Quantum state

Quite literally, quantum state describes the state of a quantum system. In quantum mechanics this is described using a mathematical representation such as a state vector (also called a wave function for some quantum mechanical systems) or a density operator.

Dirac invented a powerful and intuitive mathematical notation to talk about states, known as the bra and 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 mere notation |1s> which describes the hydrogenoïd bound state 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 in the real space. Other representations, like the projection in momentum (or reciprocal) space, are possible. The different representations are many facets of a single object, the quantum state

It is instructive to consider the most useful quantum states of the harmonic oscillator:

• The Fock state |n> (n an integer) which describes a state of definite energy.
• The coherent state |α> (α a complex number) which describes a state with minimum uncertainty in phase and amplitude.
• The thermal state which describes a state of thermal equilibrium.

The first two states are pure quantum states, i.e., they can be described by a Dirac ket vector, while the latter is a mixed quantum state, i.e., a statistical mixture of pure states. A mixed state needs a statistical description in addition to the quantum description, this is provided by the density matrix which extends quantum mechanics to quantum statistical mechanics. Below these three quantum states are represented on the vivid ladder of harmonic oscillator states. Each step of the ladder is a Fock state, that is raised and lowered respectively through the application on the state of the creation operator a^† and annihilation operator a. The coherent state is a coherent superposition of Fock states with the distribution sketched on the schema. The thermal state is an incoherent superposition with sketched distribution. Those distributions are the diagonal elements of the density matrix of the states. Coherent superposition means that the off-diagonal elements values depend on those of the diagonal. Incoherent superposition means off-diagonal elements are independent of the diagonal (generally they are even just zero).

Some have posited that the quantum state vector of the universe as a whole can be considered, and that the probability of this state vector changes through time. If such a state vector is considered, the probability of the entire state vector could be used as a measure of the complexity of the system.

Quantum states
Sketchy representation of the quantum states (Fock, coherent and thermal) of the harmonic oscillator

External link

• Quantum states of the light field