5.3: Time–independent Perturbations
- Page ID
- 28769
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\ket}[1]{\left| #1 \right>}\)
\(\newcommand{\bra}[1]{\left< #1 \right|}\)
\(\newcommand{\braket}[2]{\left< #1 \vphantom{#2} \right| \left. #2 \vphantom{#1} \right>}\)
\(\newcommand{\braopket}[3]{\left< #1 \vphantom{#2}\vphantom{#3} \right| #2 \vphantom{#1}\vphantom{#3} \left| #3 \vphantom{#1}\vphantom{#2} \right>}\)
\(\newcommand{\qmvec}[1]{\mathbf{\vec{#1}}}\)
\(\newcommand{\op}[1]{\hat{\mathbf{#1}}}\)
\(\newcommand{\expect}[1]{\langle #1 \rangle}\)
\(\newcommand{\dfn}[1]{\emph{\textbf{#1}}}\)
The results obtained in the last section can also be applied to the case where the perturbation, \(\hat{V}\), is actually independent of time (strictly, ‘switched on’ at t=0).
Again, starting the system in eigenstate \(|k \rangle\) of \(\hat{H}_0\) we obtain,
\[c_m(t)\Delta c_m(t) = (i\hbar )^{−1} V_{mk} \int^t_0 \text{ exp}(i\omega_{mk}t) dt \\ = \frac{V_{mk}}{\hbar\omega_{mk}} [1 − \text{ exp}(i\omega_{mk}t)] \nonumber\]
for \(m \neq k\), giving for the transition probability
\[p_m(t) = |\Delta c_m(t)|^2 = \frac{|V_{mk}|^2}{\hbar^2}\frac{\sin^2 (\omega_{mk}t/2)}{(\omega_{mk}/2)^2}. \nonumber\]
For sufficiently large values of \(t\), the function
\[f(t, \omega_{mk}) \equiv \frac{\sin^2 (\omega_{mk}t/2)}{(\omega_{mk}/2)^2} \nonumber\]
consists essentially of a large peak, centerd on \(\omega_{mk} = 0\), of height \(t^2\) and width \(\approx 4\pi /t\), as indicated in Fig. 4. Thus there is only a significant transition probability if \(E_m \approx E_k\). That is, if \(|\omega_{mk}| < 2\pi /t\).
Figure \(\PageIndex{1}\): Transition probability as a function of applied harmonic perturbation frequency
Note that we are assuming that the system was prepared in some eigenstate of \(\hat{H}_0\) which is not an eigenstate of \(\hat{V}\): if it were, then the matrix element \(V_{nm}\) would be zero and \(p_m(t) = 0\). Thus although the analysis treats the perturbation as time independent, it is applied to cases where the perturbation is switched on at \(t = 0\). Moreover only perturbations which are incompatible with the Hamiltonian can induce transitions.