Stationary state

This article covers the concept of quantum mechanics. For the meaning in classical physics, see equilibrium (systems theory) # Stationary state .

In quantum mechanics, a stationary state is a solution to the time-independent Schrödinger equation . It is an eigenstate of the Hamilton operator of the physical system under consideration. Its energy  is an eigenvalue of this operator. In Dirac notation , the equation applies to the stationary state: ${\displaystyle |\psi \rangle }$  ${\displaystyle H}$ ${\displaystyle E}$

${\displaystyle H|\psi \rangle =E|\psi \rangle .}$

In spatial representation , a stationary state has the form:

${\displaystyle \langle \mathbf {r} |\psi \rangle =\psi (\mathbf {r} ,t)=\psi (\mathbf {r} ,t=0)\cdot \exp \left({-{\frac {\mathrm {i} }{\hbar }}Et}\right)}$

With

• ${\displaystyle \psi ()}$, the wave function
• ${\displaystyle \mathbf {r} }$, the position vector
• ${\displaystyle \exp }$, the exponential function
• ${\displaystyle \mathrm {i} }$, the imaginary unit
• ${\displaystyle \hbar }$, the reduced Planck's constant

The square of the absolute value (the probability distribution that is decisive for physical measurements ) of the wave function is therefore independent of time . ${\displaystyle \textstyle |\langle \mathbf {r} |\psi \rangle |^{2}}$ ${\displaystyle t}$

More generally, stationary states of a (not necessarily closed) quantum system are those states for which the density matrix of the system is constant over time. This includes the above eigenstates, for which applies ${\displaystyle {\hat {\rho }}}$

${\displaystyle {\frac {\partial {\hat {\rho }}}{\partial t}}={\frac {\mathrm {i} }{\hbar }}\left[{\hat {\rho }},{\hat {H}}\right]=0}$

just like the stationary states of open quantum systems, their dynamics by a Lindblad master equation

${\displaystyle {\frac {\partial {\hat {\rho }}}{\partial t}}={\frac {\mathrm {i} }{\hbar }}{\mathcal {L}}(\rho )}$

is given and for which the states in the kernel of the Liouville operator are stationary, d. H. the states with . ${\displaystyle {\mathcal {L}}}$ ${\displaystyle \rho _{\mathrm {s} }}$ ${\displaystyle {\mathcal {L}}(\rho _{\mathrm {s} })=0}$

Web links

• 3D visualization of stationary atomic states

Individual evidence

1. ^ Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë: Quantum Mechanics , 2 volumes, 2nd edition. De Gruyter , Berlin 1999, ISBN 3-11-016458-2