用变分法研究势场V(r)中存在束缚态的条件

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

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)=-V0aδ(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/-h2>0.68,0.84,1,1.95;对V5,无论e取何值,都存在束缚态;对V6,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。

In quantum physics, the 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）=-V0aδ（r-a）,V2（r）=-V0e^-r/e,V3（r）=-V0ae^-r/a/r,V4（r）=-V0e^-r^2/a^2,V5（r）=-e^2/r,and V6（r）=-A/r^2 as examples to demonstrate the usage of this method. The derived conditions for V1 to V4 are μγa^2/h^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.