Results of Stationary Perturbation Theory

Review: Results of Stationary Perturbation Theory

For reference, we recapitulate some elementary results from perturbation theory. Assume that the Hamiltonian of a system may be written as the sum of two parts

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = H_0 + H^\prime }

and that the eigenstates and eigenvalues of $H_{0}$ are known:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0 | n^{(0)} \rangle = E_n^{(0)} | n^{(0)} {\rangle} }

If it is not possible to find the eigenvalues of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} exactly, it is possible to write power series expressions for them that converge over some interval. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^\prime} is time independent, the problem is stationary and the appropriate perturbation theory is Rayleigh- Schrodinger stationary state perturbation theory, described in most texts in quantum mechanics. We write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_n = E_n^{(0)} + E_n^{(1)} + E_n^{(2)} + \cdots }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | n \rangle = | n^{(0)} \rangle + | n^{(1)} \rangle + \cdots }

and express the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (i+1)^{\rm th}} order perturbation in terms of $E^{(i)}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | n^{(i)} {\rangle}} . The energies are given by

$E_{n}^{(m)}=\langle n^{(0)}|H^{\prime }|n^{(m-1)}{\rangle }$

We shall only use the lowest two orders here. The first order results are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_n^{(1)} = \langle n^{(0)} | H^\prime | n^{(0)}{\rangle} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | n^{(1)} \rangle = \sum'_{m} \frac{| m \rangle \langle m | H^\prime | n {\rangle}}{E_n - E_m} }

The symbol $\sum ^{\prime }$ indicates that the term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = n} is excluded. It is understood that the sum extends over continuum states. Note that the state function is nor properly normalized, but that the error is quadratic in $H^{\prime }$.

The second order results are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} E_n^{(2)} &=& \sum'_{n} \frac{ | \langle m | H^\prime | n \rangle |^2}{E_n - E_m} \\ | n^{(2)} {\rangle}&=& \sum'_{m} | m \rangle {\left[\frac{ \langle m | H^\prime | n {\rangle}}{E_n-E_m}{\left[1-\frac{ \langle n| H^\prime | n {\rangle}}{E_n - E_m} \right]} + \sum'_{p} \frac{ \langle m| H^\prime | p \rangle \langle p | H^\prime | n {\rangle}}{(E_n -E_m)(E_n-E_p)}\right]} \end{array}}

In second order perturbation theory the effect of a coupling of $n$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^\prime} is to push the levels apart, independent of the value of $H_{nm}^{\prime }$. Consequently, states coupled by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^\prime} always repel each other.