States

A '''quantum state''' is any possible state in which a [[quantum mechanics|quantum mechanical system]] can be. A fully specified quantum state can be described by a ''state vector'', a [[wavefunction]], or a complete set of [[quantum number]]s for a specific system. A partially known quantum state, such as an [[statistical ensemble|ensemble]] with some quantum numbers fixed, can be described by a [[density operator]]. Paul A. M. Dirac invented a powerful and intuitive mathematical notation to describe quantum states, known as [[bra-ket notation]]. == Basis states == Any quantum state $|\backslash psi\backslash rangle$ can be expressed in terms of a sum of ''[[Orthonormal basis|basis states]]'' (also called ''basis kets''), $|k\_i\backslash rangle$ $|\; \backslash psi\; \backslash rangle\; =\; \backslash sum\_i\; c\_i\; |\; k\_i\; \backslash rangle$ where $c\_i$ are the coefficients representing the [[probability amplitude]], such that the absolute square of the probability amplitude, $\backslash left\; |\; c\_i\; \backslash right\; |\; ^2$ is the [[probability]] of a [[measurement in quantum mechanics|measurement]] in terms of the basis states yielding the state $|k\_i\backslash rangle$. The normalization condition mandates that the total sum of probabilities is equal to one, $\backslash sum\_i\; \backslash left\; |\; c\_i\; \backslash right\; |\; ^2\; =\; 1$. The simplest understanding of basis states is obtained by examining the [[quantum harmonic oscillator]]. In this system, each basis state $|n\backslash rangle$ has an energy $E\_n\; =\; \backslash hbar\; \backslash omega\; \backslash left(n\; +\; \{\backslash begin\{matrix\}\backslash frac\{1\}\{2\}\backslash end\{matrix\}\}\backslash right)$. The set of basis states can be extracted using a construction operator $a^\{\backslash dagger\}$ and a destruction operator $a$ in what is called the [[Quantum harmonic oscillator#Ladder operator method|ladder operator method]]. == Superposition of states == If a quantum mechanical state $|\backslash psi\backslash rangle$ can be reached by more than one path, then $|\backslash psi\backslash rangle$ is said to be a linear superposition of states. In the case of two paths, if the states after passing through path $\backslash alpha$ and path $\backslash beta$ are $|\backslash alpha\backslash rangle\; =\; \backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}\; |0\backslash rangle\; +\; \backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}\; |1\backslash rangle$, and $|\backslash beta\backslash rangle\; =\; \backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}\; |0\backslash rangle\; -\; \backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}\; |1\backslash rangle$, then $|\backslash psi\backslash rangle$ is defined as the normalized linear sum of these two states. If the two paths are equally likely, this yields $|\backslash psi\backslash rangle\; =\; \backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}|\backslash alpha\backslash rangle\; +\; \backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}|\backslash beta\backslash rangle\; =\; \backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}(\backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}|0\backslash rangle\; +\; \backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}|1\backslash rangle)\; +\; \backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}(\backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}|0\backslash rangle\; -\; \backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}|1\backslash rangle)\; =\; |0\backslash rangle$. Note that in the states $|\backslash alpha\backslash rangle$ and $|\backslash beta\backslash rangle$, the two states $|0\backslash rangle$ and $|1\backslash rangle$ each have a probability of $\backslash begin\{matrix\}\backslash frac\{1\}\{2\}\backslash end\{matrix\}$, as obtained by the absolute square of the probability amplitudes, which are $\backslash begin\{matrix\}\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}$ and $\backslash begin\{matrix\}\backslash pm\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash end\{matrix\}$. 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, $|0\backslash rangle$ is said to constructively interfere, and $|1\backslash rangle$ 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 $\backslash langle\; a\; \backslash rangle$ of a measurement $A$ on a pure quantum state is given by $\backslash langle\; a\; \backslash rangle\; =\; \backslash langle\; \backslash psi\; |\; A\; |\; \backslash psi\; \backslash rangle\; =\; \backslash sum\_i\; a\_i\; \backslash langle\; \backslash psi\; |\; \backslash alpha\_i\; \backslash rangle\; \backslash langle\; \backslash alpha\_i\; |\; \backslash psi\; \backslash rangle\; =\; \backslash sum\_i\; a\_i\; |\; \backslash langle\; \backslash alpha\_i\; |\; \backslash psi\; \backslash rangle\; |^2\; =\; \backslash sum\_i\; a\_i\; P(\backslash alpha\_i)$ where $|\backslash alpha\_i\backslash rangle$ are basis kets for the operator $A$, and $P(\backslash alpha\_i)$ is the probability of $|\; \backslash psi\; \backslash rangle$ being measured in state $|\backslash alpha\_i\backslash rangle$. In order to describe a statistical distribution of pure states, or ''mixed state'', the [[density operator]] (or density matrix), $\backslash rho$, is used. This extends [[quantum mechanics]] to [[quantum statistical mechanics]]. The density operator is defined as $\backslash rho\; =\; \backslash sum\_s\; p\_s\; |\; \backslash psi\_s\; \backslash rangle\; \backslash langle\; \backslash psi\_s\; |$ where $p\_s$ is the fraction of each ensemble in pure state $|\backslash psi\_s\backslash rangle$. The ensemble average of a measurement $A$ on a mixed state is given by $\backslash left\; [\; A\; \backslash right\; ]\; =\; \backslash langle\; \backslash overline\{A\}\; \backslash rangle\; =\; \backslash sum\_s\; p\_s\; \backslash langle\; \backslash psi\_s\; |\; A\; |\; \backslash psi\_s\; \backslash rangle\; =\; \backslash sum\_s\; \backslash sum\_i\; p\_s\; a\_i\; |\; \backslash langle\; \backslash alpha\_i\; |\; \backslash psi\_s\; \backslash rangle\; |^2\; =\; tr(\backslash rho\; A)$ 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 harmonic oscillator]] * [[Bra-ket notation]] * [[Orthonormal basis]] * [[Wavefunction]] * [[Probability amplitude]] * [[Density operator]] * [[Qubit]] [[Category:Handbook of Quantum Information]]