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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 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. 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 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).
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