陳文利，史艷維，魚　翔

(西安培華學(xué)院通識(shí)教育中心，陜西 西安　710125)

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·基礎(chǔ)學(xué)科·

改良的Manning-Rosen勢(shì)Schr?dinger方程散射態(tài)解

陳文利，史艷維，魚翔

(西安培華學(xué)院通識(shí)教育中心，陜西 西安710125)

對(duì)改良的Manning-Rosen勢(shì)的徑向薛定諤方程任意l波散射態(tài)進(jìn)行解析求解，得到按“K/2π標(biāo)度”的相移公式和歸一化的超幾何函數(shù)表示的徑向波函數(shù)。通過(guò)對(duì)散射振幅在極點(diǎn)解析性質(zhì)的研究得到束縛態(tài)能級(jí), 而特征值的數(shù)值結(jié)果與先前結(jié)果的比較進(jìn)一步驗(yàn)證了本文結(jié)果的精確性。

改良 Manning-Rosen勢(shì)場(chǎng) ;散射態(tài); 近似解析解

隨著量子理論的發(fā)展，對(duì)于不同勢(shì)場(chǎng)含有離心項(xiàng)l(l+1)/r2的徑向薛定諤方程的求解一直是研究的熱點(diǎn)，然而，能夠解析求解的勢(shì)場(chǎng)大多被約束在s(l=0)態(tài)。近年來(lái)，研究者提出了近似求解方法在大多可解勢(shì)場(chǎng)中得以應(yīng)用, 諸如指數(shù)近似辦法[1]、 Pekeris近似辦法[2]和改良的指數(shù)近似辦法[3]等近似表示離心項(xiàng)。有些近似辦法被應(yīng)用到超幾何函數(shù)方法[4]中，有些學(xué)者利用超對(duì)稱[5]、形狀不變形方法[6]求解薛定諤方程解析解，這些方法進(jìn)一步地豐富了求解的計(jì)算方法。在Manning-Rosen 勢(shì)基礎(chǔ)上,有顯著改良 Manning-Rosen 勢(shì)場(chǎng)被提出來(lái)[7]，它的表達(dá)式為

(1)

其中：De是離解能；re是勢(shì)函數(shù)的最小值點(diǎn)； α是可調(diào)節(jié)的勢(shì)參數(shù)。該勢(shì)場(chǎng)被廣泛的應(yīng)用于分子物理和化學(xué)物理。對(duì)于改良Manning-Rosen 勢(shì)場(chǎng)，Jia等計(jì)算了Klein-Gordon方程束縛態(tài)解析解[8]、自旋對(duì)稱的狄拉克方程[9-10]，Oluwadare等計(jì)算了魔自旋對(duì)稱Klein-Gordon和狄拉克[11]，Dong等利用近似辦法求解了束縛態(tài)解析解，推導(dǎo)出了近似特征值方程[12]；然而散射態(tài)的求解一直很少被涉及。在我們先前工作的基礎(chǔ)上，本文利用更合適的近似公式求解改良Manning-Rosen 勢(shì)場(chǎng)散射態(tài)波函數(shù)解析解，同時(shí)，利用散射態(tài)波函數(shù)漸進(jìn)行為求出歸一化常數(shù)N和相移θl的解析表達(dá)式。 最后，給定不同的勢(shì)參數(shù)和量子數(shù)，數(shù)值求解特征值，并和先前求解數(shù)據(jù)及MATHEMAITC 程序包所得數(shù)據(jù)進(jìn)行對(duì)比。

1　散射態(tài)的近似解析解

含改良Manning-Rosen勢(shì)的徑向薛定諤方程可表示為

(2)

由于離心項(xiàng)l(l+1)/r2的存在，方程(2)的求解大都被要求在s波(l=0)條件下求解。 像我們先前工作一樣[1-2，13]，大多作者提出對(duì)離心項(xiàng)做合適的近似，繼而將薛定諤方程的求解推廣到任意l波。對(duì)于改良的Manning-Rosen勢(shì)本文應(yīng)用文獻(xiàn)[8]中的Greene-Aldrich近似表達(dá)式

(3)

近似表示離心項(xiàng)，當(dāng)c0=0時(shí)，近似表達(dá)式就變成慣用的Greene-Aldrich 近似。將方程(3)代入方程(2)可得

(4)

對(duì)上式自變量作指數(shù)變換，即引入新變量x=1-e-αr(r∈(0,),x∈(0,1))，并代入方程(4)化簡(jiǎn)得到量綱一方程

(5)

滿足邊界條件方程(5)的徑向波函數(shù)可按“K/2π標(biāo)度”歸一化的超幾何函數(shù)表示[14-16]。設(shè)波函數(shù)的形式為

u(x)=(1-x)-ikxδF(x)，

(6)

其中

(7)

把方程(6)代入方程(5)可得

(8)

方程(8)為超幾何微分方程，其解可表示為

F(x)=C12F1(a;b;c;x)+C2x1-c2F1(a-c+1;b-c+1;2-c;x)，

(9)

其中參數(shù)

c=2δ。

(10)

u(r)=N[1-e-αr]δeikαr2F1(a;b;c;1-e-αr),

(11)

其中，N是歸一化常數(shù)。

下面利用散射態(tài)波函數(shù)漸進(jìn)行為確定歸一化常數(shù)N 和相移θl的解析表達(dá)式, 從方程(10)可以得到參數(shù)a,b,c滿足

(12)

利用超幾何函數(shù)的變換公式

(13)

(14)

令

(15)

其中θ為常數(shù)。把方程(15)代入方程(14)可得

(16)

(17)

(18)

(19)

對(duì)于s波(l=0)，離心項(xiàng)l(l+1)/r2等于零；因此，對(duì)于改良的Manning-Rosen勢(shì)s波的歸一化常數(shù)N0和相移近似解析解θ0，只需取方程(18)、(19)公式中l(wèi)=0即可得。

根據(jù)散射態(tài)和束縛態(tài)的關(guān)系，結(jié)合方程(18)中伽馬函數(shù)的性質(zhì)，可得

(20)

結(jié)合方程(7)，解析求解方程(20)，即得到特征值表達(dá)式

(21)

本文求解出的特征值方程(21)和文獻(xiàn)[12]中特征值方程(20)化簡(jiǎn)相應(yīng)的參數(shù)所得的特征值方程本質(zhì)是相同的。

2　數(shù)值結(jié)果

為了說(shuō)明本文結(jié)果的精度，把公式(21)計(jì)算所得特征值和先前文獻(xiàn)[13]中的數(shù)值及公認(rèn)最接近精確值的Lucha,et al程序包所得數(shù)據(jù)作對(duì)比(見表1)。對(duì)所得數(shù)據(jù)進(jìn)行對(duì)比分析可知，本文所得的特征值比先前計(jì)算所得數(shù)據(jù)更好地逼近精確值。

表1　2p, 3p, 3d, 4p, 4d, 4f, 5p, 5d, 5f, 5g, 6d和6f態(tài)特征值數(shù)值表, 其中勢(shì)參數(shù)取為De=15

表1(續(xù))

3　結(jié)論

對(duì)短勢(shì)場(chǎng)改良的Manning-Rosen勢(shì)利用合適的近似辦法近似表示離心項(xiàng)，求解了任意l波薛定諤方程散射態(tài)解析解，得到了相應(yīng)的相移公式和按“K/2π標(biāo)度”歸一化的超幾何函數(shù)表示的徑向波函數(shù)，印證我們所計(jì)算出的特征值方程和先前計(jì)算出的表達(dá)式本質(zhì)是相同的。為了進(jìn)一步說(shuō)明近似精度，我們數(shù)值計(jì)算了特征值方程，所得的特征值數(shù)據(jù)較先前數(shù)據(jù)更好地逼近了MATHEMATICA程序包所得數(shù)據(jù)。

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(編校：葉超)

The Scattering States of Schr?dinger Equataion with the Improved Manning-Rosen Potential

CHEN Wenli, SHI Yanwei, YU Xiang

(GeneralEducationCenter,Xi’anPeihuaUniversity,Xi’an710125China)

We obtianed the scattering state analytical solutions of the Schr?dinger equation for the improved Manning-Rosen potential. The explicitly calculation formula of phase shift is derived and the normalized radial wave functions of scattering states on the“K/2π scale” are presented. The energy levels of the bound states are also obtained by studying analytical properties of scattering amplitude. All data calculated by the above approximate analytical formulae are compared with those obtained by the previous. Numerical results show the accuracy of our results.

the improved Manning-Rosen potential; scattering states; approximately analytical solutions

2016-02-12

陜西省教育廳科學(xué)研究項(xiàng)目(15JK2093)。

陳文利(1981—)，男，講師，碩士研究生，主要研究方向?yàn)橛?jì)算物理。

O365

A

1673-159X(2016)04-0039-5

10.3969/j.issn.1673-159X.2016.04.008