# 盖尔曼–劳定理

## 原理的表述

${\displaystyle |\Psi _{0}\rangle }$ ${\displaystyle H_{0}}$  的一个本征态，能量为 ${\displaystyle E_{0}}$  。定义相互作用的哈密顿量为 ${\displaystyle H=H_{0}+gV}$ ，其中 ${\displaystyle g}$  是耦合常数， ${\displaystyle V}$  是相互作用项，定义带参量的哈密顿 ${\displaystyle H_{\epsilon }=H_{0}+e^{-\epsilon |t|}gV}$  ，可以看到，当 ${\displaystyle |t|\rightarrow \infty }$  时，${\displaystyle H_{\epsilon }=H_{0}}$ 。而当${\displaystyle t=0}$ 时，${\displaystyle H_{\epsilon }=H}$ 。令 ${\displaystyle U_{\epsilon I}}$  为对应于${\displaystyle H_{\epsilon }}$ 相互作用繪景（用下标I表示）下的时间演化算符。盖尔曼–劳定理说的是，若 ${\displaystyle \epsilon \rightarrow 0^{+}}$  时，

${\displaystyle |\Psi _{\epsilon }^{(\pm )}\rangle ={\frac {U_{\epsilon I}(0,\pm \infty )|\Psi _{0}\rangle }{\langle \Psi _{0}|U_{\epsilon I}(0,\pm \infty )|\Psi _{0}\rangle }}}$

## 证明

${\displaystyle i\hbar \partial _{t_{1}}U_{\epsilon }(t_{1},t_{2})=H_{\epsilon }(t_{1})U_{\epsilon }(t_{1},t_{2})}$

${\displaystyle U_{\epsilon }(t_{1},t_{2})=1+{\frac {1}{i\hbar }}\int _{t_{2}}^{t_{1}}dt'(H_{0}+e^{\epsilon (\theta -|t'|)}V)U_{\epsilon }(t',t_{2}).}$

${\displaystyle U_{\epsilon }(t_{1},t_{2})=1+{\frac {1}{i\hbar }}\int _{\theta +t_{2}}^{\theta +t_{1}}dt'(H_{0}+e^{\epsilon t'}V)U_{\epsilon }(t'-\theta ,t_{2}).}$

${\displaystyle \partial _{\theta }U_{\epsilon }(t_{1},t_{2})=\epsilon g\partial _{g}U_{\epsilon }(t_{1},t_{2})=\partial _{t_{1}}U_{\epsilon }(t_{1},t_{2})+\partial _{t_{2}}U_{\epsilon }(t_{1},t_{2}).}$

${\displaystyle -i\hbar \partial _{t_{1}}U_{\epsilon }(t_{2},t_{1})=U_{\epsilon }(t_{2},t_{1})H_{\epsilon }(t_{1})}$

${\displaystyle i\hbar \epsilon g\partial _{g}U_{\epsilon }(t_{1},t_{2})=H_{\epsilon }(t_{1})U_{\epsilon }(t_{1},t_{2})-U_{\epsilon }(t_{1},t_{2})H_{\epsilon }(t_{2}).}$

${\displaystyle H_{\epsilon I}}$ ${\displaystyle U_{\epsilon I}}$  之间的关系式形式上与上式相同，事实上，将上式两边各左乘 ${\displaystyle e^{iH_{0}t_{1}/\hbar }}$ ，右乘 ${\displaystyle e^{iH_{0}t_{2}/\hbar }}$  ，并利用关系

${\displaystyle U_{\epsilon I}(t_{1},t_{2})=e^{iH_{0}t_{1}/\hbar }U_{\epsilon }(t_{1},t_{2})e^{-iH_{0}t_{2}/\hbar }.}$

${\displaystyle \left(H_{\epsilon ,t=0}-E_{0}+i\hbar \epsilon g\partial _{g}\right)U_{\epsilon I}(0,\infty )|\Psi _{0}\rangle =0.}$

${\displaystyle \left(H_{\epsilon ,t=0}-E_{0}\pm i\hbar \epsilon g\partial _{g}\right)U_{\epsilon I}(0,\pm \infty )|\Psi _{0}\rangle =0.}$

${\displaystyle i\hbar \epsilon g\partial _{g}\left(U|\Psi _{0}\rangle \right)=(H_{\epsilon }-E_{0})U|\Psi _{0}\rangle .}$

{\displaystyle {\begin{aligned}i\hbar \epsilon g\partial _{g}|\Psi _{\epsilon }^{-}\rangle &={\frac {1}{\langle \Psi _{0}|U|\Psi _{0}\rangle }}(H_{\epsilon }-E_{0})U|\Psi _{0}\rangle -{\frac {U|\Psi _{0}\rangle }{{\langle \Psi _{0}|U|\Psi _{0}\rangle }^{2}}}\langle \Psi _{0}|H_{\epsilon }-E_{0}|\Psi _{0}\rangle \\&=(H_{\epsilon }-E_{0})|\Psi _{\epsilon }^{-}\rangle -|\Psi _{\epsilon }^{-}\rangle \langle \Psi _{0}|H_{\epsilon }-E_{0}|\Psi _{\epsilon }^{-}\rangle \\&=\left[H_{\epsilon }-E^{-}\right]|\Psi _{\epsilon }^{-}\rangle .\end{aligned}}}

${\displaystyle \left[H_{\epsilon }-E^{-}-i\hbar \epsilon g\partial _{g}\right]|\Psi _{\epsilon }^{-}\rangle =0.}$

${\displaystyle \left[H_{\epsilon }-E^{\pm }\pm i\hbar \epsilon g\partial _{g}\right]|\Psi _{\epsilon }^{\pm }\rangle =0}$

## 参考文献

1. 尹道乐，尹澜. 2. 凝聚态量子理论. ISBN 9787301161609.
2. ^ M. Gell-Mann and F. Low: "Bound States in Quantum Field Theory", Phys. Rev. 84, 350 (1951)
3. L.G. Molinari: "Another proof of Gell-Mann and Low's theorem", J. Math. Archive.is存檔，存档日期2013-02-23Phys. 48, 052113 (2007) Archive.is存檔，存档日期2013-02-23
• K. Hepp: Lecture Notes in Physics (Springer-Verlag, New York, 1969), Vol. 2.
• G. Nenciu and G. Rasche: "Adiabatic theorem and Gell-Mann-Low formula", Helv. Phys. Acta 62, 372 (1989).
• A.L. Fetter and J.D. Walecka: "Quantum Theory of Many-Particle Systems", McGraw–Hill (1971)