Quantum state

A quantum state is any possible state in which a quantum mechanics can be. A fully specified quantum state can be described by a ''state vector'', a wavefunction, or a complete set of quantum numbers for a specific system. A partially known quantum state, such as a statistical ensemble with some quantum numbers fixed, can be described by a density operator. == Bra-ket notation == Paul Dirac invented a powerful and intuitive mathematical notation to describe quantum states, known as bra-ket notation. For instance, one can refer to an |''excited atom''> or to

|\!\!\uparrow\rangle

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 wavefunction, is a special representation of the quantum state, namely, its projection into position space. Other representations, like the projection into momentum space, are possible. The various representations are simply different expressions of a single physical quantum state. == Basis states == Any quantum state

|\psi\rangle

can be expressed in terms of a sum of ''Orthonormal basis'' (also called ''basis kets''),

|k_i\rangle

| \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 quantum mechanics in terms of the basis states yielding the state

|k_i\rangle

. The normalization condition mandates that the total sum of probabilities is equal to one,

\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

|n\rangle

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

a^{\dagger}

and a destruction operator

a

in what is called the Quantum harmonic oscillator#Ladder operator method. == Superposition of states == If a quantum mechanical state

|\psi\rangle

can be reached by more than one path, then

|\psi\rangle

is said to be a linear superposition of states. In the case of two paths, if the states after passing through path

\alpha

and path

\beta

are

|\alpha\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |0\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |1\rangle

, and

|\beta\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |0\rangle - \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |1\rangle

, then

|\psi\rangle

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

|\psi\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|\alpha\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|\beta\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}(\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|0\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|1\rangle) + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}(\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|0\rangle - \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|1\rangle) = |0\rangle

. Note that in the states

|\alpha\rangle

and

|\beta\rangle

, the two states

|0\rangle

and

|1\rangle

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}\pm\frac{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,

|0\rangle

is said to constructively interfere, and

|1\rangle

is said to destructively interfere. For more about superposition of states, see the double-slit experiment. == 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

\langle a \rangle = \langle \psi | A | \psi  \rangle = \sum_i a_i \langle \psi | \alpha_i \rangle \langle \alpha_i | \psi \rangle = \sum_i a_i | \langle \alpha_i | \psi \rangle |^2 = \sum_i a_i P(\alpha_i)

where

|\alpha_i\rangle

are basis kets for the operator

A

, and

P(\alpha_i)

is the probability of

| \psi \rangle

being measured in state

|\alpha_i\rangle

. In order to describe a statistical distribution of pure states, or ''mixed state'', the density operator (or density matrix),

\rho

, is used. This extends quantum mechanics to quantum statistical mechanics. The density operator is defined as

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

where

p_s

is the fraction of each ensemble in pure state

|\psi_s\rangle

. 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. == See also == * Quantum mechanics * Quantum harmonic oscillator * Bra-ket notation * Orthonormal basis * Wavefunction * Density operator * Qubit Quantum mechanics

Quantum state

The page is nice but it's a bit too technical for non-physics (like me!). Can it be rewritten with more user-friendliness, without perhaps sacrificing completeness? It wouls really be nice to see some more on what quantumphysical characteristics of particles define their quantum states. Technical point: The description of the quantum state as formal and non-physical, as against 'real' measurements, is highly debatable. In many accounts of quantum measurement the quantum state (density matrix) is the _only_ physical reality, the results of a measurement also being expressed in terms of a quantum state. Since everything in the Universe is quantum, including the measuring devices, the idea that the result of a measurement is somehow more 'real' can only be a shorthand for the particular type of quantum state which is produced by interactions with a measuring device and the environment. The page as it stands gives only an old-fashioned Copenhagen-like account of measurement, which by itself is incomplete and unsatisfactory since it doesn't describe what constitutes a measurement and how the system interacts with what's measuring it.

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