diff options
Diffstat (limited to 'latex/know/concept/time-independent-perturbation-theory/source.md')
| -rw-r--r-- | latex/know/concept/time-independent-perturbation-theory/source.md | 12 |
1 files changed, 7 insertions, 5 deletions
diff --git a/latex/know/concept/time-independent-perturbation-theory/source.md b/latex/know/concept/time-independent-perturbation-theory/source.md index a3167cd..48504f4 100644 --- a/latex/know/concept/time-independent-perturbation-theory/source.md +++ b/latex/know/concept/time-independent-perturbation-theory/source.md @@ -3,8 +3,8 @@ # Time-independent perturbation theory -*Time-independent perturbation theory*, sometimes also called -*stationary state perturbation theory*, is a specific application of +**Time-independent perturbation theory**, sometimes also called +**stationary state perturbation theory**, is a specific application of perturbation theory to the time-independent Schrödinger equation in quantum physics, for Hamiltonians of the following form: @@ -27,8 +27,8 @@ $$\begin{aligned} &= E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + ... \end{aligned}$$ -Where $E_n^{(1)}$ and $\ket*{\psi_n^{(1)}}$ are called the *first-order -corrections*, and so on for higher orders. We insert this into the +Where $E_n^{(1)}$ and $\ket*{\psi_n^{(1)}}$ are called the **first-order +corrections**, and so on for higher orders. We insert this into the Schrödinger equation: $$\begin{aligned} @@ -72,6 +72,7 @@ $$\begin{aligned} The approach to solving the other two equations varies depending on whether this $\hat{H}_0$ has a degenerate spectrum or not. + ## Without degeneracy We start by assuming that there is no degeneracy, in other words, each @@ -178,6 +179,7 @@ $$\begin{aligned} In practice, it is not particulary useful to calculate more corrections. + ## With degeneracy If $\varepsilon_n$ is $D$-fold degenerate, then its eigenstate could be @@ -295,7 +297,7 @@ The trick is to find a Hermitian operator $\hat{L}$ (usually using symmetries of the system) which commutes with both $\hat{H}_0$ and $\hat{H}_1$: $$\begin{aligned} - [\hat{L}, \hat{H}_0] = [\hat{L}, \hat{H}_1] = 0 + \comm*{\hat{L}}{\hat{H}_0} = \comm{\hat{L}}{\hat{H}_1} = 0 \end{aligned}$$ So that it shares its eigenstates with $\hat{H}_0$ (and $\hat{H}_1$), |