Postulates of quantum mechanics

This article treats postulates of quantum mechanics . The description of the microscopic world that the quantum Mécanique provides supports on a radically new vision, and is opposed in that to the traditional Mécanique. It rests on Postulat S.

Introduction

The implications of this new vision are so complex, if deep and so unusual (compared to our own experience) that most of the Scientific community decided to elude them, and is satisfied to use the theory, which provided the most precise forecasts to date.

Holding of this approach, known as of the school of Copenhagen, about this speech holds:

It is important to notice as of now that these postulates do not have any direction (méta-) physical: they do not describe the universe. They are purely formal, operational, in what they describe the adequate operations, but without allowing to interpret them, nor a fortiori to explain why they make it possible to describe the phenomena and to even predict them. This is why one could say:

“If somebody says to you that it included/understood quantum mechanics, it is a liar”

He acts of a radical impossibility, related to the physical absence of bond between the postulates and reality, and not of “a simple” ignorance which could be filled inside the framework of current quantum mechanics.

In short, quantum mechanics is perfectly valid as of now (while waiting for an always possible surprise…), but incomprehensible without complement still to be made.

In parallel, part of the scientific community not being able to accept the approach of the school of Copenhagen tried to create “an other” quantum mechanics which would be in agreement with the “natural” principles on which any applied science should rest: the reproducibility of an experiment and the principle of Determinism.

To this end, of many theories as serious as eccentric were born. The first solution suggested was that of the hidden variables (theory which supposes that “missing” information so that the system behaves in an absolute deterministic way is carried by variables of which we do not have knowledge). At present, it is impossible to solve all the systematic ones using a theory of this form.

Another solution with these problems is the fact of accepting quantum mechanics and these “problem of determinism”, but, in opposition to the school of Copenhagen, not to accept the fundamental character of the postulates of quantum mechanics. With this intention, the members of this school carried their analysis on the fundamental “axioms” which support the applied sciences. This analysis bore these fruits, and this school reformulates these axioms in manner that a science or mechanics based on this “axiomatic logic” is in agreement with quantum mechanics. This solution is far from known in the nonscientific world and still has a great number of detractors. The speeches of the detractors and the answers of the protagonists of this solution can be summarized as follows:

; Detractors: This solution does nothing but move the problem, because instead of having a quantum mechanics based on five postulates “left nowhere”, you found a solution so that it is based on three axioms “left nowhere”.

; Protagonists: Firstly it is necessary to include/understand at which point any science is based on axiomatic fundamental which governs the data acquisition experimental and the treatment of these data. Indeed, the idea of Causality, Determinism, Reproductivité of an experiment are fundamental concepts without which it would be impossible with the human spirit to create a science. And these concepts are axioms! These axioms were formulated during the Antiquité and accepted we them until now without any doubt. However, with the arrival of modern physics and the study of the Elementary particles, these axioms generate paradoxes, it is thus clear that we cannot thus more accept them such as it is, it thus becomes necessary to reformulate them. We did not move the problem, because we reduced six postulates and an axiom in three axioms. Lastly, these three new axioms are much more “natural” that the six postulates of quantum mechanics.

Mathematical formulation

The mathematical formulation of quantum mechanics, in its general use, largely calls upon the Notation bra-ket of Dirac, which makes it possible to represent in a concise way the operations on the spaces of Hilbert used in analyzes functional. This formulation is often allotted to John von Neumann.

That is to say a separable Space $\backslash \; mathcal\; \{H\}$ of Hilbert. The states are the projective rays of $\backslash \; mathcal\; \{H\}$. An operator is a linear transformation of a dense subspace of $\backslash \; mathcal\; \{H\}$ towards $\backslash \; mathcal\; \{H\}$. If this operator is continuous, then this transformation can be prolonged in a single way to a linear transformation limited of $\backslash \; mathcal\; \{H\}$ towards $\backslash \; mathcal\; \{H\}$. By tradition, the observable things are identified with operators, although it is debatable, particularly in the presence of the Symétrie S. This is why some prefer the formulation of state of density.

Within this framework, the Principe of uncertainty of Heisenberg becomes a theorem about the commutative operators not . Moreover, one can treat observable continuous and discrete ones; in the first case, the space of Hilbert is a space of integrable functions of wave of square.

Postulates

Postulate I

Définition of the state quantique

See also: quantum state

The knowledge of the state of a quantum system is completely contained, at the moment T, in a vector normalisable of the space of the states $\backslash \; mathcal\; \{H\}$. It is usually noted in the form of a ket $|\; \backslash \; psi\; (T)\; \backslash \; rangle$.

Postulate II

Mesure: definition of a observable

See also: Observable

To any observable property, for example the position, the energy or the Spin, correspond a Opérateur linear Hermitien acting on the Vecteur S of a Espace of Hilbert $\backslash \; mathcal\; \{H\}$. This operator is named Observable.

The operators associated with the observable properties are defined by rules of construction which rest on a principle of correspondence:

; The operator of position: $\backslash \; hat\; \{\backslash \; mathbf\; \{Q\}\}\; =\; \backslash \; mathbf\; \{R\}$

; The operator of potential energy traditional or electromagnetic: $\backslash \; hat\; \{V\}\; (\backslash \; mathbf\; \{R\})\; =\; V\_\; \{Cl\}\; (\backslash \; mathbf\; \{R\})$

; The operator of Momentum: $\backslash \; hat\; \{\backslash \; mathbf\; \{P\}\}\; (\backslash \; mathbf\; \{R\})\; =\; -\; I\; \backslash \; hbar\; \backslash \; nabla$, where $\backslash \; nabla$ indicates the gradient of the coordinates $\backslash \; mathbf\; \{R\}$

; The operator of Angular momentum: $\backslash \; hat\; \{\backslash \; mathbf\; \{L\}\}\; (\backslash \; mathbf\; \{R\})\; =\; \backslash \; hat\; \{\backslash \; mathbf\; \{Q\}\}\; \backslash \; times\; \backslash \; hat\; \{\backslash \; mathbf\; \{P\}\}\; =\; -\; I\; \backslash \; hbar\; \backslash \; mathbf\; \{R\}\; \backslash \; times\; \backslash \; nabla$

; The kinetic operator of energy: $\backslash \; hat\; \{K\}\; (\backslash \; mathbf\; \{R\})\; =\; \backslash \; frac\; \{\backslash \; hat\; \{\backslash \; mathbf\; \{P\}\}\; \backslash \; cdot\; \backslash \; hat\; \{\backslash \; mathbf\; \{P\}\}\}\; \{2m\}\; =\; -\; \backslash \; frac\; \{\backslash \; hbar^2\}\; \{2m\}\; \backslash \; nabla^2$

; The operator of total energy, called Hamiltonian: $\backslash \; hat\; \{H\}\; =\; \backslash \; hat\; \{K\}\; +\; \backslash \; hat\; \{V\}\; =\; \backslash \; hat\; \{K\}\; (\backslash \; mathbf\; \{R\})\; +\; V\_\; \{Cl\}\; (\backslash \; mathbf\; \{R\})$

; The operator action of the system, called Lagrangian: $\backslash \; hat\; \{L\}\; =\; \backslash \; hat\; \{K\}\; -\; \backslash \; hat\; \{V\}$

Postulate III

Mesure: possible values of a observable

The measurement of a physical size represented by observable the has can provide only one of the eigenvalues of A.

The clean vectors and the eigenvalues of this operator have a special significance: the eigenvalues are the values being able to result from an ideal measurement of this property, the clean vectors being the quantum state of the system during this measurement. By using the Notation bra-ket, this postulate can be written as follows:

$\backslash \; hat\; \{has\}\; |\; \backslash \; alpha\_n\; \backslash \; rangle\; =\; a\_n\; |\; \backslash alpha\_n\; \backslash rangle$

where $\backslash \; hat\; \{has\}$, $|\; \backslash \; alpha\_n\; \backslash \; rangle$ and $a\_n$ indicates, respectively, the observable one, the clean vector and the corresponding eigenvalue.

The clean states of very observable $\backslash \; hat\; \{has\}$ are complete and form a orthonormée Base in the space of Hilbert.

That means that any vector $|\; \backslash \; psi\; (T)\; \backslash \; rangle$ can break up in a single way on the basis of these clean vectors ($|\; \backslash \; phi\_i\; \backslash \; rangle$):

$|\; \backslash \; psi\; \backslash \; rangle\; =\; c\_1\; |\; \backslash \; phi\_1\; \backslash \; rangle\; +\; c\_2\; |\; \backslash \; phi\_2\; \backslash \; rangle\; +\dots \; +\; c\_n\; |\; \backslash phi\_n\; \backslash rangle$

Postulate IV

Mesure: probability of obtaining a value of Observable

The measurement of a physical size represented by observable the has , carried out on the quantum state (standardized) $|\; \backslash \; psi\; (T)\; \backslash \; rangle$, gives a_n result, with the P_n probability equalizes with |c_n|².

The Produces scalar of a state and of another vector (which it belongs or not to $\backslash \; mathcal\; \{H\}$) provides an amplitude of probability, whose square corresponds to a Probabilité or a density of probability in the following way:

• For a system made up of only one particle, the function of wave $\backslash \; Psi\_\; \backslash \; alpha\; (\backslash \; mathbf\; \{R\})\; =\; \backslash \; langle\; \backslash \; mathbf\; \{R\}\; |\; \backslash \; alpha\; \backslash \; rangle$ is the amplitude of probability that the particle is with the position $\backslash \; mathbf\; \{R\}$. The probability $P\_\; \backslash \; alpha\; (\backslash \; mathbf\; \{R\})$ of finding the particle between $\backslash \; mathbf\; \{R\}$ and $\backslash \; mathbf\; \{R\}\; +\; D\; \backslash \; mathbf\; \{R\}$ is:

$P\_\; \backslash \; alpha\; (\backslash \; mathbf\; \{R\})\; =\; ^2\; d^3\; \backslash \; mathbf\; \{R\}\; \backslash \; equiv\; ^2\; d^3\; \backslash \; mathbf\; \{R\}$ Thus $\backslash \; rho\_\; \backslash \; alpha\; (\backslash \; mathbf\; \{R\})\; =^2$ is a density of probability.

Si the system is in a state $|\backslash \; alpha\; \backslash \; rangle$, then the amplitude of probability $C\_\; \{\backslash \; beta\; \backslash \; alpha\}\; \backslash ,$ and the probability $P\_\; \{\backslash \; beta\; \backslash \; alpha\}\; \backslash ,$ to find it in any other state $|\backslash \; beta\; \backslash \; rangle$ is:

$C\_\; \{\backslash \; beta\; \backslash \; alpha\}\; =\; \backslash \; langle\; \backslash \; beta|\backslash \; alpha\; \backslash \; rangle$.

$P\_\; \{\backslash \; beta\; \backslash \; alpha\}\; =\; ^2$.

Nor $|\backslash \; alpha\; \backslash \; rangle$, nor $|\backslash \; beta\; \backslash \; rangle$ should not be necessarily a clean state of a quantum operator.

Dans the possibility where a system can evolve to a state $|\backslash \; alpha,\; T\; \backslash \; rangle$ with time $t$ by several ways different, then, in so far as one does not take measurement to determine which way was actually followed, $|\backslash \; alpha,\; T\; \backslash \; rangle$ is a linear Combinaison states $|\backslash \; alpha\_j,\; T\; \backslash \; rangle$ where $j$ specifies the way:

$|\backslash \; alpha,\; T\; \backslash \; rangle\; =\; \backslash \; sum\; \{w\_j\; |\backslash \; alpha\_j,\; T\; \backslash \; rangle\}$

where $w\_j\; \backslash ,$ is the coefficient of the linear combination.

The amplitude $C\_\; \{\backslash \; beta\; \backslash \; alpha\}\; (T)\; =$ becomes then the sum of the amplitudes $C\_\; \{\backslash \; beta\; \backslash \; alpha\_j\}\; (T)$ and the probability $P\_\; \{\backslash \; beta\; \backslash \; alpha\}\; (T)\; \backslash ,$ contains terms of interference:

$P\_\; \{\backslash \; beta\; \backslash \; alpha\}\; (T)\; =\; ^2\; =\; \{\backslash \; left|\backslash \; sum\; \{w\_j\; \backslash \; langle\; \backslash \; beta\; |\backslash \; alpha\_j,\; T\; \backslash \; rangle\}\; \backslash \; right|\}\; ^2\; =\; \{\backslash \; left|\backslash \; sum\; \{w\_j\; C\_\; \{\backslash \; beta\; \backslash \; alpha\_j\}\; (T)\}\backslash \; right|\}^2$

But if a measurement determined that the way $k$ was followed, then the coefficients become $w\_j\; \backslash \; rightarrow\; \backslash \; delta\_\; \{jk\}$ and the preceding sums are reduced in the only one term.

En supposing that the system is in a state $|\backslash \; alpha\; \backslash \; rangle$, then the theoretical prediction of the median value of the measurement of observable the $\backslash \; hat\; \{has\}$ is given by:

$\{\backslash \; langle\; \backslash \; hat\; \{has\}\; \backslash \; rangle\}\; \_\; \backslash \; alpha\; =\; \backslash \; langle\; \backslash \; alpha|\backslash \; hat\; \{has\}|\backslash \; alpha\; \backslash \; rangle$

Postulate V

Mesure: reduction of the package of wave; obtaining a single value; projection of the state quantique

See also: Reduction of the package of wave

If the measurement of the physical size has, at the moment T, on a system represented by the vector $|\; \backslash \; psi\; \backslash \; rangle$ gives like result the eigenvalue $a\_n\; \backslash ,$, then the state of the system immediately after measurement is the clean subspace associated with $a\_n\; \backslash ,$: $|\; \backslash alpha\_n\; \backslash rangle$.

This postulate is also called " postulate of Reduction of the package of wave ".

Postulate VI

temporal Évolution of the state quantique

The state $\backslash \; left|\backslash \; Phi,\; T\; \backslash \; right\; \backslash \; rangle$ of any quantum system not-relativist is a solution of the equation of Schrödinger dependant on time:

$i\; \backslash \; hbar\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; T\}\; \backslash \; left|\backslash \; Phi,\; T\; \backslash \; right\; \backslash \; rangle\; =\; \backslash \; hat\; \{H\}\; \backslash \; left|\backslash \; Phi,\; T\; \backslash \; right\; \backslash \; rangle$

The sixth postulate is the equation of Schrödinger. This equation is the dynamic equation of quantum mechanics. It means simply that it is the operator “energy total” of the system or Hamiltonien, which is person in charge of the evolution of the system in time. Indeed, the form of the equation shows that by applying the Hamiltonian to function of wave of the system, one obtains his derivative compared to the time i.e. how it varies in time.

This equation is valid only within the nonrelativistic framework.

See too

• three axioms of quantum mechanics

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