Form of quantum physics

Expression of some observable

The relations of commutation between the Observable S result from the principle of correspondence between the Hamiltonian Mécanique and the quantum Mécanique. Their expressions can then be found starting from a mathematical analysis.

Evolution in time

Equation of Schrödinger

• For an unspecified state: the state evolves/moves according to the equation of Schrödinger depend on partial time

$i\; \backslash \; hbar\; \backslash \; frac\; \{\backslash \}\; \{\backslash \; partial\; T\}\; \backslash \; left|\backslash \; psi\; (T)\; \backslash \; right\; \backslash \; rangle\; =\; \backslash \; hat\; \{H\}\; \backslash \; left|\backslash \; psi\; (T)\; \backslash \; right\; \backslash \; rangle$

• For a clean state of energy, i.e. answering the equation with the eigenvalues

$\backslash \; hat\; \{H\}\; \backslash \; left|\backslash \; psi\_0\; \backslash \; right\; \backslash \; rangle\; =\; E\; \backslash \; left|\backslash \; psi\_0\; \backslash \; right\; \backslash \; rangle$ at the initial moment t=0, the evolution at the later moments (t>0) will be: $\backslash \; left|\backslash \; psi\; (T)\; \backslash \; right\; \backslash \; rangle\; =\; e^\; \{-\; \backslash \; frac\; \{I\; \backslash ,\; E\; \backslash ,\; T\}\; \{\backslash \; hbar\}\}\; \backslash \; left|\backslash psi\_0\backslash right\backslash rangle$

Expression of some Hamiltonians

Propagator of the equation of Schrödinger

See also: Propagating of the equation of Schrödinger

Starting from the Exponential concept of of matrix, one can find the solution formal of the equation of Schrödinger. This solution is written:

$\backslash \; left|\backslash \; psi\; (T)\; \backslash \; right\; \backslash \; rangle\; =\; U\; (T,\; t\_0)\; \backslash \; left|\backslash \; psi\; (t\_0)\; \backslash \; right\; \backslash \; rangle,$ with

$U\; (T,\; t\_0)\; =\; U\; (t-t\_0)\; =\; \backslash \; exp\; \backslash \; left\; (-\; I\; \backslash \; frac\; \{H\}\; \{\backslash \; hbar\}\; (t-t\_0)\; \backslash \; right)$ if H does not depend explicitly on time, and

$U\; (T,\; t\_0)\; =\; \backslash \; exp\; \backslash \; left\; (-\; I\; \backslash \; frac\; \{\backslash \; int\_\; \{t\_0\}\; ^t\; H\; (you)\; dt\text{'}\}\; \{\backslash \; hbar\}\; \backslash \; right)$ in the general case.

Representation of Heisenberg

See also: Representation of Heisenberg

If the Hamiltonian does not depend explicitly on time, in the traditional representation called Représentation of Schrödinger, the observable ones do not depend on time and the state depends on time. By a unit transformation, one can pass to the Représentation of Heisenberg, where the state is independent of time and the observable ones depend on time following the equation below:

$\{D\; \backslash \; over\; \{dt\}\}\; A=\; \{1\; \backslash \; over\; \{I\; \backslash \; hbar\}\}\; +\; \backslash \; left\; (\backslash \; right)\; \_\; \backslash \; text\; \{clarifies\}$

Law of the black body

According to the Law of Stefan-Boltzmann, the flow of energy Φ emitted by the black body varies according to the absolute temperature T (in Kelvin) according to

$\backslash \; Phi\; =\; \backslash \; sigma\; T^4$

where σ is the Constante of Stefan-Boltzmann

The density flux of energy D Φ for a wavelength λ given is given by the Loi of Planck:

$\backslash \; frac\; \{D\; \backslash \; Phi\}\; \{D\; \backslash \; lambda\}\; =\; \backslash \; frac\; \{2\; \backslash \; pi\; c^2\; H\}\; \{\backslash \; lambda^5\}\; \backslash \; cdot\; \backslash \; frac\; \{1\}\; \{e^\; \{hc/\backslash \; lambda\; kT\}\; -\; 1\}$

where C is the Speed of light in the vacuum, H is the Constante of Planck and K is the Boltzmann constant. The maximum of this spectrum is given by the Loi of Wien:

$\backslash \; lambda\_\; \{max\}\; =\; \backslash \; frac\; \{hc\}\; \{4,965\; \backslash \; cdot\; kT\}\; =\; \backslash \; frac\; \{2,898\; \backslash \; cdot\; 10^\; \{-\; 3\}\}\; \{T\}$.

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