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 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 hydrogenoďd 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).
In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called "statevectors") residing in a complex separable Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system).
For instance, the well-known model of the quantumharmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.
Quantum electrodynamics is a quantum theory of electrons, positrons, and the electromagnetic field, and served as a role model for subsequent quantum field theories.
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