See also: state (homonymy)

In Mechanical quantum, the state of a system describes all the aspects of the physical system. It is represented by a mathematical object which gives the maximum of information possible on the system, with an aim of envisaging the results of the experiments which one can carry out. A basic difference between the traditional Mechanical and the quantum Mécanique is due to the way in which is described the state of a physical Système. There is habit to say that a quantum system can be in several states at the same time.

Concept of state in traditional mechanics

In Mechanical traditional, the most elementary system is a not material of Masse Mr. the state of this system is completely described by the data of the vector position $\backslash \; vec\; \{r\_\; \{M\}\}$ and of the Quantité of movement $\backslash \; vec\; \{p\_\; \{M\}\}\; =m.\; \backslash \; vec\; \{v\_\; \{M\}\}$ of the material point. In other words, the state of the system is located by a point in a vector Space of Dimension 6 (3 for the vector position and 3 for the Flight Path Vector). If one knows the state of the system at a given moment, as well as the force S which act on him thereafter, the fundamental equation of dynamics enables us to calculate the state of the system at one later moment.

The space of the states depends on the system considered: if one considers either one but NR material points, the state of the system will be located by a point in a vector space of dimension 6N, because it is necessary to keep the trace of the position and the impulse of each one of the NR material points; in the case of a continuous medium, like a Fluid or a Solid deformable, the state of the system is described by three fields: of deformation, mass density and Flight Path Vectors. There still, the state of one system at the moment $t+\; \backslash \; tau$ can result from the state of the system at the moment $t$.

Concept of state in quantum mechanics - state and measurement

For a quantum system, it is necessary to distinguish the observable physical sizes from the state of the system itself. In traditional mechanics, this difference does not exist: $\backslash \; vec\; R$ and $\backslash \; vec\; p$ represents the state of the system as well as the physical sizes position and impulse. Thus, if one wants to know the state of a traditional system, it is enough to measure the two physical sizes impulse and position, and the state is completely given. For a system more complex than a material point, the number of sizes to be measured is more important, but it is in theory always possible to determine the state of the system. Contrary, in quantum mechanics, the state of a physical system is in general never completely measurable.

Let us consider for example the quantum system simplest possible, i.e. a particle without spin. This system is very similar to the material point of traditional mechanics. Indeed the observable physical sizes , or more simply the observable are the same ones in both cases: the position, the impulse, kinetic energy, potential energy, total energy,… Let us suppose that our particle is prepared in a state I which one wishes to determine. One can conceive an experiment allowing to determine the position, for example in illuminant the particle, which will give like result: “the particle is in $\backslash \; vec\; r\_1$”. One could then conclude from it that the position of a particle prepared in state I is $\backslash \; vec\; r\_1$. However, if one reiterates the experiment on the same particle again prepared in state I, we will obtain another position $\backslash \; vec\; r\_2\; \backslash \; neq\; \backslash \; vec\; r\_1$! It would be the same with any other observable size.

In the course of time, the physicists acquired the conviction that this experimental fact was not due to an experimental uncertainty during measurement, or with a random preparation of once on the other (in a state i'$\backslash \; neq$i), but rather than it was not possible to determine the state by measurement, such a complete and specifies is it.

It is not however vain to hope to gain information on state I of the particle. One can reach that point while carrying out a great number of measurements such as that described above, where the same particle is prepared a great number of times in state I and its position is each time measured. By treating statistically the values $\backslash \; vec\; r\_1,\; \backslash ;\; \backslash \; vec\; r\_2,\; \backslash \; dots$ obtained, it appears that certain are obtained more frequently than others. Thus, if the value obtained at the conclusion of each individual measurement is not known, it is possible to make probabilistic forecasts on the results of measurement. A state thus determines the probability distribution associated with each of observable of the system, and reciprocally it is given if one can predict the experimental probability distributions. It is then possible to define statistical sizes such as the value Moyenne of the position or the standard deviation of the position (noted and $\backslash \; Delta\; \backslash \; vec\; r$ respectively in the case of the position, and $$ and $\backslash \; Delta\; O$ in the case of an observable O unspecified, $\backslash \; Delta\; O$ being also called uncertainty on the size O when the particle is in state I ).

An interesting particular case corresponds in states for which uncertainty $\backslash \; Delta\; O=0$ for one of observable O. One finds the forecast of traditional mechanics then: in one of these states, the measurement of O will always give the same result. There to distinguish these states from the other states, they are called clean states of observable O . The clean states of total energy are of the interest not to evolve/move during time: if a particle is in a state of energy given, it will remain in this state thereafter.

Lastly, a last particular case makes it possible to specify what is implied when it is said that a particle is in two states simultaneously. One can imagine a state where the probability distribution of the size O is pricked on two values. In fact, the particle is actually only in one state, but this quantum state will give two possible results during a measurement of O.

Mathematical representation of a state

In Mechanical quantum, one represents the state of a system by a vector in a vector Space hilbertien; space to be considered depend on the studied system. The Notation bra-ket makes it possible to represent the quantum states in a concise and simple way.

When one associates two systems to do only one of them, the space of the states of this made up system is the tensorial Produit of spaces of the states associated with the two subsystems.

One finds the Déterminisme traditional mechanics in the evolution of the quantum state. I.e. one can calculate in a deterministic way how the state of a system will evolve/move during time (thanks to the equation of Schrödinger), except when there is a measurement state of our system, in which case the evolution is not deterministic any more, but probabilistic.

It is a major difference with the traditional mechanics, which rises from the postulate of Réduction of the package of wave and which makes it possible to give a probabilistic interpretation to the quantum states.

Let us note that there exist other mathematical representations of the quantum state of a system, the Matrice density being a generalization of the representation exposed here.

Probabilistic interpretation of the quantum states

Let us suppose that a quantum system is in a state $|\backslash \; psi\; \backslash \; rangle$ and that one wants to measure a Observable $\backslash \; hat\; \{has\}$ system (energy, Position, Spin,…). The clean vectors of $\backslash \; hat\; \{has\}$ are noted $|a\_i\; \backslash \; rangle$ and the eigenvalues corresponding $\backslash \; alpha\_i$, that one will suppose not Dégénéré are to simplify. As the principle of Réduction of the package of wave postulates it, the measurement of has can give like result only one of the $\backslash \; alpha\_i$, and the probability of obtaining the result $\backslash \; alpha\_i$ is $^2$. Let us suppose that measurement gives for result $\backslash \; alpha\_p$, the system passed during the measurement and in an instantaneous way of the state $|\backslash \; psi\; \backslash \; rangle$ with the state $|a\_p\; \backslash \; rangle$.

Interpretation consequently is seen that one can make scalar produced $\backslash \; langle\; has|\backslash \; psi\; \backslash \; rangle$, where $|\backslash \; rangle$ has is an unspecified state: indeed, by supposing the existence of observable a of which $|\backslash \; rangle$ has would be one of the clean states, one can say that the Probabilité of finding the system in the state $|\backslash \; rangle$ has (implied: if measurement were made) is $^2$. For this reason, the scalar product is called amplitude of probability .

The fact that the measurement of one of the properties of the system changes the state of this system makes that one cannot cloner the unknown quantum state of a system. Indeed, one could think that by taking two systems made of same the Atome S and by measuring the state in which is system 1, one can then place system 2 in the same state as system 1 and thus to have a certified copy of it, but it would be necessary for that to take several measurements on the system 1, whose state will be irremediably changed as of the first measurement. The Théorème of noncloning is the base of the techniques of quantum Cryptographie. The quantum Teleportation, as for it, seeks to transport in a destructive way the state of system 1 on system 2.

Superposition of states, intrication and paradox of the mesure

If |a  > and |b  > are two possible states of the system, i.e belong to the space of the states, the definition of a vector Space makes that all linear Combinaison of these states will be a possible state of the system. To illustrate the strangeness of these superpositions of states , Erwin Schrödinger has advanced the Paradoxe Chat of Schrödinger: that to think of a cat which one would have put in the superposition of state

$\backslash \; sqrt\; \{2\}\; /2.\; |\; \backslash \; textit\; \{alive\}\; \backslash \; rangle\; +\; \backslash \; sqrt\; \{2\}\; /2.\; |\; \backslash \; textit\; \{dead\}\; \backslash \; rangle\; \backslash ?$

Is this the fact that there is an observer present to try to make a measurement which reduces the package of wave instantaneously to one and only one of the possible results of the mesure ? Which is the characteristic which makes that a quantum system does not obey any more the deterministic equation of Schrödinger at the time of the interaction with a measuring device (or more generally with a macroscopic object)?

In short, the principle of Réduction of the package of wave disturbs and contributes much to the reputation of quantum mechanics to be against-intuitive. After having followed the school of Copenhagen during tens of years, the majority of the physicists thinks today that the Intrication and the Décohérence play a great part in the explanation of the phenomenon of reduction of the package of wave. In experiments, it becomes possible today to carry out the experiments of thought beginning of the 20th century and groups of research try to carry out small “ cats of Schrödinger ”, i.e. objects Mésoscopique S placed in a superposition of states to study their evolution. To describe the quantum state of an intricate object, the approach presented here starting from the Notation bra-ket is not sufficient, and it is advisable to use the formalism of the Matrice density.

See too

• Postulates of quantum mechanics

• Matrix density
• Superposition of states
• Cat of Schrödinger
• Problem of quantum measurement