用变分法研究势场V(r)中存在束缚态的条件

用变分法研究中心力场V(r)中存在束缚态的条件,分别对势场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, V6(r)=-A／r2进行了研究,得到的结果是：对V1 至V4分别求μV0a2／2>068,084,1,195;对V5,无论e取何值,都存在束缚态;对V6,无论A取何值均不能形成束缚态。对比解定态方程得到的精确条件看出,用变分法得到的是充分条件。

Study on conditions for the existence of bound states in central field with variational method

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>068,084,1,195 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.