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fix the mu in the solutions (#226)
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dou-du committed Feb 13, 2023
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6 changes: 3 additions & 3 deletions notebook/quantum-mechanics/asymmetricwell.ipynb
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" the two lowest eigenvalues as a function of the parameter $\\mu$.\n",
" One can see that these two eigenvalues are closest to each other\n",
" at $\\mu = 0$. For the entire range of plotted $\\mu$ values, the red line is always higher\n",
" than the blue line, i.e. the eigenvalues change continuously. However, the nature of the state changes as we change the value of mu! For values $mu << 0$, the lowest eigenstate is localized in the right well, while for $mu >>0$ is is localized in the left well. For values of $mu$ close to zero, the two states mix and we get a symmetric and an antisymmetric solution. The $\\mu x$ term can be considered as a perturbation to the original double well potential, which results in the avoided crossing. Please check the detailed\n",
" than the blue line, i.e. the eigenvalues change continuously. However, the nature of the state changes as we change the value of mu! For values $\\mu << 0$, the lowest eigenstate is localized in the right well, while for $\\mu >>0$ is is localized in the left well. For values of $\\mu$ close to zero, the two states mix and we get a symmetric and an antisymmetric solution. The $\\mu x$ term can be considered as a perturbation to the original double well potential, which results in the avoided crossing. Please check the detailed\n",
" information in the background theory section.\n",
" </div> \n",
" </details>\n",
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" <details>\n",
" <summary style=\"color: red\">Solution</summary>\n",
" <div style=\"border:blue; border-width:3px; border-style:outset;\">\n",
" By tuning the $\\mu$ slider we observe that the values of the eigenvalues change continuously with the size of the perturbation. Also in this case, when the value of $mu$ would tend to make two eigenvalues identical, an anticrossing effect takes place (e.g. between the second and third state at $mu \\sim 0.085$), and we see the appearance of a symmetric and an antisymmetric eigenstate. \n",
" By tuning the $\\mu$ slider we observe that the values of the eigenvalues change continuously with the size of the perturbation. Also in this case, when the value of $\\mu$ would tend to make two eigenvalues identical, an anticrossing effect takes place (e.g. between the second and third state at $\\mu \\sim 0.085$), and we see the appearance of a symmetric and an antisymmetric eigenstate. \n",
" </div>\n",
" </details>\n",
" \n",
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"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.0"
"version": "3.9.12"
},
"voila": {
"authors": "Dou Du and Giovanni Pizzi"
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