# 微擾理論 (量子力學)

## 一階修正

${\displaystyle H_{0}|n^{(0)}\rangle =E_{n}^{(0)}|n^{(0)}\rangle \quad ,\quad n=1,2,3,\cdots }$

${\displaystyle H=H_{0}+\lambda V}$

${\displaystyle \left(H_{0}+\lambda V\right)|n\rangle =E_{n}|n\rangle }$

${\displaystyle E_{n}=E_{n}^{(0)}+\lambda E_{n}^{(1)}+\lambda ^{2}E_{n}^{(2)}+\cdots }$
${\displaystyle |n\rangle =|n^{(0)}\rangle +\lambda |n^{(1)}\rangle +\lambda ^{2}|n^{(2)}\rangle +\cdots }$

${\displaystyle E_{n}^{(k)}={\frac {1}{k!}}{\frac {d^{k}E_{n}}{d\lambda ^{k}}}}$
${\displaystyle |n^{(k)}\rangle ={\frac {1}{k!}}{\frac {d^{k}|n\rangle }{d\lambda ^{k}}}}$

${\displaystyle \lambda =0}$時，${\displaystyle E_{n}}$${\displaystyle |n\rangle }$分別約化為零微擾值，級數的第一個項目，${\displaystyle E_{n}^{(0)}}$${\displaystyle |n^{(0)}\rangle }$。由於微擾很微弱，含微擾系統的能級和量子態應該不會與它們的零微擾值相差太多，高階項目應該會很快地變小。

${\displaystyle {\begin{matrix}\left(H_{0}+\lambda V\right)\left(|n^{(0)}\rangle +\lambda |n^{(1)}\rangle +\cdots \right)\qquad \qquad \qquad \qquad \\\qquad \qquad \qquad =\left(E_{n}^{(0)}+\lambda E_{n}^{(1)}+\lambda ^{2}E_{n}^{(2)}+\cdots \right)\left(|n^{(0)}\rangle +\lambda |n^{(1)}\rangle +\cdots \right)\end{matrix}}}$

${\displaystyle H_{0}|n^{(1)}\rangle +V|n^{(0)}\rangle =E_{n}^{(0)}|n^{(1)}\rangle +E_{n}^{(1)}|n^{(0)}\rangle }$(1)

${\displaystyle \langle n^{(0)}|}$ 內積於這方程式：

${\displaystyle \langle n^{(0)}|H_{0}|n^{(1)}\rangle +\langle n^{(0)}|V|n^{(0)}\rangle =\langle n^{(0)}|E_{n}^{(0)}|n^{(1)}\rangle +\langle n^{(0)}|E_{n}^{(1)}|n^{(0)}\rangle }$

${\displaystyle E_{n}^{(1)}=\langle n^{(0)}|V|n^{(0)}\rangle }$

${\displaystyle |n^{(0)}\rangle =\sum _{k}|k^{(0)}\rangle \langle k^{(0)}|n^{(0)}\rangle }$

${\displaystyle \sum _{k}|k^{(0)}\rangle \langle k^{(0)}|={\boldsymbol {1}}}$

{\displaystyle {\begin{aligned}V|n^{(0)}\rangle &=\left(|n^{(0)}\rangle \,\langle n^{(0)}|\right)V|n^{(0)}\rangle +\left(\sum _{k\neq n}|k^{(0)}\rangle \langle k^{(0)}|\right)V|n^{(0)}\rangle \\&=E_{n}^{(1)}|n^{(0)}\rangle +\sum _{k\neq n}|k^{(0)}\rangle \langle k^{(0)}|V|n^{(0)}\rangle \\\end{aligned}}}

${\displaystyle \left(E_{n}^{(0)}-H_{0}\right)|n^{(1)}\rangle =\sum _{k\neq n}|k^{(0)}\rangle \langle k^{(0)}|V|n^{(0)}\rangle }$(2)

${\displaystyle \langle m^{(0)}|,\,m\neq n}$ 內積於這方程式：

${\displaystyle \langle m^{(0)}|\left(E_{n}^{(0)}-H_{0}\right)|n^{(1)}\rangle =\sum _{k\neq n}\langle m^{(0)}|k^{(0)}\rangle \langle k^{(0)}|V|n^{(0)}\rangle =\langle m^{(0)}|V|n^{(0)}\rangle }$

${\displaystyle \langle m^{(0)}|n^{(1)}\rangle ={\frac {\langle m^{(0)}|V|n^{(0)}\rangle }{\left(E_{n}^{(0)}-E_{m}^{(0)}\right)}}}$(3)

${\displaystyle |n^{(1)}\rangle =\sum _{k}c_{k}|k^{(0)}\rangle }$

${\displaystyle |n^{(1)}\rangle =\sum _{k\neq n}\langle k^{(0)}|n^{(1)}\rangle |k^{(0)}\rangle =\sum _{k\neq n}{\frac {\langle k^{(0)}|V|n^{(0)}\rangle }{E_{n}^{(0)}-E_{k}^{(0)}}}|k^{(0)}\rangle }$(4)

${\displaystyle \langle n^{(0)}|n^{(0)}\rangle =1}$

${\displaystyle \langle n|n\rangle =\langle n^{(0)}|n^{(0)}\rangle +\lambda \langle n^{(0)}|n^{(1)}\rangle +\lambda \langle n^{(1)}|n^{(0)}\rangle }$

${\displaystyle \langle n^{(0)}|n^{(1)}\rangle =\langle n^{(1)}|n^{(0)}\rangle =0}$

${\displaystyle \langle n|n\rangle =1}$

## 二階與更高階修正

${\displaystyle E_{n}=E_{n}^{(0)}+\langle n^{(0)}|V|n^{(0)}\rangle +\sum _{k\neq n}{\frac {|\langle k^{(0)}|V|n^{(0)}\rangle |^{2}}{E_{n}^{(0)}-E_{k}^{(0)}}}+\cdots }$
${\displaystyle |n\rangle =|n^{(0)}\rangle +\sum _{k\neq n}|k^{(0)}\rangle {\frac {\langle k^{(0)}|V|n^{(0)}\rangle }{E_{n}^{(0)}-E_{k}^{(0)}}}+\sum _{k\neq n}\sum _{\ell \neq n}|k^{(0)}\rangle {\frac {\langle k^{(0)}|V|\ell ^{(0)}\rangle \langle \ell ^{(0)}|V|n^{(0)}\rangle }{(E_{n}^{(0)}-E_{k}^{(0)})(E_{n}^{(0)}-E_{\ell }^{(0)})}}}$
${\displaystyle -\sum _{k\neq n}|k^{(0)}\rangle {\frac {\langle n^{(0)}|V|n^{(0)}\rangle \langle k^{(0)}|V|n^{(0)}\rangle }{(E_{n}^{(0)}-E_{k}^{(0)})^{2}}}-{\frac {1}{2}}\sum _{k\neq n}|n^{(0)}\rangle {\frac {\langle n^{(0)}|V|k^{(0)}\rangle \langle k^{(0)}|V|n^{(0)}\rangle }{(E_{k}^{(0)}-E_{n}^{(0)})^{2}}}}$

${\displaystyle E_{n}^{(3)}=\sum _{k\neq n}\sum _{m\neq n}{\frac {\langle n^{(0)}|V|m^{(0)}\rangle \langle m^{(0)}|V|k^{(0)}\rangle \langle k^{(0)}|V|n^{(0)}\rangle }{\left(E_{m}^{(0)}-E_{n}^{(0)}\right)\left(E_{k}^{(0)}-E_{n}^{(0)}\right)}}}$
${\displaystyle -\langle n^{(0)}|V|n^{(0)}\rangle \sum _{m\neq n}{\frac {|\langle n^{(0)}|V|m^{(0)}\rangle |^{2}}{\left(E_{m}^{(0)}-E_{n}^{(0)}\right)^{2}}}}$

## 簡併

${\displaystyle |n\rangle =\sum _{k\in D}\alpha _{nk}|k^{(0)}\rangle +\lambda |n^{(1)}\rangle }$

${\displaystyle V|n^{(0)}\rangle =\epsilon _{n}|n^{(0)}\rangle ,\qquad \forall \;|n^{(0)}\rangle \in D}$

${\displaystyle {\begin{bmatrix}&\cdots &\\\vdots &\langle k^{(0)}|V|l^{(0)}\rangle &\vdots \\&\cdots &\end{bmatrix}}={\begin{bmatrix}&\cdots &\\\vdots &V_{kl}&\vdots \\&\cdots &\end{bmatrix}},\qquad \forall \;|k^{(0)}\rangle ,|l^{(0)}\rangle \in D}$

${\displaystyle \left(E_{n}^{(0)}-H_{0}\right)|n^{(1)}\rangle =\sum _{k\not \in D}\langle k^{(0)}|V|n^{(0)}\rangle |k^{(0)}\rangle }$

${\displaystyle |n^{(1)}\rangle =\sum _{k\not \in D}{\frac {\langle k^{(0)}|V|n^{(0)}\rangle }{E_{n}^{(0)}-E_{k}^{(0)}}}|k^{(0)}\rangle }$

## 參考文獻

1. ^ E. Schrödinger, Annalen der Physik, Vierte Folge, Band 80, p. 437 (1926)
2. ^ J. W. S. Rayleigh, Theory of Sound, 2nd edition Vol. I, pp 115-118, Macmillan, London (1894)
3. ^ L. D. Landau, E. M. Lifschitz, Quantum Mechanics: Non-relativistic Theory", 3rd ed.