chm 421/621 statistical mechanicshelios.iiserb.ac.in/courses/chm421/lectures/lec31.pdfz exp ~2k2 2m...

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CHM 421/621 Statistical Mechanics Lecture 31

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  • CHM 421/621 Statistical Mechanics

    Lecture 31

  • EnsemblesLecture Plan

    Ideal gas revisited : Quantum particles in a 3-d box

    Classical equipartition theorem

    Quantum haromonic oscillator

  • EnsemblesIdeal gas revisited: Quantum particles in a 3-d box

    ✏(~k) =~2k22m

    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

    k↵ =⇡n↵L

    (↵ = x, y, z)

    n↵ = 1, 2, . . .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

    Allowed energies for

    1 particle

    Gas is at constant temperature T and constant volume V=L3

    ✏AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOepMhszPrzKwQQn7CiwdFvPo73vwbJ8keNLGgoajqprsrSgU31ve/vcLa+sbmVnG7tLO7t39QPjxqGpVphg2mhNLtiBoUXGLDciuwnWqkSSSwFY1uZ37rCbXhSj7YcYphQgeSx5xR66R2F1PDhZK9csWv+nOQVRLkpAI56r3yV7evWJagtExQYzqBn9pwQrXlTOC01M0MppSN6AA7jkqaoAkn83un5MwpfRIr7UpaMld/T0xoYsw4iVxnQu3QLHsz8T+vk9n4OpxwmWYWJVssijNBrCKz50mfa2RWjB2hTHN3K2FDqimzLqKSCyFYfnmVNC+qgV8N7i8rtZs8jiKcwCmcQwBXUIM7qEMDGAh4hld48x69F+/d+1i0Frx85hj+wPv8AUzRkCA=AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOepMhszPrzKwQQn7CiwdFvPo73vwbJ8keNLGgoajqprsrSgU31ve/vcLa+sbmVnG7tLO7t39QPjxqGpVphg2mhNLtiBoUXGLDciuwnWqkSSSwFY1uZ37rCbXhSj7YcYphQgeSx5xR66R2F1PDhZK9csWv+nOQVRLkpAI56r3yV7evWJagtExQYzqBn9pwQrXlTOC01M0MppSN6AA7jkqaoAkn83un5MwpfRIr7UpaMld/T0xoYsw4iVxnQu3QLHsz8T+vk9n4OpxwmWYWJVssijNBrCKz50mfa2RWjB2hTHN3K2FDqimzLqKSCyFYfnmVNC+qgV8N7i8rtZs8jiKcwCmcQwBXUIM7qEMDGAh4hld48x69F+/d+1i0Frx85hj+wPv8AUzRkCA=AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOepMhszPrzKwQQn7CiwdFvPo73vwbJ8keNLGgoajqprsrSgU31ve/vcLa+sbmVnG7tLO7t39QPjxqGpVphg2mhNLtiBoUXGLDciuwnWqkSSSwFY1uZ37rCbXhSj7YcYphQgeSx5xR66R2F1PDhZK9csWv+nOQVRLkpAI56r3yV7evWJagtExQYzqBn9pwQrXlTOC01M0MppSN6AA7jkqaoAkn83un5MwpfRIr7UpaMld/T0xoYsw4iVxnQu3QLHsz8T+vk9n4OpxwmWYWJVssijNBrCKz50mfa2RWjB2hTHN3K2FDqimzLqKSCyFYfnmVNC+qgV8N7i8rtZs8jiKcwCmcQwBXUIM7qEMDGAh4hld48x69F+/d+1i0Frx85hj+wPv8AUzRkCA=AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOepMhszPrzKwQQn7CiwdFvPo73vwbJ8keNLGgoajqprsrSgU31ve/vcLa+sbmVnG7tLO7t39QPjxqGpVphg2mhNLtiBoUXGLDciuwnWqkSSSwFY1uZ37rCbXhSj7YcYphQgeSx5xR66R2F1PDhZK9csWv+nOQVRLkpAI56r3yV7evWJagtExQYzqBn9pwQrXlTOC01M0MppSN6AA7jkqaoAkn83un5MwpfRIr7UpaMld/T0xoYsw4iVxnQu3QLHsz8T+vk9n4OpxwmWYWJVssijNBrCKz50mfa2RWjB2hTHN3K2FDqimzLqKSCyFYfnmVNC+qgV8N7i8rtZs8jiKcwCmcQwBXUIM7qEMDGAh4hld48x69F+/d+1i0Frx85hj+wPv8AUzRkCA=

    Number of states with

    energy less than

    See Lec. 10N(✏) =

    1

    8⇥ 4

    3⇡

    ✓2m✏

    ~2

    ◆ 32

    ⇥ L3

    ⇡3=

    4

    3p⇡

    p2⇡m✏

    h

    !3V

    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

    N(kBT ) =4

    3p⇡

    V

    ⇤3⇡ 0.75 V

    ⇤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

    Therefore

  • EnsemblesIdeal gas revisited: Quantum particles in a 3-d box

    To use the Boltzmann approximation for the particles we must have

    Z ⇡ zN

    N !AAACAXicbVDLSsNAFJ34rPUVdSO4GS2Cq5KIoMuiG1elgn1gE8tkOmmHTjLDzESsIW78FTcuFHHrX7jzb5y2WWjrgQuHc+7l3nsCwajSjvNtzc0vLC4tF1aKq2vrG5v21nZD8URiUseccdkKkCKMxqSuqWakJSRBUcBIMxhcjPzmHZGK8vhaDwXxI9SLaUgx0kbq2Ls30ENCSH4PvVAinD7cVrO0up917JJTdsaAs8TNSQnkqHXsL6/LcRKRWGOGlGq7jtB+iqSmmJGs6CWKCIQHqEfahsYoIspPxx9k8NAoXRhyaSrWcKz+nkhRpNQwCkxnhHRfTXsj8T+vnejwzE9pLBJNYjxZFCYMag5HccAulQRrNjQEYUnNrRD3kQlCm9CKJgR3+uVZ0jguu07ZvTopVc7zOApgDxyAI+CCU1ABl6AG6gCDR/AMXsGb9WS9WO/Wx6R1zspndsAfWJ8/GnaWqA==AAACAXicbVDLSsNAFJ34rPUVdSO4GS2Cq5KIoMuiG1elgn1gE8tkOmmHTjLDzESsIW78FTcuFHHrX7jzb5y2WWjrgQuHc+7l3nsCwajSjvNtzc0vLC4tF1aKq2vrG5v21nZD8URiUseccdkKkCKMxqSuqWakJSRBUcBIMxhcjPzmHZGK8vhaDwXxI9SLaUgx0kbq2Ls30ENCSH4PvVAinD7cVrO0up917JJTdsaAs8TNSQnkqHXsL6/LcRKRWGOGlGq7jtB+iqSmmJGs6CWKCIQHqEfahsYoIspPxx9k8NAoXRhyaSrWcKz+nkhRpNQwCkxnhHRfTXsj8T+vnejwzE9pLBJNYjxZFCYMag5HccAulQRrNjQEYUnNrRD3kQlCm9CKJgR3+uVZ0jguu07ZvTopVc7zOApgDxyAI+CCU1ABl6AG6gCDR/AMXsGb9WS9WO/Wx6R1zspndsAfWJ8/GnaWqA==AAACAXicbVDLSsNAFJ34rPUVdSO4GS2Cq5KIoMuiG1elgn1gE8tkOmmHTjLDzESsIW78FTcuFHHrX7jzb5y2WWjrgQuHc+7l3nsCwajSjvNtzc0vLC4tF1aKq2vrG5v21nZD8URiUseccdkKkCKMxqSuqWakJSRBUcBIMxhcjPzmHZGK8vhaDwXxI9SLaUgx0kbq2Ls30ENCSH4PvVAinD7cVrO0up917JJTdsaAs8TNSQnkqHXsL6/LcRKRWGOGlGq7jtB+iqSmmJGs6CWKCIQHqEfahsYoIspPxx9k8NAoXRhyaSrWcKz+nkhRpNQwCkxnhHRfTXsj8T+vnejwzE9pLBJNYjxZFCYMag5HccAulQRrNjQEYUnNrRD3kQlCm9CKJgR3+uVZ0jguu07ZvTopVc7zOApgDxyAI+CCU1ABl6AG6gCDR/AMXsGb9WS9WO/Wx6R1zspndsAfWJ8/GnaWqA==AAACAXicbVDLSsNAFJ34rPUVdSO4GS2Cq5KIoMuiG1elgn1gE8tkOmmHTjLDzESsIW78FTcuFHHrX7jzb5y2WWjrgQuHc+7l3nsCwajSjvNtzc0vLC4tF1aKq2vrG5v21nZD8URiUseccdkKkCKMxqSuqWakJSRBUcBIMxhcjPzmHZGK8vhaDwXxI9SLaUgx0kbq2Ls30ENCSH4PvVAinD7cVrO0up917JJTdsaAs8TNSQnkqHXsL6/LcRKRWGOGlGq7jtB+iqSmmJGs6CWKCIQHqEfahsYoIspPxx9k8NAoXRhyaSrWcKz+nkhRpNQwCkxnhHRfTXsj8T+vnejwzE9pLBJNYjxZFCYMag5HccAulQRrNjQEYUnNrRD3kQlCm9CKJgR3+uVZ0jguu07ZvTopVc7zOApgDxyAI+CCU1ABl6AG6gCDR/AMXsGb9WS9WO/Wx6R1zspndsAfWJ8/GnaWqA==

    Gas is at constant temperature T and constant volume V=L3

    N(kBT ) >> N

    V

    ⇤3>> N

    AAACFHicbVDLSsNAFJ3UV42vqEs3g0WpCCVRQVel1I0LKRX6giaWyXTSDp08mJkIJeQj3Pgrblwo4taFO//GaZuFth64cDjn3pl7jxsxKqRpfmu5peWV1bX8ur6xubW9Y+zutUQYc0yaOGQh77hIEEYD0pRUMtKJOEG+y0jbHV1P/PYD4YKGQUOOI+L4aBBQj2IkldQzTmvFUa8KGyfwuFyGNWjbuu1xhJNWmti36p0+uj9PZ2bPKJglcwq4SKyMFECGes/4svshjn0SSMyQEF3LjKSTIC4pZiTV7ViQCOERGpCuogHyiXCS6VEpPFJKH3ohVxVIOFV/TyTIF2Lsu6rTR3Io5r2J+J/XjaV35SQ0iGJJAjz7yIsZlCGcJAT7lBMs2VgRhDlVu0I8RCoTqXLUVQjW/MmLpHVWssySdXdRqFSzOPLgAByCIrDAJaiAG1AHTYDBI3gGr+BNe9JetHftY9aa07KZffAH2ucPKfubJg==AAACFHicbVDLSsNAFJ3UV42vqEs3g0WpCCVRQVel1I0LKRX6giaWyXTSDp08mJkIJeQj3Pgrblwo4taFO//GaZuFth64cDjn3pl7jxsxKqRpfmu5peWV1bX8ur6xubW9Y+zutUQYc0yaOGQh77hIEEYD0pRUMtKJOEG+y0jbHV1P/PYD4YKGQUOOI+L4aBBQj2IkldQzTmvFUa8KGyfwuFyGNWjbuu1xhJNWmti36p0+uj9PZ2bPKJglcwq4SKyMFECGes/4svshjn0SSMyQEF3LjKSTIC4pZiTV7ViQCOERGpCuogHyiXCS6VEpPFJKH3ohVxVIOFV/TyTIF2Lsu6rTR3Io5r2J+J/XjaV35SQ0iGJJAjz7yIsZlCGcJAT7lBMs2VgRhDlVu0I8RCoTqXLUVQjW/MmLpHVWssySdXdRqFSzOPLgAByCIrDAJaiAG1AHTYDBI3gGr+BNe9JetHftY9aa07KZffAH2ucPKfubJg==AAACFHicbVDLSsNAFJ3UV42vqEs3g0WpCCVRQVel1I0LKRX6giaWyXTSDp08mJkIJeQj3Pgrblwo4taFO//GaZuFth64cDjn3pl7jxsxKqRpfmu5peWV1bX8ur6xubW9Y+zutUQYc0yaOGQh77hIEEYD0pRUMtKJOEG+y0jbHV1P/PYD4YKGQUOOI+L4aBBQj2IkldQzTmvFUa8KGyfwuFyGNWjbuu1xhJNWmti36p0+uj9PZ2bPKJglcwq4SKyMFECGes/4svshjn0SSMyQEF3LjKSTIC4pZiTV7ViQCOERGpCuogHyiXCS6VEpPFJKH3ohVxVIOFV/TyTIF2Lsu6rTR3Io5r2J+J/XjaV35SQ0iGJJAjz7yIsZlCGcJAT7lBMs2VgRhDlVu0I8RCoTqXLUVQjW/MmLpHVWssySdXdRqFSzOPLgAByCIrDAJaiAG1AHTYDBI3gGr+BNe9JetHftY9aa07KZffAH2ucPKfubJg==AAACFHicbVDLSsNAFJ3UV42vqEs3g0WpCCVRQVel1I0LKRX6giaWyXTSDp08mJkIJeQj3Pgrblwo4taFO//GaZuFth64cDjn3pl7jxsxKqRpfmu5peWV1bX8ur6xubW9Y+zutUQYc0yaOGQh77hIEEYD0pRUMtKJOEG+y0jbHV1P/PYD4YKGQUOOI+L4aBBQj2IkldQzTmvFUa8KGyfwuFyGNWjbuu1xhJNWmti36p0+uj9PZ2bPKJglcwq4SKyMFECGes/4svshjn0SSMyQEF3LjKSTIC4pZiTV7ViQCOERGpCuogHyiXCS6VEpPFJKH3ohVxVIOFV/TyTIF2Lsu6rTR3Io5r2J+J/XjaV35SQ0iGJJAjz7yIsZlCGcJAT7lBMs2VgRhDlVu0I8RCoTqXLUVQjW/MmLpHVWssySdXdRqFSzOPLgAByCIrDAJaiAG1AHTYDBI3gGr+BNe9JetHftY9aa07KZffAH2ucPKfubJg==

    =) ⇤3

  • EnsemblesIdeal gas revisited: Quantum particles in a 3-d box

    Consider that such a condition is met we can write

    Z ⇡ zN

    N !AAACAXicbVDLSsNAFJ34rPUVdSO4GS2Cq5KIoMuiG1elgn1gE8tkOmmHTjLDzESsIW78FTcuFHHrX7jzb5y2WWjrgQuHc+7l3nsCwajSjvNtzc0vLC4tF1aKq2vrG5v21nZD8URiUseccdkKkCKMxqSuqWakJSRBUcBIMxhcjPzmHZGK8vhaDwXxI9SLaUgx0kbq2Ls30ENCSH4PvVAinD7cVrO0up917JJTdsaAs8TNSQnkqHXsL6/LcRKRWGOGlGq7jtB+iqSmmJGs6CWKCIQHqEfahsYoIspPxx9k8NAoXRhyaSrWcKz+nkhRpNQwCkxnhHRfTXsj8T+vnejwzE9pLBJNYjxZFCYMag5HccAulQRrNjQEYUnNrRD3kQlCm9CKJgR3+uVZ0jguu07ZvTopVc7zOApgDxyAI+CCU1ABl6AG6gCDR/AMXsGb9WS9WO/Wx6R1zspndsAfWJ8/GnaWqA==AAACAXicbVDLSsNAFJ34rPUVdSO4GS2Cq5KIoMuiG1elgn1gE8tkOmmHTjLDzESsIW78FTcuFHHrX7jzb5y2WWjrgQuHc+7l3nsCwajSjvNtzc0vLC4tF1aKq2vrG5v21nZD8URiUseccdkKkCKMxqSuqWakJSRBUcBIMxhcjPzmHZGK8vhaDwXxI9SLaUgx0kbq2Ls30ENCSH4PvVAinD7cVrO0up917JJTdsaAs8TNSQnkqHXsL6/LcRKRWGOGlGq7jtB+iqSmmJGs6CWKCIQHqEfahsYoIspPxx9k8NAoXRhyaSrWcKz+nkhRpNQwCkxnhHRfTXsj8T+vnejwzE9pLBJNYjxZFCYMag5HccAulQRrNjQEYUnNrRD3kQlCm9CKJgR3+uVZ0jguu07ZvTopVc7zOApgDxyAI+CCU1ABl6AG6gCDR/AMXsGb9WS9WO/Wx6R1zspndsAfWJ8/GnaWqA==AAACAXicbVDLSsNAFJ34rPUVdSO4GS2Cq5KIoMuiG1elgn1gE8tkOmmHTjLDzESsIW78FTcuFHHrX7jzb5y2WWjrgQuHc+7l3nsCwajSjvNtzc0vLC4tF1aKq2vrG5v21nZD8URiUseccdkKkCKMxqSuqWakJSRBUcBIMxhcjPzmHZGK8vhaDwXxI9SLaUgx0kbq2Ls30ENCSH4PvVAinD7cVrO0up917JJTdsaAs8TNSQnkqHXsL6/LcRKRWGOGlGq7jtB+iqSmmJGs6CWKCIQHqEfahsYoIspPxx9k8NAoXRhyaSrWcKz+nkhRpNQwCkxnhHRfTXsj8T+vnejwzE9pLBJNYjxZFCYMag5HccAulQRrNjQEYUnNrRD3kQlCm9CKJgR3+uVZ0jguu07ZvTopVc7zOApgDxyAI+CCU1ABl6AG6gCDR/AMXsGb9WS9WO/Wx6R1zspndsAfWJ8/GnaWqA==AAACAXicbVDLSsNAFJ34rPUVdSO4GS2Cq5KIoMuiG1elgn1gE8tkOmmHTjLDzESsIW78FTcuFHHrX7jzb5y2WWjrgQuHc+7l3nsCwajSjvNtzc0vLC4tF1aKq2vrG5v21nZD8URiUseccdkKkCKMxqSuqWakJSRBUcBIMxhcjPzmHZGK8vhaDwXxI9SLaUgx0kbq2Ls30ENCSH4PvVAinD7cVrO0up917JJTdsaAs8TNSQnkqHXsL6/LcRKRWGOGlGq7jtB+iqSmmJGs6CWKCIQHqEfahsYoIspPxx9k8NAoXRhyaSrWcKz+nkhRpNQwCkxnhHRfTXsj8T+vnejwzE9pLBJNYjxZFCYMag5HccAulQRrNjQEYUnNrRD3kQlCm9CKJgR3+uVZ0jguu07ZvTopVc7zOApgDxyAI+CCU1ABl6AG6gCDR/AMXsGb9WS9WO/Wx6R1zspndsAfWJ8/GnaWqA==

    Gas is at constant temperature T and constant volume V=L3

    z =V

    ⇡3

    1Z

    0

    dkx

    1Z

    0

    dky

    1Z

    0

    dkz exp

    ✓�� ~

    2k2

    2m

    =V

    ⇡3

    0

    @1Z

    0

    dkx exp

    ✓�� ~

    2k2x2m

    ◆1

    A3

    =V

    8⇡3

    ✓2⇡mkBT

    ~2

    ◆ 32

    =V

    ⇤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

    Z =1

    N !

    ✓V

    ⇤3

    ◆N

    AAACHHicbVC7SgNBFJ31GeNr1dJmNAixCbtG0EYI2lhIiGAemE3C7GQ2GTL7YOauEJb9EBt/xcZCERsLwb9x8ig08cCFwzn3ztx73EhwBZb1bSwsLi2vrGbWsusbm1vb5s5uTYWxpKxKQxHKhksUEzxgVeAgWCOSjPiuYHV3cDXy6w9MKh4GdzCMWMsnvYB7nBLQUscs3uML7HiS0MROk/JBih3BPMhPpFqaODf6sS5pF1NH8l4fjtvljpmzCtYYeJ7YU5JDU1Q65qfTDWnsswCoIEo1bSuCVkIkcCpYmnVixSJCB6THmpoGxGeqlYyPS/GRVrrYC6WuAPBY/T2REF+poe/qTp9AX816I/E/rxmDd95KeBDFwAI6+ciLBYYQj5LCXS4ZBTHUhFDJ9a6Y9omOBXSeWR2CPXvyPKmdFGyrYN+e5kqX0zgyaB8dojyy0RkqoWtUQVVE0SN6Rq/ozXgyXox342PSumBMZ/bQHxhfP5GdoRI=AAACHHicbVC7SgNBFJ31GeNr1dJmNAixCbtG0EYI2lhIiGAemE3C7GQ2GTL7YOauEJb9EBt/xcZCERsLwb9x8ig08cCFwzn3ztx73EhwBZb1bSwsLi2vrGbWsusbm1vb5s5uTYWxpKxKQxHKhksUEzxgVeAgWCOSjPiuYHV3cDXy6w9MKh4GdzCMWMsnvYB7nBLQUscs3uML7HiS0MROk/JBih3BPMhPpFqaODf6sS5pF1NH8l4fjtvljpmzCtYYeJ7YU5JDU1Q65qfTDWnsswCoIEo1bSuCVkIkcCpYmnVixSJCB6THmpoGxGeqlYyPS/GRVrrYC6WuAPBY/T2REF+poe/qTp9AX816I/E/rxmDd95KeBDFwAI6+ciLBYYQj5LCXS4ZBTHUhFDJ9a6Y9omOBXSeWR2CPXvyPKmdFGyrYN+e5kqX0zgyaB8dojyy0RkqoWtUQVVE0SN6Rq/ozXgyXox342PSumBMZ/bQHxhfP5GdoRI=AAACHHicbVC7SgNBFJ31GeNr1dJmNAixCbtG0EYI2lhIiGAemE3C7GQ2GTL7YOauEJb9EBt/xcZCERsLwb9x8ig08cCFwzn3ztx73EhwBZb1bSwsLi2vrGbWsusbm1vb5s5uTYWxpKxKQxHKhksUEzxgVeAgWCOSjPiuYHV3cDXy6w9MKh4GdzCMWMsnvYB7nBLQUscs3uML7HiS0MROk/JBih3BPMhPpFqaODf6sS5pF1NH8l4fjtvljpmzCtYYeJ7YU5JDU1Q65qfTDWnsswCoIEo1bSuCVkIkcCpYmnVixSJCB6THmpoGxGeqlYyPS/GRVrrYC6WuAPBY/T2REF+poe/qTp9AX816I/E/rxmDd95KeBDFwAI6+ciLBYYQj5LCXS4ZBTHUhFDJ9a6Y9omOBXSeWR2CPXvyPKmdFGyrYN+e5kqX0zgyaB8dojyy0RkqoWtUQVVE0SN6Rq/ozXgyXox342PSumBMZ/bQHxhfP5GdoRI=AAACHHicbVC7SgNBFJ31GeNr1dJmNAixCbtG0EYI2lhIiGAemE3C7GQ2GTL7YOauEJb9EBt/xcZCERsLwb9x8ig08cCFwzn3ztx73EhwBZb1bSwsLi2vrGbWsusbm1vb5s5uTYWxpKxKQxHKhksUEzxgVeAgWCOSjPiuYHV3cDXy6w9MKh4GdzCMWMsnvYB7nBLQUscs3uML7HiS0MROk/JBih3BPMhPpFqaODf6sS5pF1NH8l4fjtvljpmzCtYYeJ7YU5JDU1Q65qfTDWnsswCoIEo1bSuCVkIkcCpYmnVixSJCB6THmpoGxGeqlYyPS/GRVrrYC6WuAPBY/T2REF+poe/qTp9AX816I/E/rxmDd95KeBDFwAI6+ciLBYYQj5LCXS4ZBTHUhFDJ9a6Y9omOBXSeWR2CPXvyPKmdFGyrYN+e5kqX0zgyaB8dojyy0RkqoWtUQVVE0SN6Rq/ozXgyXox342PSumBMZ/bQHxhfP5GdoRI=

  • EnsemblesClassical equipartition theorem

    Consider a classical system with energy that be written as

    E(s1, s2, s3, . . . , sM , sM+1, . . . , ) =MX

    k=1

    cks2k + f(sM+1, sM+2, . . .)

    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

    i.e. with M variables appearing through a quadratic dependence.

    The classical partition function can then be written as

    Z(N,V, T ) =

    Zds1

    Zds2 · · ·

    ZdsM · · · exp(��

    MX

    k=1

    cks2k)⇥ exp(��f(sM+1, sM+2, . . .))

    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

    =MY

    k=1

    ✓Zdsk exp(��cks2k)

    ◆⇥ Zrest

    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

    Zrest =

    ZdsM+1

    ZdsM+2 · · · exp (��f(sM+1, sM+2, . . .))

    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

    where

  • EnsemblesClassical equipartition theorem

    Z(N,V, T ) =MY

    k=1

    ✓⇡

    �ck

    ◆ 12

    ⇥ ZrestAAACQHicbZDLSyNBEMZ7XB/Z7Kpx97iXxiBEkDAjwu5FCO5lL4oLJhHTcejp1CRNeh501wihmT9tL/4J3vbsZQ+KePW0ncfBV0HDx/dVUdW/KFfSoO//9ZY+LK+srlU+Vj99Xt/YrG196Zis0ALaIlOZPo+4ASVTaKNEBee5Bp5ECrrR+Oc0716BNjJLz3CSQz/hw1TGUnB0VljrXjRO9jp7Z7v0kLJcZ4PQjg+D8vKYMgUxNlisubAsl6VlESCnIhyXTMvhCHcv52FQ2v2SMpQJGHoRWg0Gy7BW95v+rOhbESxEnSzqNKzdsEEmigRSFIob0wv8HPuWa5RCQVllhYGcizEfQs/JlLttfTsDUNId5wxonGn3UqQz9/mE5YkxkyRynQnHkXmdTc33sl6B8Y++lWleIKRiviguFMWMTmnSgdQgUE2c4EJLdysVI+6ooGNedRCC119+Kzr7zcBvBr8P6q2jBY4K+Ua2SYME5DtpkV/klLSJIH/ILbkj996198978B7nrUveYuYreVHe038K+K+fAAACQHicbZDLSyNBEMZ7XB/Z7Kpx97iXxiBEkDAjwu5FCO5lL4oLJhHTcejp1CRNeh501wihmT9tL/4J3vbsZQ+KePW0ncfBV0HDx/dVUdW/KFfSoO//9ZY+LK+srlU+Vj99Xt/YrG196Zis0ALaIlOZPo+4ASVTaKNEBee5Bp5ECrrR+Oc0716BNjJLz3CSQz/hw1TGUnB0VljrXjRO9jp7Z7v0kLJcZ4PQjg+D8vKYMgUxNlisubAsl6VlESCnIhyXTMvhCHcv52FQ2v2SMpQJGHoRWg0Gy7BW95v+rOhbESxEnSzqNKzdsEEmigRSFIob0wv8HPuWa5RCQVllhYGcizEfQs/JlLttfTsDUNId5wxonGn3UqQz9/mE5YkxkyRynQnHkXmdTc33sl6B8Y++lWleIKRiviguFMWMTmnSgdQgUE2c4EJLdysVI+6ooGNedRCC119+Kzr7zcBvBr8P6q2jBY4K+Ua2SYME5DtpkV/klLSJIH/ILbkj996198978B7nrUveYuYreVHe038K+K+fAAACQHicbZDLSyNBEMZ7XB/Z7Kpx97iXxiBEkDAjwu5FCO5lL4oLJhHTcejp1CRNeh501wihmT9tL/4J3vbsZQ+KePW0ncfBV0HDx/dVUdW/KFfSoO//9ZY+LK+srlU+Vj99Xt/YrG196Zis0ALaIlOZPo+4ASVTaKNEBee5Bp5ECrrR+Oc0716BNjJLz3CSQz/hw1TGUnB0VljrXjRO9jp7Z7v0kLJcZ4PQjg+D8vKYMgUxNlisubAsl6VlESCnIhyXTMvhCHcv52FQ2v2SMpQJGHoRWg0Gy7BW95v+rOhbESxEnSzqNKzdsEEmigRSFIob0wv8HPuWa5RCQVllhYGcizEfQs/JlLttfTsDUNId5wxonGn3UqQz9/mE5YkxkyRynQnHkXmdTc33sl6B8Y++lWleIKRiviguFMWMTmnSgdQgUE2c4EJLdysVI+6ooGNedRCC119+Kzr7zcBvBr8P6q2jBY4K+Ua2SYME5DtpkV/klLSJIH/ILbkj996198978B7nrUveYuYreVHe038K+K+fAAACQHicbZDLSyNBEMZ7XB/Z7Kpx97iXxiBEkDAjwu5FCO5lL4oLJhHTcejp1CRNeh501wihmT9tL/4J3vbsZQ+KePW0ncfBV0HDx/dVUdW/KFfSoO//9ZY+LK+srlU+Vj99Xt/YrG196Zis0ALaIlOZPo+4ASVTaKNEBee5Bp5ECrrR+Oc0716BNjJLz3CSQz/hw1TGUnB0VljrXjRO9jp7Z7v0kLJcZ4PQjg+D8vKYMgUxNlisubAsl6VlESCnIhyXTMvhCHcv52FQ2v2SMpQJGHoRWg0Gy7BW95v+rOhbESxEnSzqNKzdsEEmigRSFIob0wv8HPuWa5RCQVllhYGcizEfQs/JlLttfTsDUNId5wxonGn3UqQz9/mE5YkxkyRynQnHkXmdTc33sl6B8Y++lWleIKRiviguFMWMTmnSgdQgUE2c4EJLdysVI+6ooGNedRCC119+Kzr7zcBvBr8P6q2jBY4K+Ua2SYME5DtpkV/klLSJIH/ILbkj996198978B7nrUveYuYreVHe038K+K+f

    E = �d lnZd�

    =M

    2kBT + Erest

    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

    i.e every quadratic term in the energy of a classical system contributes 1/2 kBT to the average energy at a given temperature T.

    This is the classical equipartition theorem. One expects that each such term would also contribute 1/2 kB to the specific heat at constant volume.

  • EnsemblesClassical equipartition theorem

    E.g. Monatomic ideal gas

    E =NX

    i=1

    1

    2m

    �p2i,x + p

    2i,y + p

    2i,z

    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

    E =3

    2NkBT

    AAACC3icbVDLSsNAFJ34rPUVdelmaBFclaQKuhFKRXAlFfqCJoTJdNIOnWTCzEQoIXs3/oobF4q49Qfc+TdO2iy09cDA4Zx7uHOPHzMqlWV9Gyura+sbm6Wt8vbO7t6+eXDYlTwRmHQwZ1z0fSQJoxHpKKoY6ceCoNBnpOdPrnO/90CEpDxqq2lM3BCNIhpQjJSWPLPicG3n6fQmg1fQCQTC6VmW1jN4BydeE7Y9s2rVrBngMrELUgUFWp755Qw5TkISKcyQlAPbipWbIqEoZiQrO4kkMcITNCIDTSMUEumms1syeKKVIQy40C9ScKb+TqQolHIa+noyRGosF71c/M8bJCq4dFMaxYkiEZ4vChIGFYd5MXBIBcGKTTVBWFD9V4jHSLehdH1lXYK9ePIy6dZrtlWz78+rjWZRRwkcgwo4BTa4AA1wC1qgAzB4BM/gFbwZT8aL8W58zEdXjCJzBP7A+PwBMbyZ1A==AAACC3icbVDLSsNAFJ34rPUVdelmaBFclaQKuhFKRXAlFfqCJoTJdNIOnWTCzEQoIXs3/oobF4q49Qfc+TdO2iy09cDA4Zx7uHOPHzMqlWV9Gyura+sbm6Wt8vbO7t6+eXDYlTwRmHQwZ1z0fSQJoxHpKKoY6ceCoNBnpOdPrnO/90CEpDxqq2lM3BCNIhpQjJSWPLPicG3n6fQmg1fQCQTC6VmW1jN4BydeE7Y9s2rVrBngMrELUgUFWp755Qw5TkISKcyQlAPbipWbIqEoZiQrO4kkMcITNCIDTSMUEumms1syeKKVIQy40C9ScKb+TqQolHIa+noyRGosF71c/M8bJCq4dFMaxYkiEZ4vChIGFYd5MXBIBcGKTTVBWFD9V4jHSLehdH1lXYK9ePIy6dZrtlWz78+rjWZRRwkcgwo4BTa4AA1wC1qgAzB4BM/gFbwZT8aL8W58zEdXjCJzBP7A+PwBMbyZ1A==AAACC3icbVDLSsNAFJ34rPUVdelmaBFclaQKuhFKRXAlFfqCJoTJdNIOnWTCzEQoIXs3/oobF4q49Qfc+TdO2iy09cDA4Zx7uHOPHzMqlWV9Gyura+sbm6Wt8vbO7t6+eXDYlTwRmHQwZ1z0fSQJoxHpKKoY6ceCoNBnpOdPrnO/90CEpDxqq2lM3BCNIhpQjJSWPLPicG3n6fQmg1fQCQTC6VmW1jN4BydeE7Y9s2rVrBngMrELUgUFWp755Qw5TkISKcyQlAPbipWbIqEoZiQrO4kkMcITNCIDTSMUEumms1syeKKVIQy40C9ScKb+TqQolHIa+noyRGosF71c/M8bJCq4dFMaxYkiEZ4vChIGFYd5MXBIBcGKTTVBWFD9V4jHSLehdH1lXYK9ePIy6dZrtlWz78+rjWZRRwkcgwo4BTa4AA1wC1qgAzB4BM/gFbwZT8aL8W58zEdXjCJzBP7A+PwBMbyZ1A==AAACC3icbVDLSsNAFJ34rPUVdelmaBFclaQKuhFKRXAlFfqCJoTJdNIOnWTCzEQoIXs3/oobF4q49Qfc+TdO2iy09cDA4Zx7uHOPHzMqlWV9Gyura+sbm6Wt8vbO7t6+eXDYlTwRmHQwZ1z0fSQJoxHpKKoY6ceCoNBnpOdPrnO/90CEpDxqq2lM3BCNIhpQjJSWPLPicG3n6fQmg1fQCQTC6VmW1jN4BydeE7Y9s2rVrBngMrELUgUFWp755Qw5TkISKcyQlAPbipWbIqEoZiQrO4kkMcITNCIDTSMUEumms1syeKKVIQy40C9ScKb+TqQolHIa+noyRGosF71c/M8bJCq4dFMaxYkiEZ4vChIGFYd5MXBIBcGKTTVBWFD9V4jHSLehdH1lXYK9ePIy6dZrtlWz78+rjWZRRwkcgwo4BTa4AA1wC1qgAzB4BM/gFbwZT8aL8W58zEdXjCJzBP7A+PwBMbyZ1A==

    CV =3

    2NkB

    AAACAXicbVDLSsNAFL2pr1pfUTeCm8EiuCpJFXQjlHbjSirYB7QhTKaTdujkwcxEKCFu/BU3LhRx61+482+ctllo64ELh3Pu5d57vJgzqSzr2yisrK6tbxQ3S1vbO7t75v5BW0aJILRFIh6Jrocl5SykLcUUp91YUBx4nHa8cWPqdx6okCwK79Ukpk6AhyHzGcFKS6551HDb6Br1fYFJep6l1QzdorFbR65ZtirWDGiZ2DkpQ46ma371BxFJAhoqwrGUPduKlZNioRjhNCv1E0ljTMZ4SHuahjig0klnH2ToVCsD5EdCV6jQTP09keJAykng6c4Aq5Fc9Kbif14vUf6Vk7IwThQNyXyRn3CkIjSNAw2YoETxiSaYCKZvRWSEdRhKh1bSIdiLLy+TdrViWxX77qJcq+dxFOEYTuAMbLiEGtxAE1pA4BGe4RXejCfjxXg3PuatBSOfOYQ/MD5/AKublRc=AAACAXicbVDLSsNAFL2pr1pfUTeCm8EiuCpJFXQjlHbjSirYB7QhTKaTdujkwcxEKCFu/BU3LhRx61+482+ctllo64ELh3Pu5d57vJgzqSzr2yisrK6tbxQ3S1vbO7t75v5BW0aJILRFIh6Jrocl5SykLcUUp91YUBx4nHa8cWPqdx6okCwK79Ukpk6AhyHzGcFKS6551HDb6Br1fYFJep6l1QzdorFbR65ZtirWDGiZ2DkpQ46ma371BxFJAhoqwrGUPduKlZNioRjhNCv1E0ljTMZ4SHuahjig0klnH2ToVCsD5EdCV6jQTP09keJAykng6c4Aq5Fc9Kbif14vUf6Vk7IwThQNyXyRn3CkIjSNAw2YoETxiSaYCKZvRWSEdRhKh1bSIdiLLy+TdrViWxX77qJcq+dxFOEYTuAMbLiEGtxAE1pA4BGe4RXejCfjxXg3PuatBSOfOYQ/MD5/AKublRc=AAACAXicbVDLSsNAFL2pr1pfUTeCm8EiuCpJFXQjlHbjSirYB7QhTKaTdujkwcxEKCFu/BU3LhRx61+482+ctllo64ELh3Pu5d57vJgzqSzr2yisrK6tbxQ3S1vbO7t75v5BW0aJILRFIh6Jrocl5SykLcUUp91YUBx4nHa8cWPqdx6okCwK79Ukpk6AhyHzGcFKS6551HDb6Br1fYFJep6l1QzdorFbR65ZtirWDGiZ2DkpQ46ma371BxFJAhoqwrGUPduKlZNioRjhNCv1E0ljTMZ4SHuahjig0klnH2ToVCsD5EdCV6jQTP09keJAykng6c4Aq5Fc9Kbif14vUf6Vk7IwThQNyXyRn3CkIjSNAw2YoETxiSaYCKZvRWSEdRhKh1bSIdiLLy+TdrViWxX77qJcq+dxFOEYTuAMbLiEGtxAE1pA4BGe4RXejCfjxXg3PuatBSOfOYQ/MD5/AKublRc=AAACAXicbVDLSsNAFL2pr1pfUTeCm8EiuCpJFXQjlHbjSirYB7QhTKaTdujkwcxEKCFu/BU3LhRx61+482+ctllo64ELh3Pu5d57vJgzqSzr2yisrK6tbxQ3S1vbO7t75v5BW0aJILRFIh6Jrocl5SykLcUUp91YUBx4nHa8cWPqdx6okCwK79Ukpk6AhyHzGcFKS6551HDb6Br1fYFJep6l1QzdorFbR65ZtirWDGiZ2DkpQ46ma371BxFJAhoqwrGUPduKlZNioRjhNCv1E0ljTMZ4SHuahjig0klnH2ToVCsD5EdCV6jQTP09keJAykng6c4Aq5Fc9Kbif14vUf6Vk7IwThQNyXyRn3CkIjSNAw2YoETxiSaYCKZvRWSEdRhKh1bSIdiLLy+TdrViWxX77qJcq+dxFOEYTuAMbLiEGtxAE1pA4BGe4RXejCfjxXg3PuatBSOfOYQ/MD5/AKublRc=

    Answer comes out to be same for the quantum version in the non-degenerate limit.

    (Why??)

  • EnsemblesClassical equipartition theorem

    E.g. Harmonic oscillator

    E =NX

    i=1

    ✓p2i2m

    +1

    2m!2x2i

    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

    E = NkBT CV = NkBAAACD3icbVDLSgMxFM3UV62vUZdugkVxVWZE0I1QWgRXUqEvaIchk6ZtaCYZkoxQhvoFbvwVNy4UcevWnX9jph1BWw8EDufcw809QcSo0o7zZeWWlldW1/LrhY3Nre0de3evqUQsMWlgwYRsB0gRRjlpaKoZaUeSoDBgpBWMqqnfuiNSUcHrehwRL0QDTvsUI20k3z7uCmOn6eRqAi/hDRz5FVi/T1H1mz+KbxedkjMFXCRuRoogQ823P7s9geOQcI0ZUqrjOpH2EiQ1xYxMCt1YkQjhERqQjqEchUR5yfSeCTwySg/2hTSPazhVfycSFCo1DgMzGSI9VPNeKv7ndWLdv/ASyqNYE45ni/oxg1rAtBzYo5JgzcaGICyp+SvEQyQR1qbCginBnT95kTRPS65Tcm/PiuVKVkceHIBDcAJccA7K4BrUQANg8ACewAt4tR6tZ+vNep+N5qwssw/+wPr4BmkvmvQ=AAACD3icbVDLSgMxFM3UV62vUZdugkVxVWZE0I1QWgRXUqEvaIchk6ZtaCYZkoxQhvoFbvwVNy4UcevWnX9jph1BWw8EDufcw809QcSo0o7zZeWWlldW1/LrhY3Nre0de3evqUQsMWlgwYRsB0gRRjlpaKoZaUeSoDBgpBWMqqnfuiNSUcHrehwRL0QDTvsUI20k3z7uCmOn6eRqAi/hDRz5FVi/T1H1mz+KbxedkjMFXCRuRoogQ823P7s9geOQcI0ZUqrjOpH2EiQ1xYxMCt1YkQjhERqQjqEchUR5yfSeCTwySg/2hTSPazhVfycSFCo1DgMzGSI9VPNeKv7ndWLdv/ASyqNYE45ni/oxg1rAtBzYo5JgzcaGICyp+SvEQyQR1qbCginBnT95kTRPS65Tcm/PiuVKVkceHIBDcAJccA7K4BrUQANg8ACewAt4tR6tZ+vNep+N5qwssw/+wPr4BmkvmvQ=AAACD3icbVDLSgMxFM3UV62vUZdugkVxVWZE0I1QWgRXUqEvaIchk6ZtaCYZkoxQhvoFbvwVNy4UcevWnX9jph1BWw8EDufcw809QcSo0o7zZeWWlldW1/LrhY3Nre0de3evqUQsMWlgwYRsB0gRRjlpaKoZaUeSoDBgpBWMqqnfuiNSUcHrehwRL0QDTvsUI20k3z7uCmOn6eRqAi/hDRz5FVi/T1H1mz+KbxedkjMFXCRuRoogQ823P7s9geOQcI0ZUqrjOpH2EiQ1xYxMCt1YkQjhERqQjqEchUR5yfSeCTwySg/2hTSPazhVfycSFCo1DgMzGSI9VPNeKv7ndWLdv/ASyqNYE45ni/oxg1rAtBzYo5JgzcaGICyp+SvEQyQR1qbCginBnT95kTRPS65Tcm/PiuVKVkceHIBDcAJccA7K4BrUQANg8ACewAt4tR6tZ+vNep+N5qwssw/+wPr4BmkvmvQ=AAACD3icbVDLSgMxFM3UV62vUZdugkVxVWZE0I1QWgRXUqEvaIchk6ZtaCYZkoxQhvoFbvwVNy4UcevWnX9jph1BWw8EDufcw809QcSo0o7zZeWWlldW1/LrhY3Nre0de3evqUQsMWlgwYRsB0gRRjlpaKoZaUeSoDBgpBWMqqnfuiNSUcHrehwRL0QDTvsUI20k3z7uCmOn6eRqAi/hDRz5FVi/T1H1mz+KbxedkjMFXCRuRoogQ823P7s9geOQcI0ZUqrjOpH2EiQ1xYxMCt1YkQjhERqQjqEchUR5yfSeCTwySg/2hTSPazhVfycSFCo1DgMzGSI9VPNeKv7ndWLdv/ASyqNYE45ni/oxg1rAtBzYo5JgzcaGICyp+SvEQyQR1qbCginBnT95kTRPS65Tcm/PiuVKVkceHIBDcAJccA7K4BrUQANg8ACewAt4tR6tZ+vNep+N5qwssw/+wPr4BmkvmvQ=

    Quantum version (non-degenerate)

    z = exp(�� ~!2)

    1X

    n=0

    exp(��n~!)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

    =exp(�� ~!2 )

    1� exp(��~!)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

  • EnsemblesQuantum Harmonic Oscillator

    For typical diatomic vibration frequencies at room temperature we have~!kBT

    > ⇥vibAAAB+3icbVDLSsNAFJ3UV62vWJduBkvBVUlE0FUpunFZoS9oQ5hMb9qhk0mYmRRL6K+4caGIW3/EnX/jtM1CWw9cOJxzL3PmBAlnSjvOt1XY2t7Z3Svulw4Oj45P7NNyR8WppNCmMY9lLyAKOBPQ1kxz6CUSSBRw6AaT+4XfnYJULBYtPUvAi8hIsJBRoo3k2+UWrtfxoDUGTfxsyoI59u2KU3OWwJvEzUkF5Wj69tdgGNM0AqEpJ0r1XSfRXkakZpTDvDRIFSSETsgI+oYKEoHysmX2Oa4aZYjDWJoRGi/V3xcZiZSaRSZXNSJ6rNa9hfif1091eOtlTCSpBkFXD4UpxzrGiyLwkEmgms8MIVQykxXTMZGEalNXyZTgrn95k3Suaq5Tcx+vK427vI4iOkcX6BK56AY10ANqojai6Ak9o1f0Zs2tF+vd+litFqz85gz9gfX5A4u6k3k=AAAB+3icbVDLSsNAFJ3UV62vWJduBkvBVUlE0FUpunFZoS9oQ5hMb9qhk0mYmRRL6K+4caGIW3/EnX/jtM1CWw9cOJxzL3PmBAlnSjvOt1XY2t7Z3Svulw4Oj45P7NNyR8WppNCmMY9lLyAKOBPQ1kxz6CUSSBRw6AaT+4XfnYJULBYtPUvAi8hIsJBRoo3k2+UWrtfxoDUGTfxsyoI59u2KU3OWwJvEzUkF5Wj69tdgGNM0AqEpJ0r1XSfRXkakZpTDvDRIFSSETsgI+oYKEoHysmX2Oa4aZYjDWJoRGi/V3xcZiZSaRSZXNSJ6rNa9hfif1091eOtlTCSpBkFXD4UpxzrGiyLwkEmgms8MIVQykxXTMZGEalNXyZTgrn95k3Suaq5Tcx+vK427vI4iOkcX6BK56AY10ANqojai6Ak9o1f0Zs2tF+vd+litFqz85gz9gfX5A4u6k3k=AAAB+3icbVDLSsNAFJ3UV62vWJduBkvBVUlE0FUpunFZoS9oQ5hMb9qhk0mYmRRL6K+4caGIW3/EnX/jtM1CWw9cOJxzL3PmBAlnSjvOt1XY2t7Z3Svulw4Oj45P7NNyR8WppNCmMY9lLyAKOBPQ1kxz6CUSSBRw6AaT+4XfnYJULBYtPUvAi8hIsJBRoo3k2+UWrtfxoDUGTfxsyoI59u2KU3OWwJvEzUkF5Wj69tdgGNM0AqEpJ0r1XSfRXkakZpTDvDRIFSSETsgI+oYKEoHysmX2Oa4aZYjDWJoRGi/V3xcZiZSaRSZXNSJ6rNa9hfif1091eOtlTCSpBkFXD4UpxzrGiyLwkEmgms8MIVQykxXTMZGEalNXyZTgrn95k3Suaq5Tcx+vK427vI4iOkcX6BK56AY10ANqojai6Ak9o1f0Zs2tF+vd+litFqz85gz9gfX5A4u6k3k=AAAB+3icbVDLSsNAFJ3UV62vWJduBkvBVUlE0FUpunFZoS9oQ5hMb9qhk0mYmRRL6K+4caGIW3/EnX/jtM1CWw9cOJxzL3PmBAlnSjvOt1XY2t7Z3Svulw4Oj45P7NNyR8WppNCmMY9lLyAKOBPQ1kxz6CUSSBRw6AaT+4XfnYJULBYtPUvAi8hIsJBRoo3k2+UWrtfxoDUGTfxsyoI59u2KU3OWwJvEzUkF5Wj69tdgGNM0AqEpJ0r1XSfRXkakZpTDvDRIFSSETsgI+oYKEoHysmX2Oa4aZYjDWJoRGi/V3xcZiZSaRSZXNSJ6rNa9hfif1091eOtlTCSpBkFXD4UpxzrGiyLwkEmgms8MIVQykxXTMZGEalNXyZTgrn95k3Suaq5Tcx+vK427vI4iOkcX6BK56AY10ANqojai6Ak9o1f0Zs2tF+vd+litFqz85gz9gfX5A4u6k3k= ✏� ✏0 ⇡ kB⇥vib e⇥vib/T

    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

    Similarly as T -> 0

    ✏ = E/N = �@ ln z@�

    =~!2

    +~!

    e~!

    kBT � 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

  • EnsemblesQuantum Harmonic Oscillator

    Fraction of excited molecules

    fn =e��(n+

    12 )~!

    z(T )AAACIHicbVBNS8NAEN3U7/pV9ehlsQgVsSQi6EUQvXhUsFpoaplsJ+3SzSbsboQa8lO8+Fe8eFBEb/pr3H4c1Ppg4PHeDDPzgkRwbVz30ylMTc/Mzs0vFBeXlldWS2vr1zpOFcMai0Ws6gFoFFxizXAjsJ4ohCgQeBP0zgb+zR0qzWN5ZfoJNiPoSB5yBsZKrdJh2JL0mPqhApbhbbbnB2igIndHipdn+/mO3w1A+XGEHcjz7L5ytZO3SmW36g5BJ4k3JmUyxkWr9OG3Y5ZGKA0ToHXDcxPTzEAZzgTmRT/VmADrQQcblkqIUDez4YM53bZKm4axsiUNHao/JzKItO5Hge2MwHT1X28g/uc1UhMeNTMuk9SgZKNFYSqoiekgLdrmCpkRfUuAKW5vpawLNhljMy3aELy/L0+S6/2q51a9y4Pyyek4jnmySbZIhXjkkJyQc3JBaoSRB/JEXsir8+g8O2/O+6i14IxnNsgvOF/fJNyi+w==AAACIHicbVBNS8NAEN3U7/pV9ehlsQgVsSQi6EUQvXhUsFpoaplsJ+3SzSbsboQa8lO8+Fe8eFBEb/pr3H4c1Ppg4PHeDDPzgkRwbVz30ylMTc/Mzs0vFBeXlldWS2vr1zpOFcMai0Ws6gFoFFxizXAjsJ4ohCgQeBP0zgb+zR0qzWN5ZfoJNiPoSB5yBsZKrdJh2JL0mPqhApbhbbbnB2igIndHipdn+/mO3w1A+XGEHcjz7L5ytZO3SmW36g5BJ4k3JmUyxkWr9OG3Y5ZGKA0ToHXDcxPTzEAZzgTmRT/VmADrQQcblkqIUDez4YM53bZKm4axsiUNHao/JzKItO5Hge2MwHT1X28g/uc1UhMeNTMuk9SgZKNFYSqoiekgLdrmCpkRfUuAKW5vpawLNhljMy3aELy/L0+S6/2q51a9y4Pyyek4jnmySbZIhXjkkJyQc3JBaoSRB/JEXsir8+g8O2/O+6i14IxnNsgvOF/fJNyi+w==AAACIHicbVBNS8NAEN3U7/pV9ehlsQgVsSQi6EUQvXhUsFpoaplsJ+3SzSbsboQa8lO8+Fe8eFBEb/pr3H4c1Ppg4PHeDDPzgkRwbVz30ylMTc/Mzs0vFBeXlldWS2vr1zpOFcMai0Ws6gFoFFxizXAjsJ4ohCgQeBP0zgb+zR0qzWN5ZfoJNiPoSB5yBsZKrdJh2JL0mPqhApbhbbbnB2igIndHipdn+/mO3w1A+XGEHcjz7L5ytZO3SmW36g5BJ4k3JmUyxkWr9OG3Y5ZGKA0ToHXDcxPTzEAZzgTmRT/VmADrQQcblkqIUDez4YM53bZKm4axsiUNHao/JzKItO5Hge2MwHT1X28g/uc1UhMeNTMuk9SgZKNFYSqoiekgLdrmCpkRfUuAKW5vpawLNhljMy3aELy/L0+S6/2q51a9y4Pyyek4jnmySbZIhXjkkJyQc3JBaoSRB/JEXsir8+g8O2/O+6i14IxnNsgvOF/fJNyi+w==AAACIHicbVBNS8NAEN3U7/pV9ehlsQgVsSQi6EUQvXhUsFpoaplsJ+3SzSbsboQa8lO8+Fe8eFBEb/pr3H4c1Ppg4PHeDDPzgkRwbVz30ylMTc/Mzs0vFBeXlldWS2vr1zpOFcMai0Ws6gFoFFxizXAjsJ4ohCgQeBP0zgb+zR0qzWN5ZfoJNiPoSB5yBsZKrdJh2JL0mPqhApbhbbbnB2igIndHipdn+/mO3w1A+XGEHcjz7L5ytZO3SmW36g5BJ4k3JmUyxkWr9OG3Y5ZGKA0ToHXDcxPTzEAZzgTmRT/VmADrQQcblkqIUDez4YM53bZKm4axsiUNHao/JzKItO5Hge2MwHT1X28g/uc1UhMeNTMuk9SgZKNFYSqoiekgLdrmCpkRfUuAKW5vpawLNhljMy3aELy/L0+S6/2q51a9y4Pyyek4jnmySbZIhXjkkJyQc3JBaoSRB/JEXsir8+g8O2/O+6i14IxnNsgvOF/fJNyi+w==

    fn>0 = 1� f0 = 1�e�

    �2 ~!

    z(T )

    = e�⇥vib

    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

    ⇥vib(K)AAAB+HicbVBNS8NAEJ3Ur1o/GvXoZbEI9VISEfRY9CJ4qdAvaEPYbDft0s0m7G4KNfSXePGgiFd/ijf/jds2B219MPB4b4aZeUHCmdKO820VNja3tneKu6W9/YPDsn103FZxKgltkZjHshtgRTkTtKWZ5rSbSIqjgNNOML6b+50JlYrFoqmnCfUiPBQsZARrI/l2ud8cUY39bMKCWfXhwrcrTs1ZAK0TNycVyNHw7a/+ICZpRIUmHCvVc51EexmWmhFOZ6V+qmiCyRgPac9QgSOqvGxx+AydG2WAwliaEhot1N8TGY6UmkaB6YywHqlVby7+5/VSHd54GRNJqqkgy0VhypGO0TwFNGCSEs2nhmAimbkVkRGWmGiTVcmE4K6+vE7alzXXqbmPV5X6bR5HEU7hDKrgwjXU4R4a0AICKTzDK7xZT9aL9W59LFsLVj5zAn9gff4ANi2Sxw==AAAB+HicbVBNS8NAEJ3Ur1o/GvXoZbEI9VISEfRY9CJ4qdAvaEPYbDft0s0m7G4KNfSXePGgiFd/ijf/jds2B219MPB4b4aZeUHCmdKO820VNja3tneKu6W9/YPDsn103FZxKgltkZjHshtgRTkTtKWZ5rSbSIqjgNNOML6b+50JlYrFoqmnCfUiPBQsZARrI/l2ud8cUY39bMKCWfXhwrcrTs1ZAK0TNycVyNHw7a/+ICZpRIUmHCvVc51EexmWmhFOZ6V+qmiCyRgPac9QgSOqvGxx+AydG2WAwliaEhot1N8TGY6UmkaB6YywHqlVby7+5/VSHd54GRNJqqkgy0VhypGO0TwFNGCSEs2nhmAimbkVkRGWmGiTVcmE4K6+vE7alzXXqbmPV5X6bR5HEU7hDKrgwjXU4R4a0AICKTzDK7xZT9aL9W59LFsLVj5zAn9gff4ANi2Sxw==AAAB+HicbVBNS8NAEJ3Ur1o/GvXoZbEI9VISEfRY9CJ4qdAvaEPYbDft0s0m7G4KNfSXePGgiFd/ijf/jds2B219MPB4b4aZeUHCmdKO820VNja3tneKu6W9/YPDsn103FZxKgltkZjHshtgRTkTtKWZ5rSbSIqjgNNOML6b+50JlYrFoqmnCfUiPBQsZARrI/l2ud8cUY39bMKCWfXhwrcrTs1ZAK0TNycVyNHw7a/+ICZpRIUmHCvVc51EexmWmhFOZ6V+qmiCyRgPac9QgSOqvGxx+AydG2WAwliaEhot1N8TGY6UmkaB6YywHqlVby7+5/VSHd54GRNJqqkgy0VhypGO0TwFNGCSEs2nhmAimbkVkRGWmGiTVcmE4K6+vE7alzXXqbmPV5X6bR5HEU7hDKrgwjXU4R4a0AICKTzDK7xZT9aL9W59LFsLVj5zAn9gff4ANi2Sxw==AAAB+HicbVBNS8NAEJ3Ur1o/GvXoZbEI9VISEfRY9CJ4qdAvaEPYbDft0s0m7G4KNfSXePGgiFd/ijf/jds2B219MPB4b4aZeUHCmdKO820VNja3tneKu6W9/YPDsn103FZxKgltkZjHshtgRTkTtKWZ5rSbSIqjgNNOML6b+50JlYrFoqmnCfUiPBQsZARrI/l2ud8cUY39bMKCWfXhwrcrTs1ZAK0TNycVyNHw7a/+ICZpRIUmHCvVc51EexmWmhFOZ6V+qmiCyRgPac9QgSOqvGxx+AydG2WAwliaEhot1N8TGY6UmkaB6YywHqlVby7+5/VSHd54GRNJqqkgy0VhypGO0TwFNGCSEs2nhmAimbkVkRGWmGiTVcmE4K6+vE7alzXXqbmPV5X6bR5HEU7hDKrgwjXU4R4a0AICKTzDK7xZT9aL9W59LFsLVj5zAn9gff4ANi2Sxw==

    H2 6215

    HCl 4227

    Cl2 810

    I2 310Weaker the bond more thermally excited the molecule.