Quantum state

Quite literally, the quantum state describes in which state a quantum object is.

Quantum mechanics is a formal theory, i.e., one which describes nonphysical or formal quantities (namely the wavefunction in the Schrödinger picture or quantum operators in the Heisenberg picture) which for a given formalism or interpretation relate to physical observables (namely a probability).

The quantum state is consequently a purely mathematical and abstract concept and also a source of many difficulties when first apprehending the theory. Especially, the quantum state is not the state in which a quantum object is to be found, since a quantum object can only be observed in one eigenstate of the observable, whereas when it is not observed it can be in other quantum states.

Dirac invented a powerful and intuitive notation to capture this abstractness in a mathematical way known as the bra and ket notation. It is very flexible and allows for formal notations which suit the theory very well. For instance, one can refer to an |excited atom> or to 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 polynomial 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. They are the many facets of a unique concept, the quantum state.

It is instructive to consider the most useful quantum state 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 of definite phase.
• 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).

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