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Probability densities for the electron at different quantum numbers (l)

In quantum mechanics, the quantum state of a system is a set of numbers that fully describe a quantum system. Since quantum theory is non-deterministic, these numbers only relate to the likely outcome of measuring a parameter of the system, such as its energy or angular momentum (see Measurement in quantum mechanics). These numbers are called the quantum numbers of the system. Image File history File links Broom_icon. ... Image File history File links HAtomOrbitals. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ...

In classical mechanics, a particle would be described in terms of its position and momentum. In quantum mechanics however, the position and momentum of a particle cannot be exactly measured, so instead, particles are described by a set of quantum numbers that are specific to the system being described. For example, in the case of a single particle in a one dimensional box, the state of a particle can be defined by a single quantum number related to the energy of this particle. In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a very simple problem consisting of a single particle bouncing around inside of an immovable box, from which it cannot escape, and which loses no energy when it collides with...

All experimental predictions are based on the quantum state of the system and the quantum operations acting on the state. A fully specified quantum state can be described by a state vector, a wave function, or a complete set of quantum numbers for a specific system. A partially known quantum state, such as an ensemble with some quantum numbers fixed, can be described by a density matrix. In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... Quantum numbers describe values of conserved quantity in the dynamics of the quantum system. ... In physics, a statistical ensemble is a very large set of similar systems, considered all at once. ... A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. ...

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. 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. In classical mechanics, momentum (pl. ...

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.) 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. 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. A random number generator is a computational or physical device designed to generate a sequence of elements (usually numbers), such that the sequence can be used as a random one. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

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 linear combination of other probability density functions. ...

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 generally 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 outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ...

Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system changes its state. More precisely: After measuring an observable $A$ , the system will be in an eigenstate of $A$ . 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. Entanglement can refer to the process which results in felt from fibers and dust bunnies from hairs etc. ... Bells theorem is the most famous legacy of the late Irish phyisicist John Bell. ...

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. One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger picture. Conceptually (and mathematically), both approaches are equivalent; choosing one of them is a matter of convention. The Heisenberg Picture of quantum mechanics is also known as Matrix mechanics. ... Heisenbergs form for the equations of motion We have seen that in SchrÃ¶dingers scheme the dynamical variables of the system remain fixed during a period of undisturbed motion. ...

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. For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Quantum field theory (QFT) is the quantum theory of fields. ...

Formalism in quantum physics

Bra-ket notation

Paul Dirac invented a powerful and intuitive notation to describe quantum states, known as bra-ket notation. For instance, one can refer to an |excited atom> or to $|!!uparrowrangle$ for a spin-up particle, hiding the underlying complexity of the mathematical description, which is revealed when the state is projected onto a coordinate basis. For instance, the simple notation |1s> describes the first hydrogen atom bound state, but becomes a complicated function in terms of Laguerre polynomials and spherical harmonics when projected onto the basis of position vectors |r>. The resulting expression Ψ(r)=<r|1s>, which is known as the wave function, is a special representation of the quantum state, namely, its projection into position space. Other representations, such as projection into momentum space, are possible. The various representations are simply different expressions of a single physical quantum state. Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... Complexity in general usage is the opposite of simplicity. ... Electron atomic and molecular orbitals In atomic physics and quantum chemistry, the electron configuration is the arrangement of electrons in an atom, molecule, or other physical structure (, a crystal). ... Depiction of a hydrogen atom showing the diameter as about twice the Bohr model radius. ... In physics, a bound state is a composite of two or more building blocks (particles or bodies) that behaves as a single object. ... In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by These polynomials are orthogonal to each other with respect to the inner product given by Also, for each n, Ln(x) is a solution of Laguerres equation which is a second-order... In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ... Electron atomic and molecular orbitals In atomic physics and quantum chemistry, the electron configuration is the arrangement of electrons in an atom, molecule, or other physical structure (, a crystal). ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...

Basis states

Any quantum state $|psirangle$ can be expressed in terms of a sum of basis states (also called basis kets) $|k_irangle$ in the form In mathematics, an orthonormal basis of an inner product space V(i. ...

$| psi rangle = sum_i c_i | k_i rangle$

where $c i$ are the coefficients representing the probability amplitude, such that the absolute square of the probability amplitude, $left | c_i right | ^2$ is the probability of a measurement in terms of the basis states yielding the state $|k_irangle$ . The normalization condition mandates that the total sum of probabilities is equal to one, In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... Probability is the likelihood that something is the case or will happen. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ...

$sum_i left | c_i right | ^2 = 1.$

The simplest understanding of basis states is obtained by examining the quantum harmonic oscillator. In this system, each basis state $|nrangle$ has an energy $E_n = hbar omega left(n + {begin{matrix}frac{1}{2}end{matrix}}right)$ . The set of basis states can be extracted using a construction operator $hat{a}^{dagger}$ and a destruction operator $hat{a}$ in what is called the ladder operator method. The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...

Superposition of states

If a quantum mechanical state $|psirangle$ can be reached by more than one path, then $|psirangle$ is said to be a linear superposition of states. In the case of two paths, if the states after passing through path $α$ and path $β$ are

$|alpharangle = begin{matrix}frac{1}{sqrt{2}}end{matrix} |0rangle + begin{matrix}frac{1}{sqrt{2}}end{matrix} |1rangle,$ and

$|betarangle = begin{matrix}frac{1}{sqrt{2}}end{matrix} |0rangle - begin{matrix}frac{1}{sqrt{2}}end{matrix} |1rangle,$

then $|psirangle$ is defined as the normalized linear sum of these two states. If the two paths are equally likely, this yields

$|psirangle = begin{matrix}frac{1}{sqrt{2}}end{matrix}|alpharangle + begin{matrix}frac{1}{sqrt{2}}end{matrix}|betarangle = begin{matrix}frac{1}{sqrt{2}}end{matrix}(begin{matrix}frac{1}{sqrt{2}}end{matrix}|0rangle + begin{matrix}frac{1}{sqrt{2}}end{matrix}|1rangle) + begin{matrix}frac{1}{sqrt{2}}end{matrix}(begin{matrix}frac{1}{sqrt{2}}end{matrix}|0rangle - begin{matrix}frac{1}{sqrt{2}}end{matrix}|1rangle) = |0rangle.$

Note that in the states $|alpharangle$ and $|betarangle,$ the two states $|0rangle$ and $|1rangle$ each have a probability of $begin{matrix}frac{1}{2}end{matrix},$ as obtained by the absolute square of the probability amplitudes, which are $begin{matrix}frac{1}{sqrt{2}}end{matrix}$ and $begin{matrix}pmfrac{1}{sqrt{2}}end{matrix}.$ In a superposition, it is the probability amplitudes which add, and not the probabilities themselves. The pattern which results from a superposition is often called an interference pattern. In the above case, $|0rangle$ is said to constructively interfere, and $|1rangle$ is said to destructively interfere. Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ...

For more about superposition of states, see the double-slit experiment. Double-slit diffraction and interference pattern The double-slit experiment consists of letting light diffract through two slits, which produces fringes or wave-like interference patterns on a screen. ...

Pure and mixed states

A pure quantum state is a state which can be described by a single ket vector, or as a sum of basis states. A mixed quantum state is a statistical distribution of pure states.

The expectation value $langle a rangle$ of a measurement $A$ on a pure quantum state is given by

where $|alpha_irangle$ are basis kets for the operator $A$ , and $P (α i)$ is the probability of $| psi rangle$ being measured in state $|alpha_irangle.$

In order to describe a statistical distribution of pure states, or mixed state, the density matrix (or density operator), $ρ,$ is used. This extends quantum mechanics to quantum statistical mechanics. The density operator is defined as A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. ...

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

where $p s$ is the fraction of each ensemble in pure state $|psi_srangle.$ The ensemble average of a measurement $A$ on a mixed state is given by

$left [ A right ] = langle overline{A} rangle = sum_s p_s langle psi_s | A | psi_s rangle = sum_s sum_i p_s a_i | langle alpha_i | psi_s rangle |^2 = tr(rho A)$

where it is important to note that two types of averaging are occurring, one being a quantum average over the basis kets of the pure states, and the other being a statistical average over the ensemble of pure states.

Mathematical formulation

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

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 $t 2 > t 1 > 0$ .

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