# Quantum state

(Redirected from Quantum states)

A quantum state is any possible state in which a 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 numbers for a specific system. A partially known quantum state, such as a ensemble with some quantum numbers fixed, can be described by a density operator.

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## 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 [itex]|\!\!\uparrow\rangle[itex] 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 [itex]|\psi\rangle[itex] can be expressed in terms of a sum of basis states (also called basis kets), [itex]|k_i\rangle[itex]

[itex]| \psi \rangle = \sum_i c_i | k_i \rangle[itex]

where [itex]c_i[itex] are the coefficients representing the probability amplitude, such that the absolute square of the probability amplitude, [itex]\left | c_i \right | ^2[itex] is the probability of a measurement in terms of the basis states yielding the state [itex]|k_i\rangle[itex]. The normalization condition mandates that the total sum of probabilities is equal to one,

[itex]\sum_i \left | c_i \right | ^2 = 1[itex].

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

## Superposition of states

If a quantum mechanical state [itex]|\psi\rangle[itex] can be reached by more than one path, then [itex]|\psi\rangle[itex] is said to be a linear superposition of states. In the case of two paths, if the states after passing through path [itex]\alpha[itex] and path [itex]\beta[itex] are

[itex]|\alpha\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |0\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |1\rangle[itex], and

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

then [itex]|\psi\rangle[itex] is defined as the normalized linear sum of these two states. If the two paths are equally likely, this yields

[itex]|\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[itex].

Note that in the states [itex]|\alpha\rangle[itex] and [itex]|\beta\rangle[itex], the two states [itex]|0\rangle[itex] and [itex]|1\rangle[itex] each have a probability of [itex]\begin{matrix}\frac{1}{2}\end{matrix}[itex], as obtained by the absolute square of the probability amplitudes, which are [itex]\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}[itex] and [itex]\begin{matrix}\pm\frac{1}{\sqrt{2}}\end{matrix}[itex]. 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, [itex]|0\rangle[itex] is said to constructively interfere, and [itex]|1\rangle[itex] 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 [itex]\langle a \rangle[itex] of a measurement [itex]A[itex] on a pure quantum state is given by

[itex]\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)[itex]

where [itex]|\alpha_i\rangle[itex] are basis kets for the operator [itex]A[itex], and [itex]P(\alpha_i)[itex] is the probability of [itex]| \psi \rangle[itex] being measured in state [itex]|\alpha_i\rangle[itex].

In order to describe a statistical distribution of pure states, or mixed state, the density operator (or density matrix), [itex]\rho[itex], is used. This extends quantum mechanics to quantum statistical mechanics. The density operator is defined as

[itex]\rho = \sum_s p_s | \psi_s \rangle \langle \psi_s |[itex]

where [itex]p_s[itex] is the fraction of each ensemble in pure state [itex]|\psi_s\rangle[itex]. The ensemble average of a measurement [itex]A[itex] on a mixed state is given by

[itex]\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)[itex]

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.

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