Quantum state
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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 <math>|\!\!\uparrow\rangle<math> 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 <math>|\psi\rangle<math> can be expressed in terms of a sum of basis states (also called basis kets), <math>|k_i\rangle<math>
<math>| \psi \rangle = \sum_i c_i | k_i \rangle<math>
where <math>c_i<math> are the coefficients representing the probability amplitude, such that the absolute square of the probability amplitude, <math>\left | c_i \right | ^2<math> is the probability of a measurement in terms of the basis states yielding the state <math>|k_i\rangle<math>. The normalization condition mandates that the total sum of probabilities is equal to one,
<math>\sum_i \left | c_i \right | ^2 = 1<math>.
The simplest understanding of basis states is obtained by examining the quantum harmonic oscillator. In this system, each basis state <math>|n\rangle<math> has an energy <math> E_n = \hbar \omega \left(n + {\begin{matrix}\frac{1}{2}\end{matrix}}\right)<math>. The set of basis states can be extracted using a construction operator <math>a^{\dagger}<math> and a destruction operator <math>a<math> in what is called the ladder operator method.
Superposition of states
If a quantum mechanical state <math>|\psi\rangle<math> can be reached by more than one path, then <math>|\psi\rangle<math> is said to be a linear superposition of states. In the case of two paths, if the states after passing through path <math>\alpha<math> and path <math>\beta<math> are
<math>|\alpha\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |0\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |1\rangle<math>, and
<math>|\beta\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |0\rangle - \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |1\rangle<math>,
then <math>|\psi\rangle<math> is defined as the normalized linear sum of these two states. If the two paths are equally likely, this yields
<math>|\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<math>.
Note that in the states <math>|\alpha\rangle<math> and <math>|\beta\rangle<math>, the two states <math>|0\rangle<math> and <math>|1\rangle<math> each have a probability of <math>\begin{matrix}\frac{1}{2}\end{matrix}<math>, as obtained by the absolute square of the probability amplitudes, which are <math>\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}<math> and <math>\begin{matrix}\pm\frac{1}{\sqrt{2}}\end{matrix}<math>. 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, <math>|0\rangle<math> is said to constructively interfere, and <math>|1\rangle<math> 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 <math>\langle a \rangle<math> of a measurement <math>A<math> on a pure quantum state is given by
<math>\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)<math>
where <math>|\alpha_i\rangle<math> are basis kets for the operator <math>A<math>, and <math>P(\alpha_i)<math> is the probability of <math>| \psi \rangle<math> being measured in state <math>|\alpha_i\rangle<math>.
In order to describe a statistical distribution of pure states, or mixed state, the density operator (or density matrix), <math>\rho<math>, is used. This extends quantum mechanics to quantum statistical mechanics. The density operator is defined as
<math>\rho = \sum_s p_s | \psi_s \rangle \langle \psi_s |<math>
where <math>p_s<math> is the fraction of each ensemble in pure state <math>|\psi_s\rangle<math>. The ensemble average of a measurement <math>A<math> on a mixed state is given by
<math>\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)<math>
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.