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'eigenvalz'
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DPotoyan authored Nov 22, 2024
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@@ -118,13 +118,74 @@
"\n",
"- In more compact matrix notation, this becomes:\n",
"\n",
":::{admonition}\n",
":class: important\n",
"\n",
" $$\\mathbf{H}\\mathbf{c} = E\\mathbf{S}\\mathbf{c}$$\n",
"\n",
"- $\\mathbf{H}$ N by N matrix of hamiltonian elements $\\langle f_i |\\hat{H}|f_j\\rangle$\n",
"- $S$ an N by N matrix of overlap integrals $\\langle f_i|f_j\\rangle$\n",
"- $\\mathbf{c} = (c_1, c_2, ...)$ vector of N length.\n",
"- $E$ eigenvalues that can saitsfy this equation. For symmetric matrices one expects to get N possible values!\n",
"\n",
":::\n",
"\n",
"- By left-multiplying both sides by $\\mathbf{S}^{-1}$, we transform this into a standard eigenvalue problem:\n",
"\n",
" $$\\mathbf{S}^{-1}\\mathbf{H}\\mathbf{c} = E\\mathbf{I}\\mathbf{c}$$\n",
"\n",
"- Therefore, the minimum energies correspond to the eigenvalues of $\\mathbf{S}^{-1}\\mathbf{H}$, and the variational parameters that minimize the energies are the eigenvectors of $\\mathbf{S}^{-1}\\mathbf{H}$. \n"
"- Therefore, the minimum energies correspond to the eigenvalues of $\\mathbf{S}^{-1}\\mathbf{H}$, and the variational parameters that minimize the energies are the eigenvectors of $\\mathbf{S}^{-1}\\mathbf{H}$. \n",
"\n",
":::{admonition} **Breaking problem down to matrix eigenvalue eigenvector problem**\n",
":class: tip, collapse\n",
"\n",
"In the equation \n",
"\n",
"$$\\mathbf{S}^{-1}\\mathbf{H}\\mathbf{c} = E\\mathbf{I}\\mathbf{c},$$ \n",
"\n",
"**$\\mathbf{I}$** represents the identity matrix. Its role in this context is essential to express the equation as a **standard eigenvalue problem**.\n",
"\n",
"\n",
"\n",
"1. **Eigenvalue Problem Form**: \n",
"\n",
" In linear algebra, a standard eigenvalue problem is written as: \n",
" $$\\mathbf{A}\\mathbf{v} = \\lambda \\mathbf{I}\\mathbf{v},$$ \n",
" where: \n",
" - $\\mathbf{A}$ is a square matrix, \n",
" - $\\lambda$ is a scalar eigenvalue, \n",
" - $\\mathbf{I}$ is the identity matrix, and \n",
" - $\\mathbf{v}$ is the corresponding eigenvector.\n",
"\n",
" The identity matrix $\\mathbf{I}$ ensures that $\\lambda$ scales the eigenvector $\\mathbf{v}$ without altering its direction. The eigenvalue problem is about finding the values of $\\lambda$ and their associated $\\mathbf{v}$.\n",
"\n",
"2. **Connecting to $\\mathbf{S}^{-1}\\mathbf{H}\\mathbf{c} = E\\mathbf{I}\\mathbf{c}$**: \n",
" Here, $\\mathbf{S}^{-1}\\mathbf{H}$ acts as the operator $\\mathbf{A}$ in the standard eigenvalue problem. \n",
" - $\\mathbf{S}^{-1}\\mathbf{H}$ is a matrix resulting from left-multiplying $\\mathbf{H}$ by the inverse of $\\mathbf{S}$. \n",
" - $\\mathbf{c}$ represents the eigenvector. \n",
" - $E$ is the eigenvalue (corresponding to the energy in the quantum mechanical system). \n",
"\n",
" The identity matrix $\\mathbf{I}$ is explicitly included to highlight that $E$ is a scalar multiplying the vector $\\mathbf{c}$. This ensures that the left-hand side (a matrix operation) matches the right-hand side (a scaled vector). \n",
"\n",
"3. **Why $\\mathbf{S}^{-1}$ Appears**: \n",
" Initially, we had: \n",
" $$\\mathbf{H}\\mathbf{c} = E\\mathbf{S}\\mathbf{c},$$ \n",
" which cannot directly be interpreted as an eigenvalue problem because of the presence of $\\mathbf{S}$ (the overlap matrix). To transform this into a standard form, we pre-multiply both sides by $\\mathbf{S}^{-1}$: \n",
" $$\\mathbf{S}^{-1}\\mathbf{H}\\mathbf{c} = E\\mathbf{S}^{-1}\\mathbf{S}\\mathbf{c}.$$ \n",
" Since $\\mathbf{S}^{-1}\\mathbf{S} = \\mathbf{I}$, this simplifies to: \n",
" $$\\mathbf{S}^{-1}\\mathbf{H}\\mathbf{c} = E\\mathbf{I}\\mathbf{c}.$$\n",
"\n",
"4. **How to Interpret This as an Eigenvalue Problem**: \n",
" The equation now has the form of a standard eigenvalue problem: \n",
" $$\\mathbf{A}\\mathbf{v} = \\lambda \\mathbf{I}\\mathbf{v},$$ \n",
" where: \n",
" - $\\mathbf{A} = \\mathbf{S}^{-1}\\mathbf{H}$ is the effective matrix to diagonalize, \n",
" - $\\lambda = E$ are the eigenvalues, corresponding to the energy levels, \n",
" - $\\mathbf{v} = \\mathbf{c}$ are the eigenvectors, containing the coefficients of the trial wavefunctions.\n",
"\n",
"5. **Physical Interpretation**: \n",
" Solving the eigenvalue problem $\\mathbf{S}^{-1}\\mathbf{H}\\mathbf{c} = E\\mathbf{I}\\mathbf{c}$ gives the approximate energies ($E$) of the quantum system as eigenvalues and the corresponding variational parameters ($\\mathbf{c}$) as eigenvectors. The identity matrix $\\mathbf{I}$ is crucial for preserving the standard form of the eigenvalue problem, ensuring proper mathematical and physical interpretation.\n",
":::"
]
},
{
@@ -142,7 +203,7 @@
"\n",
"$$\\hat{H} = -\\frac{\\hbar^2}{2m}\\frac{d^2}{dx^2}$$\n",
"\n",
"While we can (/have) solved this problem analytically, it will be instructive to see how the variational solution works. We start by approximating $\\psi(x)$ as an expansion in two basis functions\n",
"While we can have solved this problem analytically, it will be instructive to see how the variational solution works. We start by approximating $\\psi(x)$ as an expansion in two basis functions\n",
"\n",
"$$\\psi(x) \\approx c_1x(a-x) + c_2x^2(a-x)^2$$\n",
"\n",

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