# 相空间表述

## 相空间分布

${\displaystyle \operatorname {P} [a\leq X\leq b]=\int _{a}^{b}\int _{-\infty }^{\infty }W(x,p)\,dp\,dx}$

${\displaystyle \langle {\hat {A}}\rangle =\int A(x,p)W(x,p)\,dp\,dx}$

## 星积

${\displaystyle f{\stackrel {\leftarrow }{\partial }}_{x}g={\frac {\partial f}{\partial x}}\cdot g}$
${\displaystyle f{\stackrel {\rightarrow }{\partial }}_{x}g=f\cdot {\frac {\partial g}{\partial x}}}$

${\displaystyle f\star g=f\,\exp {\left({\tfrac {i\hbar }{2}}({\stackrel {\leftarrow }{\partial }}_{x}{\stackrel {\rightarrow }{\partial }}_{p}-{\stackrel {\leftarrow }{\partial }}_{p}{\stackrel {\rightarrow }{\partial }}_{x})\right)}\,g}$

{\displaystyle {\begin{aligned}(f\star g)(x,p)&=f\left(x+{\tfrac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{p},p-{\tfrac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{x}\right)\cdot g(x,p)\\&=f(x,p)\cdot g\left(x-{\tfrac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{p},p+{\tfrac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{x}\right)\\&=f\left(x+{\tfrac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{p},p\right)\cdot g\left(x-{\tfrac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{p},p\right)\\&=f\left(x,p-{\tfrac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{x}\right)\cdot g\left(x,p+{\tfrac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{x}\right).\end{aligned}}}

${\displaystyle (f\star g)(x,p)={\frac {1}{\pi ^{2}\hbar ^{2}}}\,\int f(x+x',p+p')\,g(x+x'',p+p'')\,\exp {\left({\tfrac {2i}{\hbar }}(x'p''-x''p')\right)}\,dx'dp'dx''dp''}$

${\displaystyle H\star W=E\cdot W}$

## 时间演化

${\displaystyle {\frac {\partial f}{\partial t}}=-{\frac {1}{i\hbar }}\left(f\star H-H\star f\right),}$

${\displaystyle {\frac {\partial W}{\partial t}}=-\{\{W,H\}\}=-{\frac {2}{\hbar }}W\sin \left({{\frac {\hbar }{2}}({\stackrel {\leftarrow }{\partial }}_{x}{\stackrel {\rightarrow }{\partial }}_{p}-{\stackrel {\leftarrow }{\partial }}_{p}{\stackrel {\rightarrow }{\partial }}_{x})}\right)\ H=-\{W,H\}+O(\hbar ^{2}),}$

## 例子

### 简单谐振子

${\displaystyle H={\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}.}$

{\displaystyle {\begin{aligned}H\star W&=\left({\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}\right)\star W\\&=\left({\frac {1}{2}}m\omega ^{2}\left(x+{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{p}\right)^{2}+{\frac {1}{2m}}\left(p-{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{x}\right)^{2}\right)~W\\&=\left({\frac {1}{2}}m\omega ^{2}\left(x^{2}-{\frac {\hbar ^{2}}{4}}{\stackrel {\rightarrow }{\partial }}_{p}^{2}\right)+{\frac {1}{2m}}\left(p^{2}-{\frac {\hbar ^{2}}{4}}{\stackrel {\rightarrow }{\partial }}_{x}^{2}\right)\right)~W\\&\,\,\,\,\,+{\frac {i\hbar }{2}}\left(m\omega ^{2}x{\stackrel {\rightarrow }{\partial }}_{p}-{\frac {p}{m}}{\stackrel {\rightarrow }{\partial }}_{x}\right)~W\\&=E\cdot W.\end{aligned}}}

${\displaystyle {\frac {\hbar }{2}}\left(m\omega ^{2}x{\stackrel {\rightarrow }{\partial }}_{p}-{\frac {p}{m}}{\stackrel {\rightarrow }{\partial }}_{x}\right)\cdot W=0}$

${\displaystyle W(x,p)=F\left({\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}\right)\equiv F(u).}$

${\displaystyle F_{n}(u)={\frac {(-1)^{n}}{\pi \hbar }}L_{n}\left(4{\frac {u}{\hbar \omega }}\right)e^{-2u/\hbar \omega }}$

${\displaystyle E_{n}=\hbar \omega \left(n+{\frac {1}{2}}\right)}$

${\displaystyle W(x,p;t)=W(m\omega x\cos \omega t-p\sin \omega t,~p\cos \omega t+\omega mx\sin \omega t;0)~.}$

### 自由粒子角动量

${\displaystyle W(\mathbf {x} ,\mathbf {p} ;t)={\frac {1}{(\pi \hbar )^{3}}}\exp {\left(-\alpha ^{2}r^{2}-{\frac {p^{2}}{\alpha ^{2}\hbar ^{2}}}\left(1+\left({\frac {t}{\tau }}\right)^{2}\right)+{\frac {2t}{\hbar \tau }}\mathbf {x} \cdot \mathbf {p} \right)}~,}$

${\displaystyle W\longrightarrow {\frac {1}{(\pi \hbar )^{3}}}\exp \left[-\alpha ^{2}\left(\mathbf {x} -{\frac {\mathbf {p} t}{m}}\right)^{2}\right]\,.}$

${\displaystyle K_{rad}={\frac {\alpha ^{2}\hbar ^{2}}{2m}}\left({\frac {3}{2}}-{\frac {1}{1+(t/\tau )^{2}}}\right)}$
${\displaystyle K_{ang}={\frac {\alpha ^{2}\hbar ^{2}}{2m}}{\frac {1}{1+(t/\tau )^{2}}}~.}$

## 注释

1. ^ 即采用动量算符或是位置算符本征矢为系统态矢的基矢
2. ^ 参见经典极限
3. ^ 参见
4. ^
5. ^ 可能是
6. ^ 因而，通过高斯函数可以构造双曲函数形式的星积 [7]
${\displaystyle \exp \left(-{a}(x^{2}+p^{2})\right)~\star ~\exp \left(-{b}(x^{2}+p^{2})\right)={1 \over 1+\hbar ^{2}ab}\exp \left(-{a+b \over 1+\hbar ^{2}ab}(x^{2}+p^{2})\right)}$
或是
${\displaystyle \delta (x)~\star ~\delta (p)={2 \over h}\exp \left(2i{xp \over \hbar }\right)}$
7. ^ 当通过不确定性原理对于定域情况给定限制时，尼尔斯·玻尔极力否认微观情况下会存在量子轨道。然而通过形式上的相空间轨道，维格纳函数的演化却可以利用路径积分方法[21]或者[22]严格解出，但两种方法都存在一定的缺陷。
8. ^ 这一相对表示自由波包在位置空间中传播。

## 参考文献

1. Groenewold, H. J. On the principles of elementary quantum mechanics. Physica. 1946, 12 (7): 405–460. doi:10.1016/S0031-8914(46)80059-4 （英语）.
2. Moyal, J. E. Quantum mechanics as a statistical theory. Mathematical Proceedings of the Cambridge Philosophical Society (Cambridge University Press). 1949, 45 (01): 99–124. doi:10.1017/S0305004100000487 （英语）.
3. ^ Weyl, H. Quantenmechanik und gruppentheorie. Zeitschrift für Physik. 1927, 46 (1-2): 1–46. doi:10.1007/BF02055756 （德语）.
4. Wigner, E. On the quantum correction for thermodynamic equilibrium. Physical Review. 1932, 40 (5): 749–759. doi:10.1103/PhysRev.40.749 （英语）.
5. ^ Ali, S. T.; Engliš, M. Quantization methods: a guide for physicists and analysts. Reviews in Mathematical Physics. 2005, 17 (04): 391–490. doi:10.1142/S0129055X05002376 （英语）.
6. Curtright, T. L.; Zachos, C. K. Quantum Mechanics in Phase Space. Asia Pacific Physics Newsletter. 2012, 1: 37. doi:10.1142/S2251158X12000069 （英语）.
7. Zachos, C.; Fairlie, D.; Curtright, T. Quantum mechanics in phase space: an overview with selected papers. World Scientific. 2005. ISBN 978-981-238-384-6 （英语）.
8. ^ Cohen, L. Generalized Phase-Space Distribution Functions. Journal of Mathematical Physics. 1966, 7 (5): 781. Bibcode:1966JMP.....7..781C. doi:10.1063/1.1931206 （英语）.
9. Agarwal, G. S.; Wolf, E. Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. II. Quantum mechanics in phase space. Physical Review D. 1970, 2 (10): 2187–2205. doi:10.1103/PhysRevD.2.2187 （英语）.
10. ^ Sudarshan, E. C. G. Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Physical Review Letters. 1963, 10 (7): 277–279. doi:10.1103/PhysRevLett.10.277 （英语）.
11. ^ Glauber, R. J. Coherent and incoherent states of the radiation field. Physical Review. 1963, 131 (6): 2766–2788. doi:10.1103/PhysRev.131.2766 （英语）.
12. ^ Husimi, K. Some formal properties of the density matrix. Nippon Sugaku-Buturigakkwai Kizi Dai 3 Ki. 1940, 22 (4): 264–314 （英语）.
13. ^ Agarwal, G. S.; Wolf, E. Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators. Physical Review D. 1970, 2 (10): 2161–2186. doi:10.1103/PhysRevD.2.2161 （英语）.
14. ^ Cahill, K. E.; Glauber, R. J. Ordered expansions in boson amplitude operators. Physical Review. 1969, 177 (5): 1857–1881. doi:10.1103/PhysRev.177.1857 （英语）.
15. ^ Cahill, K. E.; Glauber, R. J. Density operators and quasiprobability distributions. Physical Review. 1969, 177 (5): 1882–1902. doi:10.1103/PhysRev.177.1882 （英语）.
16. ^ Lax, M. Quantum Noise. XI. Multitime correspondence between quantum and classical stochastic processes. Physical Review. 1968, 172 (2): 350–361. doi:10.1103/PhysRev.172.350 （英语）.
17. ^ Baker Jr., G. A. Formulation of quantum mechanics based on the quasi-probability distribution induced on phase space. Physical Review. 1958, 109 (6): 2198–2206. doi:10.1103/PhysRev.109.2198 （英语）.
18. ^ Fairlie, D. B. The formulation of quantum mechanics in terms of phase space functions. Mathematical Proceedings of the Cambridge Philosophical Society. 1964, 60 (3): 581. doi:10.1017/S0305004100038068 （英语）.
19. Curtright, T.; Fairlie, D.; Zachos, C. Features of time-independent Wigner functions. Physical Review D. 1998, 58 (2). Bibcode:1998PhRvD..58b5002C. . doi:10.1103/PhysRevD.58.025002 （英语）.
20. ^ Mehta, C. L. Phase‐Space Formulation of the Dynamics of Canonical Variables. Journal of Mathematical Physics. 1964, 5 (5): 677–686. doi:10.1063/1.1704163 （英语）.
21. ^ Marinov, M. S. A new type of phase-space path integral. Physics Letters A. 1991, 153 (1): 5-11 [2016-01-29]. （原始内容存档于2015-09-24） （英语）.
22. ^ Krivoruchenko, M. I.; Faessler, A. Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics. Journal of mathematical physics. 2007, 48 (5): 052107. doi:10.1063/1.2735816 （英语）.
23. ^ Curtright, T. L. Time-dependent Wigner Functions. Department of Physics, College of Arts & Sciences, University Miami. [2016-01-29]. （原始内容存档于2020-10-27） （英语）.
24. ^ Dahl, J. P.; Schleich, W. P. Concepts of radial and angular kinetic energies. Physical Review A. 2002, 65 (2): 022109. doi:10.1103/PhysRevA.65.022109 （英语）.