In quantum physics, variational method can be used to study the conditions for the existence of bound states in a central field. We take the following six kinds of potentials, V1(r)=-V0a δ(r-a), V2(r)=-V0e-r／a, V3(r)=-V0ae-r／a／r, V4(r)=-V0e-r2／a2, V5(r)=-e2／r, and V6(r)=-A／r2 as examples to demonstrate the usage of this method. The derived conditions for V1 to V4 are μγa2／2>068,084,1,195 respectively; for V5, the bound states can exist for e with arbitrary value, while for V6, there is no bound states for any value of A. We also get the precise conditions by solving the Schroedinger equation for these potentials. A comparison of the results obtained by these two methods shows that the conditions derived by the variational method are sufficient, but not necessary conditions.
