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

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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 $|\backslash !\backslash !\backslash uparrow\backslash rangle$ 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

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Any quantum state $|\backslash psi\backslash rangle$ can be expressed in terms of a sum of basis states (also called basis kets) $|k\_i\backslash rangle$ in the form

$|\; \backslash psi\; \backslash rangle\; =\; \backslash sum\_i\; c\_i\; |\; k\_i\; \backslash rangle$

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 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 $\backslash hat\{a\}^\{\backslash dagger\}$ and a destruction operator $\backslash hat\{a\}$ in what is called the ladder operator method.

Superposition of states

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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

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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

$\backslash leftA\; \backslash right=\; \backslash langle\; \backslash overline\{A\}\; \backslash rangle\; =\; \backslash sum\_s\; p\_s\; \backslash langle\; \backslash psi\_s\; |\; A\; |\; \backslash psi\_s\; \backslash rangle\; =\; \backslash sum\_s\; \backslash sum\_i\; p\_s\; a\_i\; |\; \backslash langle\; \backslash alpha\_i\; |\; \backslash psi\_s\; \backslash rangle\; |^2\; =\; tr(\backslash rho\; A)$

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.

Mathematical formulation

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For a mathematical discussion on states as functionals, see GNS construction. There, the same objects are described in a C*-algebraic context.

See also

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

Quantum mechanics

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