Quantum state

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In quantum mechanics, 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 an ensemble with some quantum numbers fixed, can be described by a density operator.

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 [|\!\!\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, such as 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 basis states (also called basis kets) [|k_i\rangle] in the form

[| \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_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 + \frac\end}\right)]. The set of basis states can be extracted using a construction operator [\hat^] and a destruction operator [\hat] in what is called the 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\frac}\end |0\rangle + \begin\frac}\end |1\rangle,] and

[|\beta\rangle = \begin\frac}\end |0\rangle - \begin\frac}\end |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\frac}\end|\alpha\rangle + \begin\frac}\end|\beta\rangle = \begin\frac}\end(\begin\frac}\end|0\rangle + \begin\frac}\end|1\rangle) + \begin\frac}\end(\begin\frac}\end|0\rangle - \begin\frac}\end|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\frac\end,] as obtained by the absolute square of the probability amplitudes, which are [\begin\frac}\end] and [\begin\pm\frac}\end.] 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 \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 GNS construction. There, the same objects are described in a C*-algebraic context.

See also

• Quantum harmonic oscillator
• Bra-ket notation
• Orthonormal basis
• Wavefunction
• Probability amplitude
• Density operator
• Qubit

 

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