Expand knowledge base

1 files changed, 18 insertions, 11 deletions

diff --git a/content/know/concept/time-independent-perturbation-theory/index.pdc b/content/know/concept/time-independent-perturbation-theory/index.pdc
index 2035fc2..3be3cd5 100644
--- a/content/know/concept/time-independent-perturbation-theory/index.pdc
+++ b/content/know/concept/time-independent-perturbation-theory/index.pdc
@@ -159,22 +159,22 @@ $$\begin{aligned}
 Here it is clear why this is only valid in the non-degenerate case:
 otherwise we would divide by zero in the denominator.
 
-Next, to find the second-order correction to the energy $E_n^{(2)}$, we
-take the corresponding equation and put $\bra{n}$ in front of it:
+Next, to find the second-order energy correction $E_n^{(2)}$,
+we take the corresponding equation and put $\bra{n}$ in front of it:
 
 $$\begin{aligned}
-    \matrixel{n}{\hat{H}_1}{\psi_n^{(1)}} + \matrixel{n}{\hat{H}_0}{\psi_n^{(2)}}
-    &= E_n^{(2)} \braket{n}{n} + E_n^{(1)} \braket{n}{\psi_n^{(1)}} + \varepsilon_n \braket{n}{\psi_n^{(2)}}
+    \matrixel*{n}{\hat{H}_1}{\psi_n^{(1)}} + \matrixel*{n}{\hat{H}_0}{\psi_n^{(2)}}
+    &= E_n^{(2)} \braket{n}{n} + E_n^{(1)} \braket*{n}{\psi_n^{(1)}} + \varepsilon_n \braket*{n}{\psi_n^{(2)}}
 \end{aligned}$$
 
 Because $\hat{H}_0$ is Hermitian, we know that
-$\matrixel{n}{\hat{H}_0}{\psi_n^{(2)}} = \varepsilon_n \braket{n}{\psi_n^{(2)}}$,
+$\matrixel*{n}{\hat{H}_0}{\psi_n^{(2)}} = \varepsilon_n \braket*{n}{\psi_n^{(2)}}$,
 i.e. we apply it to the bra, which lets us eliminate two terms. Also,
 since $\ket{n}$ is normalized, we find:
 
 $$\begin{aligned}
     E_n^{(2)}
-    = \matrixel{n}{\hat{H}_1}{\psi_n^{(1)}} - E_n^{(1)} \braket{n}{\psi_n^{(1)}}
+    = \matrixel*{n}{\hat{H}_1}{\psi_n^{(1)}} - E_n^{(1)} \braket*{n}{\psi_n^{(1)}}
 \end{aligned}$$
 
 We explicitly removed the $\ket{n}$-dependence of $\ket*{\psi_n^{(1)}}$,
@@ -221,8 +221,8 @@ take the equation at order $\lambda^1$ and prepend an arbitrary
 eigenspace basis vector $\bra{n, \delta}$:
 
 $$\begin{aligned}
-    \matrixel{n, \delta}{\hat{H}_1}{n} + \matrixel{n, \delta}{\hat{H}_0}{\psi_n^{(1)}}
-    &= E_n^{(1)} \braket{n, \delta}{n} + \varepsilon_n \braket{n, \delta}{\psi_n^{(1)}}
+    \matrixel{n, \delta}{\hat{H}_1}{n} + \matrixel*{n, \delta}{\hat{H}_0}{\psi_n^{(1)}}
+    &= E_n^{(1)} \braket{n, \delta}{n} + \varepsilon_n \braket*{n, \delta}{\psi_n^{(1)}}
 \end{aligned}$$
 
 Since $\hat{H}_0$ is Hermitian, we use the same trick as before to
@@ -320,11 +320,18 @@ $\ket{n, d_1}$ and $\ket{n, d_2}$ have distinct eigenvalues
 $\ell_1 \neq \ell_2$ for $d_1 \neq d_2$:
 
 $$\begin{aligned}
-    \hat{L} \ket{n, b_1} = \ell_1 \ket{n, b_1}
+    \hat{L} \ket{n, d_1} = \ell_1 \ket{n, d_1}
     \qquad
-    \hat{L} \ket{n, b_2} = \ell_2 \ket{n, b_2}
+    \hat{L} \ket{n, d_2} = \ell_2 \ket{n, d_2}
 \end{aligned}$$
 
-When this holds for any orthogonal choice of $\ket{n, d_1}$ and
+When this condition holds for any orthogonal choice of $\ket{n, d_1}$ and
 $\ket{n, d_2}$, then these specific eigenvectors of $\hat{L}$ are the
 "good states", for any valid choice of $\hat{L}$.
+
+
+
+## References
+1.  D.J. Griffiths, D.F. Schroeter,
+    *Introduction to quantum mechanics*, 3rd edition,
+    Cambridge.