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Diffstat (limited to 'latex/know')
| -rw-r--r-- | latex/know/concept/time-independent-perturbation-theory/source.md | 86 |
1 files changed, 42 insertions, 44 deletions
diff --git a/latex/know/concept/time-independent-perturbation-theory/source.md b/latex/know/concept/time-independent-perturbation-theory/source.md index d076457..a3167cd 100644 --- a/latex/know/concept/time-independent-perturbation-theory/source.md +++ b/latex/know/concept/time-independent-perturbation-theory/source.md @@ -16,31 +16,31 @@ $$\begin{aligned} Where $\hat{H}_0$ is a Hamiltonian for which the time-independent Schrödinger equation has a known solution, and $\hat{H}_1$ is a small perturbing Hamiltonian. The eigenenergies $E_n$ and eigenstates -$\ket{\psi_n}$ of the composite problem are expanded accordingly in the +$\ket{\psi_n}$ of the composite problem are expanded in the perturbation "bookkeeping" parameter $\lambda$: $$\begin{aligned} \ket{\psi_n} - &= \ket{\psi_n^{(0)}} + \lambda \ket{\psi_n^{(1)}} + \lambda^2 \ket{\psi_n^{(2)}} + ... + &= \ket*{\psi_n^{(0)}} + \lambda \ket*{\psi_n^{(1)}} + \lambda^2 \ket*{\psi_n^{(2)}} + ... \\ E_n &= E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + ... \end{aligned}$$ -Where $E_n^{(1)}$ and $\ket{\psi_n^{(1)}}$ are called the *first-order +Where $E_n^{(1)}$ and $\ket*{\psi_n^{(1)}}$ are called the *first-order corrections*, and so on for higher orders. We insert this into the Schrödinger equation: $$\begin{aligned} \hat{H} \ket{\psi_n} - &= \hat{H}_0 \ket{\psi_n^{(0)}} - + \lambda \big( \hat{H}_1 \ket{\psi_n^{(0)}} + \hat{H}_0 \ket{\psi_n^{(1)}} \big) \\ - &\qquad + \lambda^2 \big( \hat{H}_1 \ket{\psi_n^{(1)}} + \hat{H}_0 \ket{\psi_n^{(2)}} \big) + ... + &= \hat{H}_0 \ket*{\psi_n^{(0)}} + + \lambda \big( \hat{H}_1 \ket*{\psi_n^{(0)}} + \hat{H}_0 \ket*{\psi_n^{(1)}} \big) \\ + &\qquad + \lambda^2 \big( \hat{H}_1 \ket*{\psi_n^{(1)}} + \hat{H}_0 \ket*{\psi_n^{(2)}} \big) + ... \\ E_n \ket{\psi_n} - &= E_n^{(0)} \ket{\psi_n^{(0)}} - + \lambda \big( E_n^{(1)} \ket{\psi_n^{(0)}} + E_n^{(0)} \ket{\psi_n^{(1)}} \big) \\ - &\qquad + \lambda^2 \big( E_n^{(2)} \ket{\psi_n^{(0)}} + E_n^{(1)} \ket{\psi_n^{(1)}} + E_n^{(0)} \ket{\psi_n^{(2)}} \big) + ... + &= E_n^{(0)} \ket*{\psi_n^{(0)}} + + \lambda \big( E_n^{(1)} \ket*{\psi_n^{(0)}} + E_n^{(0)} \ket*{\psi_n^{(1)}} \big) \\ + &\qquad + \lambda^2 \big( E_n^{(2)} \ket*{\psi_n^{(0)}} + E_n^{(1)} \ket*{\psi_n^{(1)}} + E_n^{(0)} \ket*{\psi_n^{(2)}} \big) + ... \end{aligned}$$ If we collect the terms according to the order of $\lambda$, we arrive @@ -48,14 +48,14 @@ at the following endless series of equations, of which in practice only the first three are typically used: $$\begin{aligned} - \hat{H}_0 \ket{\psi_n^{(0)}} - &= E_n^{(0)} \ket{\psi_n^{(0)}} + \hat{H}_0 \ket*{\psi_n^{(0)}} + &= E_n^{(0)} \ket*{\psi_n^{(0)}} \\ - \hat{H}_1 \ket{\psi_n^{(0)}} + \hat{H}_0 \ket{\psi_n^{(1)}} - &= E_n^{(1)} \ket{\psi_n^{(0)}} + E_n^{(0)} \ket{\psi_n^{(1)}} + \hat{H}_1 \ket*{\psi_n^{(0)}} + \hat{H}_0 \ket*{\psi_n^{(1)}} + &= E_n^{(1)} \ket*{\psi_n^{(0)}} + E_n^{(0)} \ket*{\psi_n^{(1)}} \\ - \hat{H}_1 \ket{\psi_n^{(1)}} + \hat{H}_0 \ket{\psi_n^{(2)}} - &= E_n^{(2)} \ket{\psi_n^{(0)}} + E_n^{(1)} \ket{\psi_n^{(1)}} + E_n^{(0)} \ket{\psi_n^{(2)}} + \hat{H}_1 \ket*{\psi_n^{(1)}} + \hat{H}_0 \ket*{\psi_n^{(2)}} + &= E_n^{(2)} \ket*{\psi_n^{(0)}} + E_n^{(1)} \ket*{\psi_n^{(1)}} + E_n^{(0)} \ket*{\psi_n^{(2)}} \\ ... &= ... @@ -63,7 +63,7 @@ $$\begin{aligned} The first equation is the unperturbed problem, which we assume has already been solved, with eigenvalues $E_n^{(0)} = \varepsilon_n$ and -eigenvectors $\ket{\psi_n^{(0)}} = \ket{n}$: +eigenvectors $\ket*{\psi_n^{(0)}} = \ket{n}$: $$\begin{aligned} \hat{H}_0 \ket{n} = \varepsilon_n \ket{n} @@ -79,28 +79,27 @@ $\varepsilon_n$ corresponds to one $\ket{n}$. At order $\lambda^1$, we rewrite the equation as follows: $$\begin{aligned} - (\hat{H}_1 - E_n^{(1)}) \ket{n} + (\hat{H}_0 - \varepsilon_n) \ket{\psi_n^{(1)}} = 0 + (\hat{H}_1 - E_n^{(1)}) \ket{n} + (\hat{H}_0 - \varepsilon_n) \ket*{\psi_n^{(1)}} = 0 \end{aligned}$$ Since $\ket{n}$ form a complete basis, we can express -$\ket{\psi_n^{(1)}}$ in terms of them: +$\ket*{\psi_n^{(1)}}$ in terms of them: $$\begin{aligned} - \ket{\psi_n^{(1)}} = \sum_{m \neq n} c_m \ket{m} + \ket*{\psi_n^{(1)}} = \sum_{m \neq n} c_m \ket{m} \end{aligned}$$ Importantly, $n$ has been removed from the summation to prevent dividing -by zero later. This is allowed, because -$\ket{\psi_n^{(1)}} - c_n \ket{n}$ also satisfies the $\lambda^1$-order +by zero later. We are allowed to do this, because +$\ket*{\psi_n^{(1)}} - c_n \ket{n}$ also satisfies the order-$\lambda^1$ equation for any value of $c_n$, as demonstrated here: $$\begin{aligned} - (\hat{H}_1 - E_n^{(1)}) \ket{n} + (\hat{H}_0 - \varepsilon_n) \ket{\psi_n^{(1)}} - (\varepsilon_n - \varepsilon_n) c_n \ket{n} = 0 + (\hat{H}_1 - E_n^{(1)}) \ket{n} + (\hat{H}_0 - \varepsilon_n) \ket*{\psi_n^{(1)}} - (\varepsilon_n - \varepsilon_n) c_n \ket{n} = 0 \end{aligned}$$ -Where we used $\hat{H}_0 \ket{n} = \varepsilon_n \ket{n}$. Inserting the -series form of $\ket{\psi_n^{(1)}}$ into the order-$\lambda^1$ equation -gives us: +Where we used $\hat{H}_0 \ket{n} = \varepsilon_n \ket{n}$. +We insert the series form of $\ket*{\psi_n^{(1)}}$ into the $\lambda^1$-equation: $$\begin{aligned} (\hat{H}_1 - E_n^{(1)}) \ket{n} + \sum_{m \neq n} c_m (\varepsilon_m - \varepsilon_n) \ket{m} = 0 @@ -135,12 +134,12 @@ $$\begin{aligned} \end{aligned}$$ We isolate this result for $c_k$ and insert it into the series form of -$\ket{\psi_n^{(1)}}$ to get the full first-order correction to the wave +$\ket*{\psi_n^{(1)}}$ to get the full first-order correction to the wave function: $$\begin{aligned} \boxed{ - \ket{\psi_n^{(1)}} + \ket*{\psi_n^{(1)}} = \sum_{m \neq n} \frac{\matrixel{m}{\hat{H}_1}{n}}{\varepsilon_n - \varepsilon_m} \ket{m} } \end{aligned}$$ @@ -166,9 +165,9 @@ $$\begin{aligned} = \matrixel{n}{\hat{H}_1}{\psi_n^{(1)}} - E_n^{(1)} \braket{n}{\psi_n^{(1)}} \end{aligned}$$ -We explicitly removed the $\ket{n}$-dependence of $\ket{\psi_n^{(1)}}$, +We explicitly removed the $\ket{n}$-dependence of $\ket*{\psi_n^{(1)}}$, so the last term is zero. By simply inserting our result for -$\ket{\psi_n^{(1)}}$, we thus arrive at: +$\ket*{\psi_n^{(1)}}$, we thus arrive at: $$\begin{aligned} \boxed{ @@ -257,9 +256,8 @@ $$\begin{aligned} This is an eigenvalue problem for $E_n^{(1)}$, where $c_d$ are the components of the eigenvectors which represent the "good" states. -Suppose that this eigenvalue problem has been solved, and that -$\ket{n, g}$ are the resulting "good" states. Then, as long as -$E_n^{(1)}$ is a non-degenerate eigenvalue of $M$: +After solving this, let $\ket{n, g}$ be the resulting "good" states. +Then, as long as $E_n^{(1)}$ is a non-degenerate eigenvalue of $M$: $$\begin{aligned} \boxed{ @@ -272,36 +270,36 @@ first-order wave function correction is also unchanged: $$\begin{aligned} \boxed{ - \ket{\psi_{n,g}^{(1)}} + \ket*{\psi_{n,g}^{(1)}} = \sum_{m \neq (n, g)} \frac{\matrixel{m}{\hat{H}_1}{n, g}}{\varepsilon_n - \varepsilon_m} \ket{m} } \end{aligned}$$ This works because the matrix $M$ is diagonal in the $\ket{n, g}$-basis, such that when $\ket{m}$ is any vector $\ket{n, \gamma}$ in the -$\ket{n}$-eigenspace (except for $\ket{n,g}$ of course, which is -explicitly excluded), then conveniently the corresponding numerator +$\ket{n}$-eigenspace (except for $\ket{n,g}$, which is +explicitly excluded), then the corresponding numerator $\matrixel{n, \gamma}{\hat{H}_1}{n, g} = M_{\gamma, g} = 0$, so the term does not contribute. If any of the eigenvalues $E_n^{(1)}$ of $M$ are degenerate, then there -is still some information missing about the components $c_d$ of the -"good" states, in which case we must find these states some other way. +is still information missing about the components $c_d$ of the +"good" states, in which case we must find them some other way. -An alternative way of determining these "good" states is also of -interest if there is no degeneracy in $M$, since such a shortcut would -allow us use the formulae from non-degenerate perturbation theory +Such an alternative way of determining these "good" states is also of +interest even if there is no degeneracy in $M$, since such a shortcut would +allow us to use the formulae from non-degenerate perturbation theory straight away. -The method is to find a Hermitian operator $\hat{L}$ (usually using -symmetry) which commutes with both $\hat{H}_0$ and $\hat{H}_1$: +The trick is to find a Hermitian operator $\hat{L}$ (usually using +symmetries of the system) which commutes with both $\hat{H}_0$ and $\hat{H}_1$: $$\begin{aligned} -= [\hat{L}, \hat{H}_1] = 0 + [\hat{L}, \hat{H}_0] = [\hat{L}, \hat{H}_1] = 0 \end{aligned}$$ So that it shares its eigenstates with $\hat{H}_0$ (and $\hat{H}_1$), -meaning at least $D$ of the vectors of the $D$-dimensional +meaning all the vectors of the $D$-dimensional $\ket{n}$-eigenspace are also eigenvectors of $\hat{L}$. The crucial part, however, is that $\hat{L}$ must be chosen such that |