance of relative phase in quantum superposition is Rabi oscillations, where the relative phase of two states varies in time due to the Schrödinger equation. In Physics, the Rabi cycle is the cyclic behaviour of a Two-state quantum system in the presence of an oscillatory driving field In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system The resulting superposition ends up oscillating back and forth between two different states.

Mixed states

See also: Density matrix

A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see quantum statistical mechanics). In Mathematical physics, especially as introduced into Statistical mechanics and Thermodynamics by J Quantum statistical mechanics is the study of Statistical ensembles of quantum mechanical systems.

A mixed state cannot be described as a ket vector. Instead, it is described by its associated density matrix (or density operator), usually denoted $ρ$ . Note that density matrices can describe both mixed and pure states, treating them on the same footing.

The density matrix is defined as

$\rho = \sum_s p_s | \psi_s \rangle \langle \psi_s |$

where $p s$ is the fraction of the ensemble in each pure state $|\psi_s\rangle.$ Here, one typically uses a one-particle formalism to describe the average behaviour of a N-particle system.

A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of ρ² is equal to 1 if the state is pure, and less than 1 if the state is mixed. In Linear algebra, the trace of an n -by- n Square matrix A is defined to be the sum of the elements on the Main diagonal Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state. In Quantum statistical mechanics, von Neumann entropy refers to the extension of classical Entropy concepts to the field of Quantum mechanics.

The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (expectation value) of a measurement corresponding to an observable $A$ is given by

where $|\alpha_i\rangle, \; a_i$ are eigenkets and eigenvalues, respectively, for the operator $A$ , and tr denotes trace. In Quantum mechanics, the expectation value is the predicted mean value of the result of an experiment It is important to note that two types of averaging are occurring, one being a quantum average over the basis kets $|\psi_s\rangle$ of the pure states, and the other being a statistical average with the probabilities $p s$ of those states.

W. r. t. these different types of averaging, i. e. to distinguish pure and/or mixed states, one often uses the expressions 'coherent' and/or 'incoherent superposition' of quantum states.

Mathematical formulation

For a mathematical discussion on states as functionals, see Gelfand-Naimark-Segal construction. In Functional analysis, given a C*-algebra A, the Gelfand-Naimark-Segal construction establishes a correspondence between cyclic *-representations of There, the same objects are described in a C*-algebraic context.

Notes

^ If you are not familiar with the concept of momentum, think of it as being the velocity of the particle. That is fully justified in this context.
^ To avoid misunderstandings: Here we mean that $Q (t)$ and $P (t)$ are measured in the same state, but not in the same run of the experiment. )
^ For concreteness' sake, you may suppose that $A = Q (t 1)$ and $B = P (t 2)$ in the above example, with $t 2 > t 1 > 0$ .

Dictionary

-noun

(physics) any of the possible states of a quantum mechanical system

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