List of quantum-mechanical systems with analytical solutions

Much insight in quantum mechanics can be gained from understanding the closed-form solutions to the time-dependent non-relativistic Schrödinger equation. It takes the form 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}},} where ψ {\displaystyle \psi } is the wave function of the system, H ^ {\displaystyle {\hat {H}}} is the Hamiltonian operator, and t {\displaystyle t} is time.

Source: Wikipedia — List of quantum-mechanical systems with analytical solutions (CC BY-SA 4.0)

List of quantum-mechanical systems with analytical solutions

Much insight in quantum mechanics can be gained from understanding the closed-form solutions to the time-dependent non-relativistic Schrödinger equation. It takes the form 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}},} where ψ {\displaystyle \psi } is the wave function of the system, H ^ {\displaystyle {\hat {H}}} is the Hamiltonian operator, and t {\displaystyle t} is time.

Source: Wikipedia "List of quantum-mechanical systems with analytical solutions" · CC BY-SA 4.0

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