diff options
Diffstat (limited to 'content/know/concept/time-independent-perturbation-theory')
| -rw-r--r-- | content/know/concept/time-independent-perturbation-theory/index.pdc | 29 |
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. |