This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

Cautions:

In this situation, the SE is given by the form $${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi _{\alpha }(\mathbf {r} ,\,t)={\hat {H}}\Psi _{\alpha }(\mathbf {r} ,\,t)=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\right)\Psi _{\alpha }(\mathbf {r} ,\,t)=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi _{\alpha }(\mathbf {r} ,\,t)+V(\mathbf {r} )\Psi _{\alpha }(\mathbf {r} ,\,t)}$$ It can be derived from (1) by considering ${\displaystyle \Psi _{\alpha }(x,t):=\langle x|\alpha \rangle }$ and ${\displaystyle {\hat {H}}:=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\hat {V}}}$