量子與經(jīng)典對(duì)應(yīng):Dirac方程中的速度算符

張治國(guó), 封文江, 鄭 偉, 陳 皓, 崔 崧

(沈陽(yáng)師范大學(xué) 物理科學(xué)與技術(shù)學(xué)院, 沈陽(yáng) 110034)

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量子與經(jīng)典對(duì)應(yīng):Dirac方程中的速度算符

張治國(guó), 封文江, 鄭 偉, 陳 皓, 崔 崧

(沈陽(yáng)師范大學(xué) 物理科學(xué)與技術(shù)學(xué)院, 沈陽(yáng) 110034)

由量子力學(xué)中的Bohr對(duì)應(yīng)原理可知,在大量子數(shù)情形下,量子力學(xué)應(yīng)過(guò)渡到經(jīng)典力學(xué)。在經(jīng)典極限下,由Heisenberg對(duì)應(yīng)原理可知,厄密算符的量子矩陣元對(duì)應(yīng)經(jīng)典物理量的Fourier展開系數(shù)。利用Heisenberg對(duì)應(yīng)原理研究相對(duì)論效應(yīng)的自由粒子和在勻磁場(chǎng)中的帶電粒子的量子經(jīng)典對(duì)應(yīng)問(wèn)題。將Heisenberg對(duì)應(yīng)原理應(yīng)用到相對(duì)論領(lǐng)域的Dirac方程,計(jì)算出自由粒子的Dirac方程中的α算符及其經(jīng)典近似,并且研究自旋1/2的帶電粒子在勻磁場(chǎng)中的Dirac方程情形。對(duì)于相對(duì)論效應(yīng)的自由粒子和在勻磁場(chǎng)中的帶電粒子,Dirac理論中的α算符將對(duì)應(yīng)經(jīng)典的速度。

Heisenberg對(duì)應(yīng)原理; Dirac方程; 速度算符

0 引 言

自量子力學(xué)誕生起,人們就開始利用波函數(shù)或Schr?dinger方程來(lái)研究量子與經(jīng)典的對(duì)應(yīng)問(wèn)題[1-8]。Bohr原理指出,在大量子數(shù)近似下量子力學(xué)應(yīng)過(guò)渡到經(jīng)典力學(xué)。在后來(lái)的研究中,人們逐漸發(fā)現(xiàn)Heisenberg對(duì)應(yīng)原理(HCP)在量子與經(jīng)典的對(duì)應(yīng)問(wèn)題的作用[9-15]。HCP以量子矩陣元為基點(diǎn),認(rèn)為在經(jīng)典極限下,量子矩陣元對(duì)應(yīng)于經(jīng)典物理量的Fourier展開系數(shù)。由HCP可知,物理量所有可能的矩陣元之和將給出經(jīng)典運(yùn)動(dòng)方程的解,即在大量子數(shù)近似下,有

(1)

式中ψn(x,t)為體系中的波函數(shù),厄密算符F應(yīng)該對(duì)應(yīng)于一個(gè)經(jīng)典物理方程的解。

在Hilbert空間中,粒子與反粒子在各自獨(dú)立的空間里運(yùn)動(dòng),因此應(yīng)該分別應(yīng)用HCP來(lái)求解Dirac方程的正能解和負(fù)能解(對(duì)應(yīng)粒子與反粒子)。本文以正能解波函數(shù)研究了相對(duì)論領(lǐng)域的Dirac方程里的速度算符,計(jì)算表明,在經(jīng)典近似下,速度算符α對(duì)應(yīng)著經(jīng)典物理的速度量。

1 Dirac方程

考慮相對(duì)論情形下自旋1/2的自由粒子的哈密頓量:

相應(yīng)Dirac方程:

(2)

式(2)的正能解可寫成形式為ψ(x,t)=ψ(x)exp(-iω(t-t0))

(3)

其中:χs為泡利算符σz的本征態(tài)(σzχs=sχs),s=±1;V為相空間歸一化常數(shù)。

由式(2)得:

(4)

2 磁場(chǎng)中的Dirac方程

相應(yīng)的Dirac方程為

(5)

假設(shè)波函數(shù)形式取為

(6)

將波函數(shù)(6)代入Dirac方程(5),有

(7a)

(7b)

(8)

將式(8)代入第1個(gè)方程(7a),得

下面計(jì)算速度算符的矩陣元。

(9)

(10a)

(10b)

3 經(jīng)典極限

在經(jīng)典近似下n→∞,(10a),(10b)中αnx(t),αny(t)分別為

(11a)

(11b)

(12)

(13a)

(13b)

4 結(jié) 論

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Quantum and classical correspondence for velocity operator in Dirac equation

ZHANGZhiguo,FENGWenjiang,ZHENGWei,CHENHao,CUISong

(College of Physical Science and Technology, Shenyang Normal University, Shenyang 110034, China)

Due to Bohr correspondence principle in the quantum mechanics, quantum mechanics go to classical mechanics in the case of large quantum number. Based upon Heisenberg correspondence principle, quantum matrix element of a Hermitian operator reduces to the coefficient of Fourier expansion of the corresponding classical quantity in the classical limit. Using Heisenberg correspondence principle, quantum-classical correspondence of the relativistic free particle and the 1/2 spin charged particles in a constant magnetic field are studied. Applying Heisenberg correspondence principle to Dirac equation in relativistic realm, the operator of free particle in the Dirac theory and quantum-classical correspondence are obtained,and 1/2 spin charged particle in a constant magnetic field in Dirac equation is also studied. For the relativistic free particle or the charged particle in a constant magnetic field, the velocity operator in the Dirac theory will reduce to the classical velocity.

Heisenberg correspondence principle；Dirac equation；velocity operator

2016-03-17。

遼寧省教育廳科學(xué)研究一般項(xiàng)目(L2014442)。

張治國(guó)(1977-),男,遼寧沈陽(yáng)人,沈陽(yáng)師范大學(xué)講師,碩士。

1673-5862(2016)04-0441-04

理論與應(yīng)用研究

O413.1

A

10.3969/ j.issn.1673-5862.2016.04.013