一維無限深勢阱不確定關系的算符理論推導法-

1、- 23 -不確定關系p x x h 是眾所熟知的普遍成 /2立的公式,然而具體情況下(如單縫衍射、一維諧振子和一維無限深勢阱等等p x x 究竟等于什么,這個問題,在通行的量子力學教科書中沒有見到過論述,致使許多初學者深感困惑.其實,用算符理論推導一維無限深勢阱的不確定關系并不復雜.當我們推出了一個與量子數有關的等式型不確定關系之后,就會感覺到不確定關系不再像原來那樣神秘了.為方便起見,在下面的討論中我們用符號p 來代替符號p x .對于一個寬度為a 的一維無限深勢阱,如果把x 軸的坐標原點取在阱壁處,并使x 軸指向另一阱壁,則根據薛定諤方程可以推出,勢阱中的波函數在阱外波函數等于零.1容易

2、推出,平均位置坐標方均位置坐標故位置坐標的均方差于是阱中粒子位置坐標的不確定范圍阱中粒子的平均動量阱中粒子的方均動量若以c p 表示動量幾率函數則粒子動量的均方差,一維無限深勢阱不確定關系的算符理論推導法羅凌霄伊曉軍(大理學院荷花校區(qū)物理系,云南大理671000摘要: 用算符理論推出了一維無限深勢阱的不確定關系.關鍵詞:一維無限深勢阱;不確定關系;算符中圖分類號:文獻標識碼:文章編號:-(-O413.1 A 16720520200405002302收稿日期:2003-12-17基金項目:云南省教育科技基金項目(03Y650A .作者簡介:羅凌霄(男,云南大理人,大理學院物理系副教授,從事基礎物

3、理教學與研究1964. n (x =2a sinn x a, (1x =ax *n (x n (x d x = a 0x 2a sin 2n x a x =a 2x 2=ax 2*n (x n (x d x = a 0x 22a sin 2n x a d x =a 2(13-12n 22.(x 2=a 0(x -x 2*n (x n (x dx = x -2x +x 2=x -x 2=a 2(112-12n 22.x =a112-12n 22. (2p 2=a0*n (x p 2x n (x dx = a 0*n (x -(ih x2n (x dx = a 02a sin n x a -(h

4、2d 2dx2(2a sin n x a dx = n 22h 2a 2.(p 2=+-(p -p 2c p c *p d p = +-(p 2-2+p 2 d p = p 2-2p 2-p 2=p 2-pp =a*n (x p x n (x dx = a 0*n (x -(ih n (x dx = a2a sin n x a -(ih d dx (2a sin n x adx =0.- 24 -故粒子動量的不確定范圍根據式和式,得(2(3這就是一維無限深勢阱的不確定關系.容易看出,p x 隨量子數n 的增大而增大.當n =時,1時間要比修理時間長得多.定理3在條件下, (2.14t 0證由定

5、理, 1 (X 遵從參數為的指數1分布而,a (依概率收斂于是由0,.Slusky 4定理(T 20= (X + (Y 依分布收斂到參數為的指數分布.1參考文獻:1Epstien, B.(1971.Sojourn time distributions for two unitp x 取最小值,(p x min =×.故1.135p x 的最小值比普遍形式的不確定關系給出的下限 參考文獻:楊福家.原子物理學.北京:高等教育出版社,1M.2000108122責任編輯馬統(tǒng)一(AviItzhak, ed.J. Gordon and Breach, 183190.2Basu, A. P. an

6、d Kirmani, S.N.U.A. (1986. Some results .involving HNBUE distributionsJ. J. Appl. Prod23, 10381044.3Ross, S. M. Stochastic Process. New York: John Wiley & Sons. 1983.4 Chung, K. L. (1974. A First Course in Probability Theory, 2nd ed. Academic Press.陳侃.壽命分布類與可靠性數學理論.北京:科5M學出版社,.1999責行編輯馬統(tǒng)一h2羅凌霄,伊

7、曉軍一維無限深勢阱不確定關系的算符理論推導法:hThe operator theory method lead to the uncertainty relation of one dimensional infinite wellLUO Ling-xiao YI Xiao-jun(Department of Physics, Dali College, Dali 671000, ChinaAbstract: The uncertainty relation of one dimensional infinite well is deduced with the operator theory

8、. Key words: one dimensional infinite well; uncertainty relation; operatorp x =n 2212-12 h =n 223-2 h2. (4p =n h a=p 2= n 2 2 h 2 a 2 (上接第頁12Some functioning characterizations for two unit repairable systemsKANG Dian-tong(Department of mathematics, Hexi University, Zhangye 734000, ChinaAbstract: In this paper, some functioning characterizations for two unit repairable systems are discussed.Key words: repairable systems; limiting distribution; Gamma