Ground state

This article deals with the term in quantum mechanics. For the meaning in classical physics, see equilibrium (system theory)#stationary state.

A stationary state $|\psi \rangle$ is a solution of the time-independent Schrödinger equation in quantum mechanics. It is an eigenstate of the Hamiltonian operator of the physical system under consideration. Its energy is an eigenvalue of this operator. In Dirac notation, the equation thus applies to the stationary state:

$H | \psi \rangle = E | \psi \rangle.$

In spatial representation, a steady state has the form:

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

with

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

The magnitude square $\textstyle |\langle \mathbf {r} |\psi \rangle |^{2}$ (the probability distribution crucial for physical measurements) of the wave function is thus independent of the time .

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

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

as well as the stationary states of open quantum systems, whose dynamics are described by a Lindblad master equation.

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

and for which the states in the core of the Liouville operator $\mathcal L$ are stationary, i.e. the states ρ $\rho _{\mathrm {s} }$ with ${\mathcal {L}}(\rho _{\mathrm {s} })=0$ .

Ground state

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