Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
List of quantum-mechanical systems with analytical solutions
open-in-new
<p>Much insight in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> can be gained from understanding the <a href="/facts/Closed-form_solution/y0xCXlSk">closed-form solutions</a> to the time-dependent non-relativistic <a href="/facts/Schr%25C3%25B6dinger_equation/3csgSIWF">Schrödinger equation</a>. It takes the form </p><p> H ^ ψ ( r , t ) = [ − ℏ 2 2 m ∇ 2 + V ( r ) ] ψ ( r , t ) = i ℏ ∂ ψ ( r , t ) ∂ t , {\displaystyle {\hat {H}}\psi {\left(\mathbf {r} ,t\right)}=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V{\left(\mathbf {r} \right)}\right]\psi {\left(\mathbf {r} ,t\right)}=i\hbar {\frac {\partial \psi {\left(\mathbf {r} ,t\right)}}{\partial t}},} </p><p>where ψ {\displaystyle \psi } is the <a href="/facts/Wave_function/KgM5ykjY">wave function</a> of the system, H ^ {\displaystyle {\hat {H}}} is the <a href="/facts/Hamiltonian_operator/5XwK2HmD">Hamiltonian operator</a>, and t {\displaystyle t} is time. <a href="/facts/Stationary_state/HBQvbb6s">Stationary states</a> of this equation are found by solving the time-independent Schrödinger equation, </p><p> [ − ℏ 2 2 m ∇ 2 + V ( r ) ] ψ ( r ) = E ψ ( r ) , {\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V{\left(\mathbf {r} \right)}\right]\psi {\left(\mathbf {r} \right)}=E\psi {\left(\mathbf {r} \right)},} </p><p>which is an eigenvalue equation. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. However, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies, or eigenvalues, can be found. These quantum-mechanical systems with analytical solutions are listed below. </p>