用变分法研究中心力场存在束缚态的条件  

Study on Conditions for the Existence of Bound States in Central Field with Variational Method

作　　者：陈岗[1] 

机构地区：[1]山东警察学院公共基础部,山东济南250014

出　　处：《山东教育学院学报》2008年第4期85-87,共3页Journal of Shandong Education Institute

摘　　要：在量子力学中,变分法可以用来研究中心力场V(r)中存在束缚态的条件。我们用变分法对以下5种中心力场:V1(r)=-γδr-a),V2(r)=-V0e-r/a,V3(r)=γer-r/aV4(r)=-er2,V5(r)=-rA2,研究了存在束缚态的条件,结果分别为(1)μγa2>0.68,(2)μV 02a2>0.84,(3)μγ2a>1,(4)e取任意值,(5)无论A取何值,均不能形成束缚态。我们还对这些势场的定态薛定谔方程,用精确求解的方法得到的条件分别为(1)μγ2a>0.5,(2)μV 02a2>0.7228,(3)μγ2a>0.84,(4)e取任意值,(5)无论A取何值,均不能形成束缚态。对比以上两种结果看出,由变分法得到的条件是充分条件,不是必要条件,变分法是简单适用的。In quantum physics, variational method can be used to study the conditions for the existence of bound states in a central field.Using this method, we have studied such conditions for five types of central field：V（r）=-γδr-a）,V2（r）=-V0e^-r/a,V3（r）=γe^-r/a/rV4（r）=-e^2/r,V5（r）=-A/r^2.Correspondingly, the derived conditions are ：（1）μγa/h^2〉0.68,（2）μV0a^2/h^2〉0.84,（3）μγa/h^2〉1,（4） e with arbitrary value, and （5）no bound states for any value of A. We have also derived the precise solutions of steady- state Schrtodinger equation for these potentials and the obtained conditions are：（1）μγa/h^2〉0.5,（2）μV0a^2/h^2〉0.07228,（3）μγa/h^2〉0.84,（4） e witharbitrary value, and （5） no bound states for any value of A. 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. It turns out that the variational method is simple and useful.

关 键 词：中心力场 束缚态 变分法 

分 类 号：O41[理学—理论物理]