CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec...Quantum...

10
CHM 421/621 Statistical Mechanics Lecture 10

Transcript of CHM 421/621 Statistical Mechanicshelios.iiserb.ac.in/~vardha/Courses/CHM421/Lectures/Lec...Quantum...

  • CHM 421/621 Statistical Mechanics

    Lecture 10

  • Introduction and ReviewLecture Plan

    Quantum Systems • Particle in a box

    • Harmonic oscillator

    • Rigid rotor

    • Many-particle quantum systems

  • Quantum SystemsConsider a quantum particle trapped in a cubic box of volume V=L3

    Eigenvalues are given by three quantum numbers

    Particle in a box

    where allowed values for each n=1,2,3,…

    Ener

    gy

    Spectrum in 1-d case3.31⇥ 10�37 J

    AAAB/3icbVDLSgNBEJz1GeNrVfDiZTAIXgw7RojHoBfxFME8IFnD7GQ2GTL7YKZXCGsEf8WLB0W8+hve/BsnyR40saChqOqmu8uLpdDgON/WwuLS8spqbi2/vrG5tW3v7NZ1lCjGayySkWp6VHMpQl4DAZI3Y8Vp4Ene8AaXY79xz5UWUXgLw5i7Ae2FwheMgpE69n6pWCK4DSLgGhPnLj0plUeP1x274BSdCfA8IRkpoAzVjv3V7kYsCXgITFKtW8SJwU2pAsEkH+XbieYxZQPa4y1DQ2r2uenk/hE+MkoX+5EyFQKeqL8nUhpoPQw80xlQ6OtZbyz+57US8M/dVIRxAjxk00V+IjFEeBwG7grFGcihIZQpYW7FrE8VZWAiy5sQyOzL86R+WiROkdycFSoXWRw5dIAO0TEiqIwq6ApVUQ0x9ICe0St6s56sF+vd+pi2LljZzB76A+vzB/I/lCA=AAAB/3icbVDLSgNBEJz1GeNrVfDiZTAIXgw7RojHoBfxFME8IFnD7GQ2GTL7YKZXCGsEf8WLB0W8+hve/BsnyR40saChqOqmu8uLpdDgON/WwuLS8spqbi2/vrG5tW3v7NZ1lCjGayySkWp6VHMpQl4DAZI3Y8Vp4Ene8AaXY79xz5UWUXgLw5i7Ae2FwheMgpE69n6pWCK4DSLgGhPnLj0plUeP1x274BSdCfA8IRkpoAzVjv3V7kYsCXgITFKtW8SJwU2pAsEkH+XbieYxZQPa4y1DQ2r2uenk/hE+MkoX+5EyFQKeqL8nUhpoPQw80xlQ6OtZbyz+57US8M/dVIRxAjxk00V+IjFEeBwG7grFGcihIZQpYW7FrE8VZWAiy5sQyOzL86R+WiROkdycFSoXWRw5dIAO0TEiqIwq6ApVUQ0x9ICe0St6s56sF+vd+pi2LljZzB76A+vzB/I/lCA=AAAB/3icbVDLSgNBEJz1GeNrVfDiZTAIXgw7RojHoBfxFME8IFnD7GQ2GTL7YKZXCGsEf8WLB0W8+hve/BsnyR40saChqOqmu8uLpdDgON/WwuLS8spqbi2/vrG5tW3v7NZ1lCjGayySkWp6VHMpQl4DAZI3Y8Vp4Ene8AaXY79xz5UWUXgLw5i7Ae2FwheMgpE69n6pWCK4DSLgGhPnLj0plUeP1x274BSdCfA8IRkpoAzVjv3V7kYsCXgITFKtW8SJwU2pAsEkH+XbieYxZQPa4y1DQ2r2uenk/hE+MkoX+5EyFQKeqL8nUhpoPQw80xlQ6OtZbyz+57US8M/dVIRxAjxk00V+IjFEeBwG7grFGcihIZQpYW7FrE8VZWAiy5sQyOzL86R+WiROkdycFSoXWRw5dIAO0TEiqIwq6ApVUQ0x9ICe0St6s56sF+vd+pi2LljZzB76A+vzB/I/lCA=AAAB/3icbVDLSgNBEJz1GeNrVfDiZTAIXgw7RojHoBfxFME8IFnD7GQ2GTL7YKZXCGsEf8WLB0W8+hve/BsnyR40saChqOqmu8uLpdDgON/WwuLS8spqbi2/vrG5tW3v7NZ1lCjGayySkWp6VHMpQl4DAZI3Y8Vp4Ene8AaXY79xz5UWUXgLw5i7Ae2FwheMgpE69n6pWCK4DSLgGhPnLj0plUeP1x274BSdCfA8IRkpoAzVjv3V7kYsCXgITFKtW8SJwU2pAsEkH+XbieYxZQPa4y1DQ2r2uenk/hE+MkoX+5EyFQKeqL8nUhpoPQw80xlQ6OtZbyz+57US8M/dVIRxAjxk00V+IjFEeBwG7grFGcihIZQpYW7FrE8VZWAiy5sQyOzL86R+WiROkdycFSoXWRw5dIAO0TEiqIwq6ApVUQ0x9ICe0St6s56sF+vd+pi2LljZzB76A+vzB/I/lCA=

    ✏nx,ny,nz =~2⇡22mL2

    �n2x + n

    2y + n

    2z

    �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

    E.g., for H atom in V=1 cm3.

    ~2⇡22mL2

    =AAACCXicbZC7SgNBFIbPxluMt1VLm8EgWIXdIGgjBG0sLCKYC2Q3YXYymwyZnV1mZoWwpLXxVWwsFLH1Dex8GyfJFpr4w8DHf87hzPmDhDOlHefbKqysrq1vFDdLW9s7u3v2/kFTxakktEFiHst2gBXlTNCGZprTdiIpjgJOW8HoelpvPVCpWCzu9TihfoQHgoWMYG2sno28UGKSecMAy24VeQnrVidZFUXo1gC6RD277FScmdAyuDmUIVe9Z395/ZikERWacKxUx3US7WdYakY4nZS8VNEEkxEe0I5BgSOq/Gx2yQSdGKePwliaJzSaub8nMhwpNY4C0xlhPVSLtan5X62T6vDCz5hIUk0FmS8KU450jKaxoD6TlGg+NoCJZOaviAyxiUab8EomBHfx5GVoViuuU3Hvzsq1qzyOIhzBMZyCC+dQgxuoQwMIPMIzvMKb9WS9WO/Wx7y1YOUzh/BH1ucP2a+YeQ==AAACCXicbZC7SgNBFIbPxluMt1VLm8EgWIXdIGgjBG0sLCKYC2Q3YXYymwyZnV1mZoWwpLXxVWwsFLH1Dex8GyfJFpr4w8DHf87hzPmDhDOlHefbKqysrq1vFDdLW9s7u3v2/kFTxakktEFiHst2gBXlTNCGZprTdiIpjgJOW8HoelpvPVCpWCzu9TihfoQHgoWMYG2sno28UGKSecMAy24VeQnrVidZFUXo1gC6RD277FScmdAyuDmUIVe9Z395/ZikERWacKxUx3US7WdYakY4nZS8VNEEkxEe0I5BgSOq/Gx2yQSdGKePwliaJzSaub8nMhwpNY4C0xlhPVSLtan5X62T6vDCz5hIUk0FmS8KU450jKaxoD6TlGg+NoCJZOaviAyxiUab8EomBHfx5GVoViuuU3Hvzsq1qzyOIhzBMZyCC+dQgxuoQwMIPMIzvMKb9WS9WO/Wx7y1YOUzh/BH1ucP2a+YeQ==AAACCXicbZC7SgNBFIbPxluMt1VLm8EgWIXdIGgjBG0sLCKYC2Q3YXYymwyZnV1mZoWwpLXxVWwsFLH1Dex8GyfJFpr4w8DHf87hzPmDhDOlHefbKqysrq1vFDdLW9s7u3v2/kFTxakktEFiHst2gBXlTNCGZprTdiIpjgJOW8HoelpvPVCpWCzu9TihfoQHgoWMYG2sno28UGKSecMAy24VeQnrVidZFUXo1gC6RD277FScmdAyuDmUIVe9Z395/ZikERWacKxUx3US7WdYakY4nZS8VNEEkxEe0I5BgSOq/Gx2yQSdGKePwliaJzSaub8nMhwpNY4C0xlhPVSLtan5X62T6vDCz5hIUk0FmS8KU450jKaxoD6TlGg+NoCJZOaviAyxiUab8EomBHfx5GVoViuuU3Hvzsq1qzyOIhzBMZyCC+dQgxuoQwMIPMIzvMKb9WS9WO/Wx7y1YOUzh/BH1ucP2a+YeQ==AAACCXicbZC7SgNBFIbPxluMt1VLm8EgWIXdIGgjBG0sLCKYC2Q3YXYymwyZnV1mZoWwpLXxVWwsFLH1Dex8GyfJFpr4w8DHf87hzPmDhDOlHefbKqysrq1vFDdLW9s7u3v2/kFTxakktEFiHst2gBXlTNCGZprTdiIpjgJOW8HoelpvPVCpWCzu9TihfoQHgoWMYG2sno28UGKSecMAy24VeQnrVidZFUXo1gC6RD277FScmdAyuDmUIVe9Z395/ZikERWacKxUx3US7WdYakY4nZS8VNEEkxEe0I5BgSOq/Gx2yQSdGKePwliaJzSaub8nMhwpNY4C0xlhPVSLtan5X62T6vDCz5hIUk0FmS8KU450jKaxoD6TlGg+NoCJZOaviAyxiUab8EomBHfx5GVoViuuU3Hvzsq1qzyOIhzBMZyCC+dQgxuoQwMIPMIzvMKb9WS9WO/Wx7y1YOUzh/BH1ucP2a+YeQ==

  • Quantum SystemsTypical energy in systems ~ kB T = 4.14 x 10-21 J (@ T = 300 K)

    => n ~ 1 x 108

    Particle in a box

    Minimum energy gap

    = 10�8 ⇥ ✏nx,ny,nzAAACEHicbVDLSsNAFJ34rPUVdelmsIgutCQi2I1QdOOygn1AE8NkOmmHTiZhZiLGkE9w46+4caGIW5fu/BunbRbaeuDC4Zx7ufceP2ZUKsv6NubmFxaXlksr5dW19Y1Nc2u7JaNEYNLEEYtEx0eSMMpJU1HFSCcWBIU+I21/eDny23dESBrxG5XGxA1Rn9OAYqS05JkH59C2brPjWg4dRUMioUNiSZn2Mu7dH3Ev1fWQe2bFqlpjwFliF6QCCjQ888vpRTgJCVeYISm7thUrN0NCUcxIXnYSSWKEh6hPuppypHe72fihHO5rpQeDSOjiCo7V3xMZCqVMQ193hkgN5LQ3Ev/zuokKam5GeZwowvFkUZAwqCI4Sgf2qCBYsVQThAXVt0I8QAJhpTMs6xDs6ZdnSeukaltV+/q0Ur8o4iiBXbAHDoENzkAdXIEGaAIMHsEzeAVvxpPxYrwbH5PWOaOY2QF/YHz+AG71nD8=AAACEHicbVDLSsNAFJ34rPUVdelmsIgutCQi2I1QdOOygn1AE8NkOmmHTiZhZiLGkE9w46+4caGIW5fu/BunbRbaeuDC4Zx7ufceP2ZUKsv6NubmFxaXlksr5dW19Y1Nc2u7JaNEYNLEEYtEx0eSMMpJU1HFSCcWBIU+I21/eDny23dESBrxG5XGxA1Rn9OAYqS05JkH59C2brPjWg4dRUMioUNiSZn2Mu7dH3Ev1fWQe2bFqlpjwFliF6QCCjQ888vpRTgJCVeYISm7thUrN0NCUcxIXnYSSWKEh6hPuppypHe72fihHO5rpQeDSOjiCo7V3xMZCqVMQ193hkgN5LQ3Ev/zuokKam5GeZwowvFkUZAwqCI4Sgf2qCBYsVQThAXVt0I8QAJhpTMs6xDs6ZdnSeukaltV+/q0Ur8o4iiBXbAHDoENzkAdXIEGaAIMHsEzeAVvxpPxYrwbH5PWOaOY2QF/YHz+AG71nD8=AAACEHicbVDLSsNAFJ34rPUVdelmsIgutCQi2I1QdOOygn1AE8NkOmmHTiZhZiLGkE9w46+4caGIW5fu/BunbRbaeuDC4Zx7ufceP2ZUKsv6NubmFxaXlksr5dW19Y1Nc2u7JaNEYNLEEYtEx0eSMMpJU1HFSCcWBIU+I21/eDny23dESBrxG5XGxA1Rn9OAYqS05JkH59C2brPjWg4dRUMioUNiSZn2Mu7dH3Ev1fWQe2bFqlpjwFliF6QCCjQ888vpRTgJCVeYISm7thUrN0NCUcxIXnYSSWKEh6hPuppypHe72fihHO5rpQeDSOjiCo7V3xMZCqVMQ193hkgN5LQ3Ev/zuokKam5GeZwowvFkUZAwqCI4Sgf2qCBYsVQThAXVt0I8QAJhpTMs6xDs6ZdnSeukaltV+/q0Ur8o4iiBXbAHDoENzkAdXIEGaAIMHsEzeAVvxpPxYrwbH5PWOaOY2QF/YHz+AG71nD8=AAACEHicbVDLSsNAFJ34rPUVdelmsIgutCQi2I1QdOOygn1AE8NkOmmHTiZhZiLGkE9w46+4caGIW5fu/BunbRbaeuDC4Zx7ufceP2ZUKsv6NubmFxaXlksr5dW19Y1Nc2u7JaNEYNLEEYtEx0eSMMpJU1HFSCcWBIU+I21/eDny23dESBrxG5XGxA1Rn9OAYqS05JkH59C2brPjWg4dRUMioUNiSZn2Mu7dH3Ev1fWQe2bFqlpjwFliF6QCCjQ888vpRTgJCVeYISm7thUrN0NCUcxIXnYSSWKEh6hPuppypHe72fihHO5rpQeDSOjiCo7V3xMZCqVMQ193hkgN5LQ3Ev/zuokKam5GeZwowvFkUZAwqCI4Sgf2qCBYsVQThAXVt0I8QAJhpTMs6xDs6ZdnSeukaltV+/q0Ur8o4iiBXbAHDoENzkAdXIEGaAIMHsEzeAVvxpPxYrwbH5PWOaOY2QF/YHz+AG71nD8=

    The gap is very small fraction of the level energy. Hence we can essentially treat the energies as continuously varying.

    �✏ = ✏nx,ny,nz � ✏nx+1,ny,nz = 2(nx + 1)~2⇡22mL2

    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

    ⇡ 2nx~2⇡22mL2

    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

  • Quantum SystemsParticle in a box

    Eigenvalues are then

    ~k =⇡

    L(nx, ny, nz)

    AAACGXicbVBNS8NAEN34bf2qevSyWAQFKYkIehFELx48VLAqNCVstpN26WYTdifFGvI3vPhXvHhQxKOe/DduPw5+PRh4vDfDzLwwlcKg6346E5NT0zOzc/OlhcWl5ZXy6tqVSTLNoc4TmeibkBmQQkEdBUq4STWwOJRwHXZPB/51D7QRibrEfgrNmLWViARnaKWg7Po94Hm3oEfUjzTjuZ+KIj8vqC8hwm0V3O6qoG/rztei3cGdoFxxq+4Q9C/xxqRCxqgF5Xe/lfAsBoVcMmManptiM2caBZdQlPzMQMp4l7WhYaliMZhmPvysoFtWadEo0bYU0qH6fSJnsTH9OLSdMcOO+e0NxP+8RobRYTMXKs0QFB8tijJJMaGDmGhLaOAo+5YwroW9lfIOswGhDbNkQ/B+v/yXXO1VPbfqXexXjk/GccyRDbJJtolHDsgxOSM1Uiec3JNH8kxenAfnyXl13katE854Zp38gPPxBWqYoIo=AAACGXicbVBNS8NAEN34bf2qevSyWAQFKYkIehFELx48VLAqNCVstpN26WYTdifFGvI3vPhXvHhQxKOe/DduPw5+PRh4vDfDzLwwlcKg6346E5NT0zOzc/OlhcWl5ZXy6tqVSTLNoc4TmeibkBmQQkEdBUq4STWwOJRwHXZPB/51D7QRibrEfgrNmLWViARnaKWg7Po94Hm3oEfUjzTjuZ+KIj8vqC8hwm0V3O6qoG/rztei3cGdoFxxq+4Q9C/xxqRCxqgF5Xe/lfAsBoVcMmManptiM2caBZdQlPzMQMp4l7WhYaliMZhmPvysoFtWadEo0bYU0qH6fSJnsTH9OLSdMcOO+e0NxP+8RobRYTMXKs0QFB8tijJJMaGDmGhLaOAo+5YwroW9lfIOswGhDbNkQ/B+v/yXXO1VPbfqXexXjk/GccyRDbJJtolHDsgxOSM1Uiec3JNH8kxenAfnyXl13katE854Zp38gPPxBWqYoIo=AAACGXicbVBNS8NAEN34bf2qevSyWAQFKYkIehFELx48VLAqNCVstpN26WYTdifFGvI3vPhXvHhQxKOe/DduPw5+PRh4vDfDzLwwlcKg6346E5NT0zOzc/OlhcWl5ZXy6tqVSTLNoc4TmeibkBmQQkEdBUq4STWwOJRwHXZPB/51D7QRibrEfgrNmLWViARnaKWg7Po94Hm3oEfUjzTjuZ+KIj8vqC8hwm0V3O6qoG/rztei3cGdoFxxq+4Q9C/xxqRCxqgF5Xe/lfAsBoVcMmManptiM2caBZdQlPzMQMp4l7WhYaliMZhmPvysoFtWadEo0bYU0qH6fSJnsTH9OLSdMcOO+e0NxP+8RobRYTMXKs0QFB8tijJJMaGDmGhLaOAo+5YwroW9lfIOswGhDbNkQ/B+v/yXXO1VPbfqXexXjk/GccyRDbJJtolHDsgxOSM1Uiec3JNH8kxenAfnyXl13katE854Zp38gPPxBWqYoIo=AAACGXicbVBNS8NAEN34bf2qevSyWAQFKYkIehFELx48VLAqNCVstpN26WYTdifFGvI3vPhXvHhQxKOe/DduPw5+PRh4vDfDzLwwlcKg6346E5NT0zOzc/OlhcWl5ZXy6tqVSTLNoc4TmeibkBmQQkEdBUq4STWwOJRwHXZPB/51D7QRibrEfgrNmLWViARnaKWg7Po94Hm3oEfUjzTjuZ+KIj8vqC8hwm0V3O6qoG/rztei3cGdoFxxq+4Q9C/xxqRCxqgF5Xe/lfAsBoVcMmManptiM2caBZdQlPzMQMp4l7WhYaliMZhmPvysoFtWadEo0bYU0qH6fSJnsTH9OLSdMcOO+e0NxP+8RobRYTMXKs0QFB8tijJJMaGDmGhLaOAo+5YwroW9lfIOswGhDbNkQ/B+v/yXXO1VPbfqXexXjk/GccyRDbJJtolHDsgxOSM1Uiec3JNH8kxenAfnyXl13katE854Zp38gPPxBWqYoIo=

    Define

    ✏(~k) =~2k22m

    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

    kx

    ky

    ✏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=

    ✏+ d✏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

    ⌦ =⇣⇡L

    ⌘3AAACDnicbVDJSgNBEO1xjXGLevTSGALxEmZU0IsQ9OJBMIJZIBNDT6cmadKz0F0jhCFf4MVf8eJBEa+evfk3dpaDJj4oeLxXRVU9L5ZCo21/WwuLS8srq5m17PrG5tZ2bme3pqNEcajySEaq4TENUoRQRYESGrECFngS6l7/cuTXH0BpEYV3OIihFbBuKHzBGRqpnSu4NwF0GT2nrgQfi66vGE/dWAzT66GrRLeHh/fH7VzeLtlj0HniTEmeTFFp577cTsSTAELkkmnddOwYWylTKLiEYdZNNMSM91kXmoaGLADdSsfvDGnBKB3qR8pUiHSs/p5IWaD1IPBMZ8Cwp2e9kfif10zQP2ulIowThJBPFvmJpBjRUTa0IxRwlANDGFfC3Ep5j5lA0CSYNSE4sy/Pk9pRybFLzu1JvnwxjSND9skBKRKHnJIyuSIVUiWcPJJn8krerCfrxXq3PiatC9Z0Zo/8gfX5A1sHm6s=AAACDnicbVDJSgNBEO1xjXGLevTSGALxEmZU0IsQ9OJBMIJZIBNDT6cmadKz0F0jhCFf4MVf8eJBEa+evfk3dpaDJj4oeLxXRVU9L5ZCo21/WwuLS8srq5m17PrG5tZ2bme3pqNEcajySEaq4TENUoRQRYESGrECFngS6l7/cuTXH0BpEYV3OIihFbBuKHzBGRqpnSu4NwF0GT2nrgQfi66vGE/dWAzT66GrRLeHh/fH7VzeLtlj0HniTEmeTFFp577cTsSTAELkkmnddOwYWylTKLiEYdZNNMSM91kXmoaGLADdSsfvDGnBKB3qR8pUiHSs/p5IWaD1IPBMZ8Cwp2e9kfif10zQP2ulIowThJBPFvmJpBjRUTa0IxRwlANDGFfC3Ep5j5lA0CSYNSE4sy/Pk9pRybFLzu1JvnwxjSND9skBKRKHnJIyuSIVUiWcPJJn8krerCfrxXq3PiatC9Z0Zo/8gfX5A1sHm6s=AAACDnicbVDJSgNBEO1xjXGLevTSGALxEmZU0IsQ9OJBMIJZIBNDT6cmadKz0F0jhCFf4MVf8eJBEa+evfk3dpaDJj4oeLxXRVU9L5ZCo21/WwuLS8srq5m17PrG5tZ2bme3pqNEcajySEaq4TENUoRQRYESGrECFngS6l7/cuTXH0BpEYV3OIihFbBuKHzBGRqpnSu4NwF0GT2nrgQfi66vGE/dWAzT66GrRLeHh/fH7VzeLtlj0HniTEmeTFFp577cTsSTAELkkmnddOwYWylTKLiEYdZNNMSM91kXmoaGLADdSsfvDGnBKB3qR8pUiHSs/p5IWaD1IPBMZ8Cwp2e9kfif10zQP2ulIowThJBPFvmJpBjRUTa0IxRwlANDGFfC3Ep5j5lA0CSYNSE4sy/Pk9pRybFLzu1JvnwxjSND9skBKRKHnJIyuSIVUiWcPJJn8krerCfrxXq3PiatC9Z0Zo/8gfX5A1sHm6s=AAACDnicbVDJSgNBEO1xjXGLevTSGALxEmZU0IsQ9OJBMIJZIBNDT6cmadKz0F0jhCFf4MVf8eJBEa+evfk3dpaDJj4oeLxXRVU9L5ZCo21/WwuLS8srq5m17PrG5tZ2bme3pqNEcajySEaq4TENUoRQRYESGrECFngS6l7/cuTXH0BpEYV3OIihFbBuKHzBGRqpnSu4NwF0GT2nrgQfi66vGE/dWAzT66GrRLeHh/fH7VzeLtlj0HniTEmeTFFp577cTsSTAELkkmnddOwYWylTKLiEYdZNNMSM91kXmoaGLADdSsfvDGnBKB3qR8pUiHSs/p5IWaD1IPBMZ8Cwp2e9kfif10zQP2ulIowThJBPFvmJpBjRUTa0IxRwlANDGFfC3Ep5j5lA0CSYNSE4sy/Pk9pRybFLzu1JvnwxjSND9skBKRKHnJIyuSIVUiWcPJJn8krerCfrxXq3PiatC9Z0Zo/8gfX5A1sHm6s=

    N(✏) =18 ⇥

    43⇡k

    3

    ⌦=

    18 ⇥

    43⇡

    �2m✏~2

    � 32

    �⇡L

    �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

    =

    p2m

    32

    3⇡2~3 ⇥ V ⇥ ✏32

    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 g(✏) =1

    V

    dN

    d✏=

    m32

    p2⇡2~3

    ⇥ ✏ 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

    Density of states:

    No. of states in the shell (above) per unit volume

  • Quantum SystemsParticle in a box

    Example, canonical partition function

    Q(N,V, T ) =X

    exp(��✏) =X

    nx,ny,nz

    exp(�� ~2

    2mL2(n2x + n

    2y + n

    2z))

    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

    =X

    kx,ky,kz

    exp

    ✓�� ~

    2k2

    2m

    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

    �! V⇡3

    Zd3k exp

    ✓�� ~

    2k2

    2m

    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

    =V m

    32

    p2⇡2~3

    1Z

    0

    d✏ ✏12 exp (��✏)

    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

    Q(N,V, T ) = V

    1Z

    0

    d✏ g(✏) exp (��✏)AAACR3icbZDPSyMxFMcz9Ve362rV417CloUW3DIjgl4EcS+eFgVbhaaWTPqmDSaZIXkjlqH967x43Zv/gpc97LLscdNawV8PAt98v+/xkk+cKekwDO+D0sLi0vJK+UPl4+qntfXqxmbbpbkV0BKpSu1FzB0oaaCFEhVcZBa4jhWcx1ffp/n5NVgnU3OGowy6mg+MTKTg6K1e9fK0/mO7vX3WoPSAtpk0yJTUEl0vvPS3BEe0zyBzUqVmMqg/ycakYFZTuMnGTEGC9W8sBuT0KafMysEQG71qLWyGs6JvRTQXNTKvk171J+unItdgUCjuXCcKM+wW3KIUCsYVljvIuLjiA+h4abgG1y1mHMb0q3f6NEmtPwbpzH0+UXDt3EjHvlNzHLrX2dR8L+vkmOx3C2myHMGIx0VJriimdAqV9qUFgcqzklxY6d9KxZBbLtCjr3gI0esvvxXtnWYUNqPT3drh0RxHmXwmX0idRGSPHJJjckJaRJBb8kB+kz/BXfAr+Bv8e2wtBfOZLfKiSsF/sGSxVg==AAACR3icbZDPSyMxFMcz9Ve362rV417CloUW3DIjgl4EcS+eFgVbhaaWTPqmDSaZIXkjlqH967x43Zv/gpc97LLscdNawV8PAt98v+/xkk+cKekwDO+D0sLi0vJK+UPl4+qntfXqxmbbpbkV0BKpSu1FzB0oaaCFEhVcZBa4jhWcx1ffp/n5NVgnU3OGowy6mg+MTKTg6K1e9fK0/mO7vX3WoPSAtpk0yJTUEl0vvPS3BEe0zyBzUqVmMqg/ycakYFZTuMnGTEGC9W8sBuT0KafMysEQG71qLWyGs6JvRTQXNTKvk171J+unItdgUCjuXCcKM+wW3KIUCsYVljvIuLjiA+h4abgG1y1mHMb0q3f6NEmtPwbpzH0+UXDt3EjHvlNzHLrX2dR8L+vkmOx3C2myHMGIx0VJriimdAqV9qUFgcqzklxY6d9KxZBbLtCjr3gI0esvvxXtnWYUNqPT3drh0RxHmXwmX0idRGSPHJJjckJaRJBb8kB+kz/BXfAr+Bv8e2wtBfOZLfKiSsF/sGSxVg==AAACR3icbZDPSyMxFMcz9Ve362rV417CloUW3DIjgl4EcS+eFgVbhaaWTPqmDSaZIXkjlqH967x43Zv/gpc97LLscdNawV8PAt98v+/xkk+cKekwDO+D0sLi0vJK+UPl4+qntfXqxmbbpbkV0BKpSu1FzB0oaaCFEhVcZBa4jhWcx1ffp/n5NVgnU3OGowy6mg+MTKTg6K1e9fK0/mO7vX3WoPSAtpk0yJTUEl0vvPS3BEe0zyBzUqVmMqg/ycakYFZTuMnGTEGC9W8sBuT0KafMysEQG71qLWyGs6JvRTQXNTKvk171J+unItdgUCjuXCcKM+wW3KIUCsYVljvIuLjiA+h4abgG1y1mHMb0q3f6NEmtPwbpzH0+UXDt3EjHvlNzHLrX2dR8L+vkmOx3C2myHMGIx0VJriimdAqV9qUFgcqzklxY6d9KxZBbLtCjr3gI0esvvxXtnWYUNqPT3drh0RxHmXwmX0idRGSPHJJjckJaRJBb8kB+kz/BXfAr+Bv8e2wtBfOZLfKiSsF/sGSxVg==AAACR3icbZDPSyMxFMcz9Ve362rV417CloUW3DIjgl4EcS+eFgVbhaaWTPqmDSaZIXkjlqH967x43Zv/gpc97LLscdNawV8PAt98v+/xkk+cKekwDO+D0sLi0vJK+UPl4+qntfXqxmbbpbkV0BKpSu1FzB0oaaCFEhVcZBa4jhWcx1ffp/n5NVgnU3OGowy6mg+MTKTg6K1e9fK0/mO7vX3WoPSAtpk0yJTUEl0vvPS3BEe0zyBzUqVmMqg/ycakYFZTuMnGTEGC9W8sBuT0KafMysEQG71qLWyGs6JvRTQXNTKvk171J+unItdgUCjuXCcKM+wW3KIUCsYVljvIuLjiA+h4abgG1y1mHMb0q3f6NEmtPwbpzH0+UXDt3EjHvlNzHLrX2dR8L+vkmOx3C2myHMGIx0VJriimdAqV9qUFgcqzklxY6d9KxZBbLtCjr3gI0esvvxXtnWYUNqPT3drh0RxHmXwmX0idRGSPHJJjckJaRJBb8kB+kz/BXfAr+Bv8e2wtBfOZLfKiSsF/sGSxVg==

    =V

    ⇤3AAAB/3icbVDLSsNAFL2pr1pfUcGNm8EiuCqJCroRim5cuKhgH9DEMplM2qGTBzMTocQs/BU3LhRx62+482+ctllo64GBwzn3cO8cL+FMKsv6NkoLi0vLK+XVytr6xuaWub3TknEqCG2SmMei42FJOYtoUzHFaScRFIcep21veDX22w9USBZHd2qUUDfE/YgFjGClpZ65hy6QEwhMslaeOTc66OP7k7xnVq2aNQGaJ3ZBqlCg0TO/HD8maUgjRTiWsmtbiXIzLBQjnOYVJ5U0wWSI+7SraYRDKt1scn+ODrXioyAW+kUKTdTfiQyHUo5CT0+GWA3krDcW//O6qQrO3YxFSapoRKaLgpQjFaNxGchnghLFR5pgIpi+FZEB1m0oXVlFl2DPfnmetI5rtlWzb0+r9cuijjLswwEcgQ1nUIdraEATCDzCM7zCm/FkvBjvxsd0tGQUmV34A+PzB/1QlXA=AAAB/3icbVDLSsNAFL2pr1pfUcGNm8EiuCqJCroRim5cuKhgH9DEMplM2qGTBzMTocQs/BU3LhRx62+482+ctllo64GBwzn3cO8cL+FMKsv6NkoLi0vLK+XVytr6xuaWub3TknEqCG2SmMei42FJOYtoUzHFaScRFIcep21veDX22w9USBZHd2qUUDfE/YgFjGClpZ65hy6QEwhMslaeOTc66OP7k7xnVq2aNQGaJ3ZBqlCg0TO/HD8maUgjRTiWsmtbiXIzLBQjnOYVJ5U0wWSI+7SraYRDKt1scn+ODrXioyAW+kUKTdTfiQyHUo5CT0+GWA3krDcW//O6qQrO3YxFSapoRKaLgpQjFaNxGchnghLFR5pgIpi+FZEB1m0oXVlFl2DPfnmetI5rtlWzb0+r9cuijjLswwEcgQ1nUIdraEATCDzCM7zCm/FkvBjvxsd0tGQUmV34A+PzB/1QlXA=AAAB/3icbVDLSsNAFL2pr1pfUcGNm8EiuCqJCroRim5cuKhgH9DEMplM2qGTBzMTocQs/BU3LhRx62+482+ctllo64GBwzn3cO8cL+FMKsv6NkoLi0vLK+XVytr6xuaWub3TknEqCG2SmMei42FJOYtoUzHFaScRFIcep21veDX22w9USBZHd2qUUDfE/YgFjGClpZ65hy6QEwhMslaeOTc66OP7k7xnVq2aNQGaJ3ZBqlCg0TO/HD8maUgjRTiWsmtbiXIzLBQjnOYVJ5U0wWSI+7SraYRDKt1scn+ODrXioyAW+kUKTdTfiQyHUo5CT0+GWA3krDcW//O6qQrO3YxFSapoRKaLgpQjFaNxGchnghLFR5pgIpi+FZEB1m0oXVlFl2DPfnmetI5rtlWzb0+r9cuijjLswwEcgQ1nUIdraEATCDzCM7zCm/FkvBjvxsd0tGQUmV34A+PzB/1QlXA=AAAB/3icbVDLSsNAFL2pr1pfUcGNm8EiuCqJCroRim5cuKhgH9DEMplM2qGTBzMTocQs/BU3LhRx62+482+ctllo64GBwzn3cO8cL+FMKsv6NkoLi0vLK+XVytr6xuaWub3TknEqCG2SmMei42FJOYtoUzHFaScRFIcep21veDX22w9USBZHd2qUUDfE/YgFjGClpZ65hy6QEwhMslaeOTc66OP7k7xnVq2aNQGaJ3ZBqlCg0TO/HD8maUgjRTiWsmtbiXIzLBQjnOYVJ5U0wWSI+7SraYRDKt1scn+ODrXioyAW+kUKTdTfiQyHUo5CT0+GWA3krDcW//O6qQrO3YxFSapoRKaLgpQjFaNxGchnghLFR5pgIpi+FZEB1m0oXVlFl2DPfnmetI5rtlWzb0+r9cuijjLswwEcgQ1nUIdraEATCDzCM7zCm/FkvBjvxsd0tGQUmV34A+PzB/1QlXA=

    ⇤ =hp

    2⇡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

  • Quantum SystemsRigid Rotor

    Eigenvalues

    ✏J = J(J + 1)~22I

    AAACEHicbZC7SgNBFIZn4y3G26qlzWAQI0LYDYI2QtBGU0UwF8jGMDs5mwyZvTAzK4RlH8HGV7GxUMTW0s63cZJsodEfBj7+cw5nzu9GnEllWV9GbmFxaXklv1pYW9/Y3DK3d5oyjAWFBg15KNoukcBZAA3FFId2JID4LoeWO7qc1Fv3ICQLg1s1jqDrk0HAPEaJ0lbPPHQgkoxrrOFzXCvVju0j7HiC0MQZukTcVdKkgq/Tnlm0ytZU+C/YGRRRpnrP/HT6IY19CBTlRMqObUWqmxChGOWQFpxYQkToiAygozEgPshuMj0oxQfa6WMvFPoFCk/dnxMJ8aUc+67u9IkayvnaxPyv1omVd9ZNWBDFCgI6W+TFHKsQT9LBfSaAKj7WQKhg+q+YDolOQ+kMCzoEe/7kv9CslG2rbN+cFKsXWRx5tIf2UQnZ6BRV0RWqowai6AE9oRf0ajwaz8ab8T5rzRnZzC76JePjG7Hjmxo=AAACEHicbZC7SgNBFIZn4y3G26qlzWAQI0LYDYI2QtBGU0UwF8jGMDs5mwyZvTAzK4RlH8HGV7GxUMTW0s63cZJsodEfBj7+cw5nzu9GnEllWV9GbmFxaXklv1pYW9/Y3DK3d5oyjAWFBg15KNoukcBZAA3FFId2JID4LoeWO7qc1Fv3ICQLg1s1jqDrk0HAPEaJ0lbPPHQgkoxrrOFzXCvVju0j7HiC0MQZukTcVdKkgq/Tnlm0ytZU+C/YGRRRpnrP/HT6IY19CBTlRMqObUWqmxChGOWQFpxYQkToiAygozEgPshuMj0oxQfa6WMvFPoFCk/dnxMJ8aUc+67u9IkayvnaxPyv1omVd9ZNWBDFCgI6W+TFHKsQT9LBfSaAKj7WQKhg+q+YDolOQ+kMCzoEe/7kv9CslG2rbN+cFKsXWRx5tIf2UQnZ6BRV0RWqowai6AE9oRf0ajwaz8ab8T5rzRnZzC76JePjG7Hjmxo=AAACEHicbZC7SgNBFIZn4y3G26qlzWAQI0LYDYI2QtBGU0UwF8jGMDs5mwyZvTAzK4RlH8HGV7GxUMTW0s63cZJsodEfBj7+cw5nzu9GnEllWV9GbmFxaXklv1pYW9/Y3DK3d5oyjAWFBg15KNoukcBZAA3FFId2JID4LoeWO7qc1Fv3ICQLg1s1jqDrk0HAPEaJ0lbPPHQgkoxrrOFzXCvVju0j7HiC0MQZukTcVdKkgq/Tnlm0ytZU+C/YGRRRpnrP/HT6IY19CBTlRMqObUWqmxChGOWQFpxYQkToiAygozEgPshuMj0oxQfa6WMvFPoFCk/dnxMJ8aUc+67u9IkayvnaxPyv1omVd9ZNWBDFCgI6W+TFHKsQT9LBfSaAKj7WQKhg+q+YDolOQ+kMCzoEe/7kv9CslG2rbN+cFKsXWRx5tIf2UQnZ6BRV0RWqowai6AE9oRf0ajwaz8ab8T5rzRnZzC76JePjG7Hjmxo=AAACEHicbZC7SgNBFIZn4y3G26qlzWAQI0LYDYI2QtBGU0UwF8jGMDs5mwyZvTAzK4RlH8HGV7GxUMTW0s63cZJsodEfBj7+cw5nzu9GnEllWV9GbmFxaXklv1pYW9/Y3DK3d5oyjAWFBg15KNoukcBZAA3FFId2JID4LoeWO7qc1Fv3ICQLg1s1jqDrk0HAPEaJ0lbPPHQgkoxrrOFzXCvVju0j7HiC0MQZukTcVdKkgq/Tnlm0ytZU+C/YGRRRpnrP/HT6IY19CBTlRMqObUWqmxChGOWQFpxYQkToiAygozEgPshuMj0oxQfa6WMvFPoFCk/dnxMJ8aUc+67u9IkayvnaxPyv1omVd9ZNWBDFCgI6W+TFHKsQT9LBfSaAKj7WQKhg+q+YDolOQ+kMCzoEe/7kv9CslG2rbN+cFKsXWRx5tIf2UQnZ6BRV0RWqowai6AE9oRf0ajwaz8ab8T5rzRnZzC76JePjG7Hjmxo=

    J=0,1,2,…I = µR2

    AAAB8nicbVBNSwMxEJ2tX7V+VT16CRbBU9ktgl6Eohe9VbEfsF1LNs22odlkSbJCWfozvHhQxKu/xpv/xrTdg7Y+GHi8N8PMvDDhTBvX/XYKK6tr6xvFzdLW9s7uXnn/oKVlqghtEsml6oRYU84EbRpmOO0kiuI45LQdjq6nfvuJKs2keDDjhAYxHggWMYKNlfxbdIm6cYruH2u9csWtujOgZeLlpAI5Gr3yV7cvSRpTYQjHWvuem5ggw8owwumk1E01TTAZ4QH1LRU4pjrIZidP0IlV+iiSypYwaKb+nshwrPU4Dm1njM1QL3pT8T/PT010EWRMJKmhgswXRSlHRqLp/6jPFCWGjy3BRDF7KyJDrDAxNqWSDcFbfHmZtGpVz616d2eV+lUeRxGO4BhOwYNzqMMNNKAJBCQ8wyu8OcZ5cd6dj3lrwclnDuEPnM8fRU6P7g==AAAB8nicbVBNSwMxEJ2tX7V+VT16CRbBU9ktgl6Eohe9VbEfsF1LNs22odlkSbJCWfozvHhQxKu/xpv/xrTdg7Y+GHi8N8PMvDDhTBvX/XYKK6tr6xvFzdLW9s7uXnn/oKVlqghtEsml6oRYU84EbRpmOO0kiuI45LQdjq6nfvuJKs2keDDjhAYxHggWMYKNlfxbdIm6cYruH2u9csWtujOgZeLlpAI5Gr3yV7cvSRpTYQjHWvuem5ggw8owwumk1E01TTAZ4QH1LRU4pjrIZidP0IlV+iiSypYwaKb+nshwrPU4Dm1njM1QL3pT8T/PT010EWRMJKmhgswXRSlHRqLp/6jPFCWGjy3BRDF7KyJDrDAxNqWSDcFbfHmZtGpVz616d2eV+lUeRxGO4BhOwYNzqMMNNKAJBCQ8wyu8OcZ5cd6dj3lrwclnDuEPnM8fRU6P7g==AAAB8nicbVBNSwMxEJ2tX7V+VT16CRbBU9ktgl6Eohe9VbEfsF1LNs22odlkSbJCWfozvHhQxKu/xpv/xrTdg7Y+GHi8N8PMvDDhTBvX/XYKK6tr6xvFzdLW9s7uXnn/oKVlqghtEsml6oRYU84EbRpmOO0kiuI45LQdjq6nfvuJKs2keDDjhAYxHggWMYKNlfxbdIm6cYruH2u9csWtujOgZeLlpAI5Gr3yV7cvSRpTYQjHWvuem5ggw8owwumk1E01TTAZ4QH1LRU4pjrIZidP0IlV+iiSypYwaKb+nshwrPU4Dm1njM1QL3pT8T/PT010EWRMJKmhgswXRSlHRqLp/6jPFCWGjy3BRDF7KyJDrDAxNqWSDcFbfHmZtGpVz616d2eV+lUeRxGO4BhOwYNzqMMNNKAJBCQ8wyu8OcZ5cd6dj3lrwclnDuEPnM8fRU6P7g==AAAB8nicbVBNSwMxEJ2tX7V+VT16CRbBU9ktgl6Eohe9VbEfsF1LNs22odlkSbJCWfozvHhQxKu/xpv/xrTdg7Y+GHi8N8PMvDDhTBvX/XYKK6tr6xvFzdLW9s7uXnn/oKVlqghtEsml6oRYU84EbRpmOO0kiuI45LQdjq6nfvuJKs2keDDjhAYxHggWMYKNlfxbdIm6cYruH2u9csWtujOgZeLlpAI5Gr3yV7cvSRpTYQjHWvuem5ggw8owwumk1E01TTAZ4QH1LRU4pjrIZidP0IlV+iiSypYwaKb+nshwrPU4Dm1njM1QL3pT8T/PT010EWRMJKmhgswXRSlHRqLp/6jPFCWGjy3BRDF7KyJDrDAxNqWSDcFbfHmZtGpVz616d2eV+lUeRxGO4BhOwYNzqMMNNKAJBCQ8wyu8OcZ5cd6dj3lrwclnDuEPnM8fRU6P7g==

    �✏ = ✏1 � ✏0 = 2~2

    2µ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

    ⇡ 4.2⇥ 10�22JAAACBnicbVDLSsNAFJ3UV62vqEsRBovgxpCEgi6LbsRVBfuAJpbJdNIOncyEmYlYQldu/BU3LhRx6ze482+cPhbaeuDC4Zx7ufeeKGVUadf9tgpLyyura8X10sbm1vaOvbvXUCKTmNSxYEK2IqQIo5zUNdWMtFJJUBIx0owGl2O/eU+kooLf6mFKwgT1OI0pRtpIHfswQGkqxQOsOD4MNE2Igp57l5/6/ghed+yy67gTwEXizUgZzFDr2F9BV+AsIVxjhpRqe26qwxxJTTEjo1KQKZIiPEA90jaUI7MvzCdvjOCxUbowFtIU13Ci/p7IUaLUMIlMZ4J0X817Y/E/r53p+DzMKU8zTTieLoozBrWA40xgl0qCNRsagrCk5laI+0girE1yJROCN//yImn4juc63k2lXL2YxVEEB+AInAAPnIEquAI1UAcYPIJn8ArerCfrxXq3PqatBWs2sw/+wPr8AQpBluc=AAACBnicbVDLSsNAFJ3UV62vqEsRBovgxpCEgi6LbsRVBfuAJpbJdNIOncyEmYlYQldu/BU3LhRx6ze482+cPhbaeuDC4Zx7ufeeKGVUadf9tgpLyyura8X10sbm1vaOvbvXUCKTmNSxYEK2IqQIo5zUNdWMtFJJUBIx0owGl2O/eU+kooLf6mFKwgT1OI0pRtpIHfswQGkqxQOsOD4MNE2Igp57l5/6/ghed+yy67gTwEXizUgZzFDr2F9BV+AsIVxjhpRqe26qwxxJTTEjo1KQKZIiPEA90jaUI7MvzCdvjOCxUbowFtIU13Ci/p7IUaLUMIlMZ4J0X817Y/E/r53p+DzMKU8zTTieLoozBrWA40xgl0qCNRsagrCk5laI+0girE1yJROCN//yImn4juc63k2lXL2YxVEEB+AInAAPnIEquAI1UAcYPIJn8ArerCfrxXq3PqatBWs2sw/+wPr8AQpBluc=AAACBnicbVDLSsNAFJ3UV62vqEsRBovgxpCEgi6LbsRVBfuAJpbJdNIOncyEmYlYQldu/BU3LhRx6ze482+cPhbaeuDC4Zx7ufeeKGVUadf9tgpLyyura8X10sbm1vaOvbvXUCKTmNSxYEK2IqQIo5zUNdWMtFJJUBIx0owGl2O/eU+kooLf6mFKwgT1OI0pRtpIHfswQGkqxQOsOD4MNE2Igp57l5/6/ghed+yy67gTwEXizUgZzFDr2F9BV+AsIVxjhpRqe26qwxxJTTEjo1KQKZIiPEA90jaUI7MvzCdvjOCxUbowFtIU13Ci/p7IUaLUMIlMZ4J0X817Y/E/r53p+DzMKU8zTTieLoozBrWA40xgl0qCNRsagrCk5laI+0girE1yJROCN//yImn4juc63k2lXL2YxVEEB+AInAAPnIEquAI1UAcYPIJn8ArerCfrxXq3PqatBWs2sw/+wPr8AQpBluc=AAACBnicbVDLSsNAFJ3UV62vqEsRBovgxpCEgi6LbsRVBfuAJpbJdNIOncyEmYlYQldu/BU3LhRx6ze482+cPhbaeuDC4Zx7ufeeKGVUadf9tgpLyyura8X10sbm1vaOvbvXUCKTmNSxYEK2IqQIo5zUNdWMtFJJUBIx0owGl2O/eU+kooLf6mFKwgT1OI0pRtpIHfswQGkqxQOsOD4MNE2Igp57l5/6/ghed+yy67gTwEXizUgZzFDr2F9BV+AsIVxjhpRqe26qwxxJTTEjo1KQKZIiPEA90jaUI7MvzCdvjOCxUbowFtIU13Ci/p7IUaLUMIlMZ4J0X817Y/E/r53p+DzMKU8zTTieLoozBrWA40xgl0qCNRsagrCk5laI+0girE1yJROCN//yImn4juc63k2lXL2YxVEEB+AInAAPnIEquAI1UAcYPIJn8ArerCfrxXq3PqatBWs2sw/+wPr8AQpBluc=

    For HCl with I=2.65 x 10-47 kg/m3

    Compared to the energy of first excitation in particle-in-a-box, this is much larger.

    Hence, quantisation of levels is more important in this case.

    Degeneracy of each level is 2J+1

  • Quantum SystemsHarmonic oscillator

    Eigenvalues v=0,1,2,…

    For HCl with

    Hence, quantisation of levels is more important in this case too.

    ✏v = (v +1

    2)h⌫

    AAACC3icbZDLSsNAFIYn9VbrrerSzdAiVISSFEE3QtGNywr2Ak0ok+lJO3QyCTOTQgnZu/FV3LhQxK0v4M63cXpZaOsPAx//OYcz5/djzpS27W8rt7a+sbmV3y7s7O7tHxQPj1oqSiSFJo14JDs+UcCZgKZmmkMnlkBCn0PbH91O6+0xSMUi8aAnMXghGQgWMEq0sXrFkguxYtzgGF/jyvjcDSShqZOltewMD7Erkl6xbFftmfAqOAsoo4UaveKX249oEoLQlBOluo4day8lUjPKISu4iYKY0BEZQNegICEoL53dkuFT4/RxEEnzhMYz9/dESkKlJqFvOkOih2q5NjX/q3UTHVx5KRNxokHQ+aIg4VhHeBoM7jMJVPOJAUIlM3/FdEhMGNrEVzAhOMsnr0KrVnXsqnN/Ua7fLOLIoxNUQhXkoEtUR3eogZqIokf0jF7Rm/VkvVjv1se8NWctZo7RH1mfPxramcs=AAACC3icbZDLSsNAFIYn9VbrrerSzdAiVISSFEE3QtGNywr2Ak0ok+lJO3QyCTOTQgnZu/FV3LhQxK0v4M63cXpZaOsPAx//OYcz5/djzpS27W8rt7a+sbmV3y7s7O7tHxQPj1oqSiSFJo14JDs+UcCZgKZmmkMnlkBCn0PbH91O6+0xSMUi8aAnMXghGQgWMEq0sXrFkguxYtzgGF/jyvjcDSShqZOltewMD7Erkl6xbFftmfAqOAsoo4UaveKX249oEoLQlBOluo4day8lUjPKISu4iYKY0BEZQNegICEoL53dkuFT4/RxEEnzhMYz9/dESkKlJqFvOkOih2q5NjX/q3UTHVx5KRNxokHQ+aIg4VhHeBoM7jMJVPOJAUIlM3/FdEhMGNrEVzAhOMsnr0KrVnXsqnN/Ua7fLOLIoxNUQhXkoEtUR3eogZqIokf0jF7Rm/VkvVjv1se8NWctZo7RH1mfPxramcs=AAACC3icbZDLSsNAFIYn9VbrrerSzdAiVISSFEE3QtGNywr2Ak0ok+lJO3QyCTOTQgnZu/FV3LhQxK0v4M63cXpZaOsPAx//OYcz5/djzpS27W8rt7a+sbmV3y7s7O7tHxQPj1oqSiSFJo14JDs+UcCZgKZmmkMnlkBCn0PbH91O6+0xSMUi8aAnMXghGQgWMEq0sXrFkguxYtzgGF/jyvjcDSShqZOltewMD7Erkl6xbFftmfAqOAsoo4UaveKX249oEoLQlBOluo4day8lUjPKISu4iYKY0BEZQNegICEoL53dkuFT4/RxEEnzhMYz9/dESkKlJqFvOkOih2q5NjX/q3UTHVx5KRNxokHQ+aIg4VhHeBoM7jMJVPOJAUIlM3/FdEhMGNrEVzAhOMsnr0KrVnXsqnN/Ua7fLOLIoxNUQhXkoEtUR3eogZqIokf0jF7Rm/VkvVjv1se8NWctZo7RH1mfPxramcs=AAACC3icbZDLSsNAFIYn9VbrrerSzdAiVISSFEE3QtGNywr2Ak0ok+lJO3QyCTOTQgnZu/FV3LhQxK0v4M63cXpZaOsPAx//OYcz5/djzpS27W8rt7a+sbmV3y7s7O7tHxQPj1oqSiSFJo14JDs+UcCZgKZmmkMnlkBCn0PbH91O6+0xSMUi8aAnMXghGQgWMEq0sXrFkguxYtzgGF/jyvjcDSShqZOltewMD7Erkl6xbFftmfAqOAsoo4UaveKX249oEoLQlBOluo4day8lUjPKISu4iYKY0BEZQNegICEoL53dkuFT4/RxEEnzhMYz9/dESkKlJqFvOkOih2q5NjX/q3UTHVx5KRNxokHQ+aIg4VhHeBoM7jMJVPOJAUIlM3/FdEhMGNrEVzAhOMsnr0KrVnXsqnN/Ua7fLOLIoxNUQhXkoEtUR3eogZqIokf0jF7Rm/VkvVjv1se8NWctZo7RH1mfPxramcs=

    �✏ = ✏1 � ✏0 = h⌫AAACGnicbZDLSgMxFIYz9VbrbdSlm2AR3FhmRNCNUNSFywr2Ap1hyKRn2tBMZkgyQhn6HG58FTcuFHEnbnwb0wteWg8Evvz/OSTnD1POlHacT6uwsLi0vFJcLa2tb2xu2ds7DZVkkkKdJjyRrZAo4ExAXTPNoZVKIHHIoRn2L0d+8w6kYom41YMU/Jh0BYsYJdpIge16V8A1wR6kivFE4PNvDFx89HNxjNPDnsgCu+xUnHHheXCnUEbTqgX2u9dJaBaD0JQTpdquk2o/J1IzymFY8jIFKaF90oW2QUFiUH4+Xm2ID4zSwVEizREaj9XfEzmJlRrEoemMie6pWW8k/ue1Mx2d+TkTaaZB0MlDUcaxTvAoJ9xhEqjmAwOESmb+immPSEK1SbNkQnBnV56HxnHFdSruzUm5ejGNo4j20D46RC46RVV0jWqojii6R4/oGb1YD9aT9Wq9TVoL1nRmF/0p6+MLkNqfSg==AAACGnicbZDLSgMxFIYz9VbrbdSlm2AR3FhmRNCNUNSFywr2Ap1hyKRn2tBMZkgyQhn6HG58FTcuFHEnbnwb0wteWg8Evvz/OSTnD1POlHacT6uwsLi0vFJcLa2tb2xu2ds7DZVkkkKdJjyRrZAo4ExAXTPNoZVKIHHIoRn2L0d+8w6kYom41YMU/Jh0BYsYJdpIge16V8A1wR6kivFE4PNvDFx89HNxjNPDnsgCu+xUnHHheXCnUEbTqgX2u9dJaBaD0JQTpdquk2o/J1IzymFY8jIFKaF90oW2QUFiUH4+Xm2ID4zSwVEizREaj9XfEzmJlRrEoemMie6pWW8k/ue1Mx2d+TkTaaZB0MlDUcaxTvAoJ9xhEqjmAwOESmb+immPSEK1SbNkQnBnV56HxnHFdSruzUm5ejGNo4j20D46RC46RVV0jWqojii6R4/oGb1YD9aT9Wq9TVoL1nRmF/0p6+MLkNqfSg==AAACGnicbZDLSgMxFIYz9VbrbdSlm2AR3FhmRNCNUNSFywr2Ap1hyKRn2tBMZkgyQhn6HG58FTcuFHEnbnwb0wteWg8Evvz/OSTnD1POlHacT6uwsLi0vFJcLa2tb2xu2ds7DZVkkkKdJjyRrZAo4ExAXTPNoZVKIHHIoRn2L0d+8w6kYom41YMU/Jh0BYsYJdpIge16V8A1wR6kivFE4PNvDFx89HNxjNPDnsgCu+xUnHHheXCnUEbTqgX2u9dJaBaD0JQTpdquk2o/J1IzymFY8jIFKaF90oW2QUFiUH4+Xm2ID4zSwVEizREaj9XfEzmJlRrEoemMie6pWW8k/ue1Mx2d+TkTaaZB0MlDUcaxTvAoJ9xhEqjmAwOESmb+immPSEK1SbNkQnBnV56HxnHFdSruzUm5ejGNo4j20D46RC46RVV0jWqojii6R4/oGb1YD9aT9Wq9TVoL1nRmF/0p6+MLkNqfSg==AAACGnicbZDLSgMxFIYz9VbrbdSlm2AR3FhmRNCNUNSFywr2Ap1hyKRn2tBMZkgyQhn6HG58FTcuFHEnbnwb0wteWg8Evvz/OSTnD1POlHacT6uwsLi0vFJcLa2tb2xu2ds7DZVkkkKdJjyRrZAo4ExAXTPNoZVKIHHIoRn2L0d+8w6kYom41YMU/Jh0BYsYJdpIge16V8A1wR6kivFE4PNvDFx89HNxjNPDnsgCu+xUnHHheXCnUEbTqgX2u9dJaBaD0JQTpdquk2o/J1IzymFY8jIFKaF90oW2QUFiUH4+Xm2ID4zSwVEizREaj9XfEzmJlRrEoemMie6pWW8k/ue1Mx2d+TkTaaZB0MlDUcaxTvAoJ9xhEqjmAwOESmb+immPSEK1SbNkQnBnV56HxnHFdSruzUm5ejGNo4j20D46RC46RVV0jWqojii6R4/oGb1YD9aT9Wq9TVoL1nRmF/0p6+MLkNqfSg==

    ⇡ 5.7⇥ 10�20JAAACBnicbVDLSsNAFJ34rPUVdSnCYBHcGJKi1GXRjbiqYB/QxDKZTtqhk8kwMxFL6MqNv+LGhSJu/QZ3/o3TNgttPXDhcM693HtPKBhV2nW/rYXFpeWV1cJacX1jc2vb3tltqCSVmNRxwhLZCpEijHJS11Qz0hKSoDhkpBkOLsd+855IRRN+q4eCBDHqcRpRjLSROvaBj4SQyQM8cyrQ1zQmCnruXXZSdkfwumOXXMedAM4TLyclkKPWsb/8boLTmHCNGVKq7blCBxmSmmJGRkU/VUQgPEA90jaUI7MvyCZvjOCRUbowSqQpruFE/T2RoVipYRyazhjpvpr1xuJ/XjvV0XmQUS5STTieLopSBnUCx5nALpUEazY0BGFJza0Q95FEWJvkiiYEb/bledIoO57reDenpepFHkcB7INDcAw8UAFVcAVqoA4weATP4BW8WU/Wi/VufUxbF6x8Zg/8gfX5AxC5lus=AAACBnicbVDLSsNAFJ34rPUVdSnCYBHcGJKi1GXRjbiqYB/QxDKZTtqhk8kwMxFL6MqNv+LGhSJu/QZ3/o3TNgttPXDhcM693HtPKBhV2nW/rYXFpeWV1cJacX1jc2vb3tltqCSVmNRxwhLZCpEijHJS11Qz0hKSoDhkpBkOLsd+855IRRN+q4eCBDHqcRpRjLSROvaBj4SQyQM8cyrQ1zQmCnruXXZSdkfwumOXXMedAM4TLyclkKPWsb/8boLTmHCNGVKq7blCBxmSmmJGRkU/VUQgPEA90jaUI7MvyCZvjOCRUbowSqQpruFE/T2RoVipYRyazhjpvpr1xuJ/XjvV0XmQUS5STTieLopSBnUCx5nALpUEazY0BGFJza0Q95FEWJvkiiYEb/bledIoO57reDenpepFHkcB7INDcAw8UAFVcAVqoA4weATP4BW8WU/Wi/VufUxbF6x8Zg/8gfX5AxC5lus=AAACBnicbVDLSsNAFJ34rPUVdSnCYBHcGJKi1GXRjbiqYB/QxDKZTtqhk8kwMxFL6MqNv+LGhSJu/QZ3/o3TNgttPXDhcM693HtPKBhV2nW/rYXFpeWV1cJacX1jc2vb3tltqCSVmNRxwhLZCpEijHJS11Qz0hKSoDhkpBkOLsd+855IRRN+q4eCBDHqcRpRjLSROvaBj4SQyQM8cyrQ1zQmCnruXXZSdkfwumOXXMedAM4TLyclkKPWsb/8boLTmHCNGVKq7blCBxmSmmJGRkU/VUQgPEA90jaUI7MvyCZvjOCRUbowSqQpruFE/T2RoVipYRyazhjpvpr1xuJ/XjvV0XmQUS5STTieLopSBnUCx5nALpUEazY0BGFJza0Q95FEWJvkiiYEb/bledIoO57reDenpepFHkcB7INDcAw8UAFVcAVqoA4weATP4BW8WU/Wi/VufUxbF6x8Zg/8gfX5AxC5lus=AAACBnicbVDLSsNAFJ34rPUVdSnCYBHcGJKi1GXRjbiqYB/QxDKZTtqhk8kwMxFL6MqNv+LGhSJu/QZ3/o3TNgttPXDhcM693HtPKBhV2nW/rYXFpeWV1cJacX1jc2vb3tltqCSVmNRxwhLZCpEijHJS11Qz0hKSoDhkpBkOLsd+855IRRN+q4eCBDHqcRpRjLSROvaBj4SQyQM8cyrQ1zQmCnruXXZSdkfwumOXXMedAM4TLyclkKPWsb/8boLTmHCNGVKq7blCBxmSmmJGRkU/VUQgPEA90jaUI7MvyCZvjOCRUbowSqQpruFE/T2RoVipYRyazhjpvpr1xuJ/XjvV0XmQUS5STTieLopSBnUCx5nALpUEazY0BGFJza0Q95FEWJvkiiYEb/bledIoO57reDenpepFHkcB7INDcAw8UAFVcAVqoA4weATP4BW8WU/Wi/VufUxbF6x8Zg/8gfX5AxC5lus=

    ⌫ = 8.65⇥ 1013HzAAACBHicbVC7TsMwFHXKq5RXgLGLRYXEFMU8uyBVsHQsEn1ITagc122tOk5kO0gl6sDCr7AwgBArH8HG3+C2GaDlSFc6Oude3XtPEHOmtOt+W7ml5ZXVtfx6YWNza3vH3t1rqCiRhNZJxCPZCrCinAla10xz2oolxWHAaTMYXk/85j2VikXiVo9i6oe4L1iPEayN1LGLnkjgJSw752eeZiFVyL1L0ckYwupDxy65jjsFXCQoIyWQodaxv7xuRJKQCk04VqqN3Fj7KZaaEU7HBS9RNMZkiPu0bajAZp+fTp8Yw0OjdGEvkqaEhlP190SKQ6VGYWA6Q6wHat6biP957UT3yn7KRJxoKshsUS/hUEdwkgjsMkmJ5iNDMJHM3ArJAEtMtMmtYEJA8y8vksaxg1wH3ZyWKldZHHlQBAfgCCBwASqgCmqgDgh4BM/gFbxZT9aL9W59zFpzVjazD/7A+vwBMiKV4Q==AAACBHicbVC7TsMwFHXKq5RXgLGLRYXEFMU8uyBVsHQsEn1ITagc122tOk5kO0gl6sDCr7AwgBArH8HG3+C2GaDlSFc6Oude3XtPEHOmtOt+W7ml5ZXVtfx6YWNza3vH3t1rqCiRhNZJxCPZCrCinAla10xz2oolxWHAaTMYXk/85j2VikXiVo9i6oe4L1iPEayN1LGLnkjgJSw752eeZiFVyL1L0ckYwupDxy65jjsFXCQoIyWQodaxv7xuRJKQCk04VqqN3Fj7KZaaEU7HBS9RNMZkiPu0bajAZp+fTp8Yw0OjdGEvkqaEhlP190SKQ6VGYWA6Q6wHat6biP957UT3yn7KRJxoKshsUS/hUEdwkgjsMkmJ5iNDMJHM3ArJAEtMtMmtYEJA8y8vksaxg1wH3ZyWKldZHHlQBAfgCCBwASqgCmqgDgh4BM/gFbxZT9aL9W59zFpzVjazD/7A+vwBMiKV4Q==AAACBHicbVC7TsMwFHXKq5RXgLGLRYXEFMU8uyBVsHQsEn1ITagc122tOk5kO0gl6sDCr7AwgBArH8HG3+C2GaDlSFc6Oude3XtPEHOmtOt+W7ml5ZXVtfx6YWNza3vH3t1rqCiRhNZJxCPZCrCinAla10xz2oolxWHAaTMYXk/85j2VikXiVo9i6oe4L1iPEayN1LGLnkjgJSw752eeZiFVyL1L0ckYwupDxy65jjsFXCQoIyWQodaxv7xuRJKQCk04VqqN3Fj7KZaaEU7HBS9RNMZkiPu0bajAZp+fTp8Yw0OjdGEvkqaEhlP190SKQ6VGYWA6Q6wHat6biP957UT3yn7KRJxoKshsUS/hUEdwkgjsMkmJ5iNDMJHM3ArJAEtMtMmtYEJA8y8vksaxg1wH3ZyWKldZHHlQBAfgCCBwASqgCmqgDgh4BM/gFbxZT9aL9W59zFpzVjazD/7A+vwBMiKV4Q==AAACBHicbVC7TsMwFHXKq5RXgLGLRYXEFMU8uyBVsHQsEn1ITagc122tOk5kO0gl6sDCr7AwgBArH8HG3+C2GaDlSFc6Oude3XtPEHOmtOt+W7ml5ZXVtfx6YWNza3vH3t1rqCiRhNZJxCPZCrCinAla10xz2oolxWHAaTMYXk/85j2VikXiVo9i6oe4L1iPEayN1LGLnkjgJSw752eeZiFVyL1L0ckYwupDxy65jjsFXCQoIyWQodaxv7xuRJKQCk04VqqN3Fj7KZaaEU7HBS9RNMZkiPu0bajAZp+fTp8Yw0OjdGEvkqaEhlP190SKQ6VGYWA6Q6wHat6biP957UT3yn7KRJxoKshsUS/hUEdwkgjsMkmJ5iNDMJHM3ArJAEtMtMmtYEJA8y8vksaxg1wH3ZyWKldZHHlQBAfgCCBwASqgCmqgDgh4BM/gFbxZT9aL9W59zFpzVjazD/7A+vwBMiKV4Q==

    � �✏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

  • Quantum SystemsElectrons

    Consider a H atom as example n=1,2,3,…

    The electronic energies and populations essentially remain constant due to this large gap. Thus, their contribution can be mostly ignored.

    ⌫ = 8.65⇥ 1013HzAAACBHicbVC7TsMwFHXKq5RXgLGLRYXEFMU8uyBVsHQsEn1ITagc122tOk5kO0gl6sDCr7AwgBArH8HG3+C2GaDlSFc6Oude3XtPEHOmtOt+W7ml5ZXVtfx6YWNza3vH3t1rqCiRhNZJxCPZCrCinAla10xz2oolxWHAaTMYXk/85j2VikXiVo9i6oe4L1iPEayN1LGLnkjgJSw752eeZiFVyL1L0ckYwupDxy65jjsFXCQoIyWQodaxv7xuRJKQCk04VqqN3Fj7KZaaEU7HBS9RNMZkiPu0bajAZp+fTp8Yw0OjdGEvkqaEhlP190SKQ6VGYWA6Q6wHat6biP957UT3yn7KRJxoKshsUS/hUEdwkgjsMkmJ5iNDMJHM3ArJAEtMtMmtYEJA8y8vksaxg1wH3ZyWKldZHHlQBAfgCCBwASqgCmqgDgh4BM/gFbxZT9aL9W59zFpzVjazD/7A+vwBMiKV4Q==AAACBHicbVC7TsMwFHXKq5RXgLGLRYXEFMU8uyBVsHQsEn1ITagc122tOk5kO0gl6sDCr7AwgBArH8HG3+C2GaDlSFc6Oude3XtPEHOmtOt+W7ml5ZXVtfx6YWNza3vH3t1rqCiRhNZJxCPZCrCinAla10xz2oolxWHAaTMYXk/85j2VikXiVo9i6oe4L1iPEayN1LGLnkjgJSw752eeZiFVyL1L0ckYwupDxy65jjsFXCQoIyWQodaxv7xuRJKQCk04VqqN3Fj7KZaaEU7HBS9RNMZkiPu0bajAZp+fTp8Yw0OjdGEvkqaEhlP190SKQ6VGYWA6Q6wHat6biP957UT3yn7KRJxoKshsUS/hUEdwkgjsMkmJ5iNDMJHM3ArJAEtMtMmtYEJA8y8vksaxg1wH3ZyWKldZHHlQBAfgCCBwASqgCmqgDgh4BM/gFbxZT9aL9W59zFpzVjazD/7A+vwBMiKV4Q==AAACBHicbVC7TsMwFHXKq5RXgLGLRYXEFMU8uyBVsHQsEn1ITagc122tOk5kO0gl6sDCr7AwgBArH8HG3+C2GaDlSFc6Oude3XtPEHOmtOt+W7ml5ZXVtfx6YWNza3vH3t1rqCiRhNZJxCPZCrCinAla10xz2oolxWHAaTMYXk/85j2VikXiVo9i6oe4L1iPEayN1LGLnkjgJSw752eeZiFVyL1L0ckYwupDxy65jjsFXCQoIyWQodaxv7xuRJKQCk04VqqN3Fj7KZaaEU7HBS9RNMZkiPu0bajAZp+fTp8Yw0OjdGEvkqaEhlP190SKQ6VGYWA6Q6wHat6biP957UT3yn7KRJxoKshsUS/hUEdwkgjsMkmJ5iNDMJHM3ArJAEtMtMmtYEJA8y8vksaxg1wH3ZyWKldZHHlQBAfgCCBwASqgCmqgDgh4BM/gFbxZT9aL9W59zFpzVjazD/7A+vwBMiKV4Q==AAACBHicbVC7TsMwFHXKq5RXgLGLRYXEFMU8uyBVsHQsEn1ITagc122tOk5kO0gl6sDCr7AwgBArH8HG3+C2GaDlSFc6Oude3XtPEHOmtOt+W7ml5ZXVtfx6YWNza3vH3t1rqCiRhNZJxCPZCrCinAla10xz2oolxWHAaTMYXk/85j2VikXiVo9i6oe4L1iPEayN1LGLnkjgJSw752eeZiFVyL1L0ckYwupDxy65jjsFXCQoIyWQodaxv7xuRJKQCk04VqqN3Fj7KZaaEU7HBS9RNMZkiPu0bajAZp+fTp8Yw0OjdGEvkqaEhlP190SKQ6VGYWA6Q6wHat6biP957UT3yn7KRJxoKshsUS/hUEdwkgjsMkmJ5iNDMJHM3ArJAEtMtMmtYEJA8y8vksaxg1wH3ZyWKldZHHlQBAfgCCBwASqgCmqgDgh4BM/gFbxZT9aL9W59zFpzVjazD/7A+vwBMiKV4Q==

    ✏n = �13.6

    n2eV

    AAACCnicbZC7SgNBFIZn4y3G26qlzWgQbAy7UdRGCNpYRjAXyMYwOzmbDJmdXWZmhbBsbeOr2FgoYusT2Pk2Ti6FJv4w8PGfczhzfj/mTGnH+bZyC4tLyyv51cLa+sbmlr29U1dRIinUaMQj2fSJAs4E1DTTHJqxBBL6HBr+4HpUbzyAVCwSd3oYQzskPcECRok2Vsfe9yBWjBsU+BIfe4EkNHVPSmdZKu7LGYZ6xy46JWcsPA/uFIpoqmrH/vK6EU1CEJpyolTLdWLdTonUjHLICl6iICZ0QHrQMihICKqdjk/J8KFxujiIpHlC47H7eyIloVLD0DedIdF9NVsbmf/VWokOLtopE3GiQdDJoiDhWEd4lAvuMglU86EBQiUzf8W0T0wa2qRXMCG4syfPQ71ccp2Se3tarFxN48ijPXSAjpCLzlEF3aAqqiGKHtEzekVv1pP1Yr1bH5PWnDWd2UV/ZH3+ADXtmUs=AAACCnicbZC7SgNBFIZn4y3G26qlzWgQbAy7UdRGCNpYRjAXyMYwOzmbDJmdXWZmhbBsbeOr2FgoYusT2Pk2Ti6FJv4w8PGfczhzfj/mTGnH+bZyC4tLyyv51cLa+sbmlr29U1dRIinUaMQj2fSJAs4E1DTTHJqxBBL6HBr+4HpUbzyAVCwSd3oYQzskPcECRok2Vsfe9yBWjBsU+BIfe4EkNHVPSmdZKu7LGYZ6xy46JWcsPA/uFIpoqmrH/vK6EU1CEJpyolTLdWLdTonUjHLICl6iICZ0QHrQMihICKqdjk/J8KFxujiIpHlC47H7eyIloVLD0DedIdF9NVsbmf/VWokOLtopE3GiQdDJoiDhWEd4lAvuMglU86EBQiUzf8W0T0wa2qRXMCG4syfPQ71ccp2Se3tarFxN48ijPXSAjpCLzlEF3aAqqiGKHtEzekVv1pP1Yr1bH5PWnDWd2UV/ZH3+ADXtmUs=AAACCnicbZC7SgNBFIZn4y3G26qlzWgQbAy7UdRGCNpYRjAXyMYwOzmbDJmdXWZmhbBsbeOr2FgoYusT2Pk2Ti6FJv4w8PGfczhzfj/mTGnH+bZyC4tLyyv51cLa+sbmlr29U1dRIinUaMQj2fSJAs4E1DTTHJqxBBL6HBr+4HpUbzyAVCwSd3oYQzskPcECRok2Vsfe9yBWjBsU+BIfe4EkNHVPSmdZKu7LGYZ6xy46JWcsPA/uFIpoqmrH/vK6EU1CEJpyolTLdWLdTonUjHLICl6iICZ0QHrQMihICKqdjk/J8KFxujiIpHlC47H7eyIloVLD0DedIdF9NVsbmf/VWokOLtopE3GiQdDJoiDhWEd4lAvuMglU86EBQiUzf8W0T0wa2qRXMCG4syfPQ71ccp2Se3tarFxN48ijPXSAjpCLzlEF3aAqqiGKHtEzekVv1pP1Yr1bH5PWnDWd2UV/ZH3+ADXtmUs=AAACCnicbZC7SgNBFIZn4y3G26qlzWgQbAy7UdRGCNpYRjAXyMYwOzmbDJmdXWZmhbBsbeOr2FgoYusT2Pk2Ti6FJv4w8PGfczhzfj/mTGnH+bZyC4tLyyv51cLa+sbmlr29U1dRIinUaMQj2fSJAs4E1DTTHJqxBBL6HBr+4HpUbzyAVCwSd3oYQzskPcECRok2Vsfe9yBWjBsU+BIfe4EkNHVPSmdZKu7LGYZ6xy46JWcsPA/uFIpoqmrH/vK6EU1CEJpyolTLdWLdTonUjHLICl6iICZ0QHrQMihICKqdjk/J8KFxujiIpHlC47H7eyIloVLD0DedIdF9NVsbmf/VWokOLtopE3GiQdDJoiDhWEd4lAvuMglU86EBQiUzf8W0T0wa2qRXMCG4syfPQ71ccp2Se3tarFxN48ijPXSAjpCLzlEF3aAqqiGKHtEzekVv1pP1Yr1bH5PWnDWd2UV/ZH3+ADXtmUs=

    �✏ = ✏2 � ✏1 =13.6⇥ 3

    4eV

    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

    ⇡ 1.63⇥ 10�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AAAB/XicbZBLSwMxFIXv+Ky1anXrwmAR3DhMFLRLwY24qmAf0BlLJs20oZmZkGTEMnTpxr/ixoUi/g13/hvTx0JbDwQ+zkm4uSeUgmvjed/O0vLK6tp6YaO4Wdra3invlho6zRRldZqKVLVCopngCasbbgRrScVIHArWDAdX47z5wJTmaXJnhpIFMeklPOKUGGt1ygc+kVKljwhh9/wM+YbHTCPs3ecnuDpCN51yxXO9idAi4BlUYKZap/zld1OaxSwxVBCt29iTJsiJMpwKNir6mWaS0AHpsbbFhNh5QT5ZZISOrNNFUarsSQyauL9f5CTWehiH9mZMTF/PZ2Pzv6ydmaga5DyRmWEJnQ6KMoFMisatoC5XjBoxtECo4vaviPaJItTY7oq2BDy/8iI0Tl3sufjWgwLswyEcA4YLuIRrqEEdKDzBC7zBu/PsvDof07qWnFlve/BHzucPaCCV4Q==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

    � �✏vibAAACAnicbVBNS8NAEN3Ur1q/op7Ey2IRPJVEBD0W9eCxgv2AJpTNdpIu3WzC7qZQQvHiX/HiQRGv/gpv/hu3bQ7a+mDg8d4MM/OClDOlHefbKq2srq1vlDcrW9s7u3v2/kFLJZmk0KQJT2QnIAo4E9DUTHPopBJIHHBoB8Obqd8egVQsEQ96nIIfk0iwkFGijdSzj7wowt4tcE2wB6li3Kj5iAWTnl11as4MeJm4BamiAo2e/eX1E5rFIDTlRKmu66Taz4nUjHKYVLxMQUrokETQNVSQGJSfz16Y4FOj9HGYSFNC45n6eyInsVLjODCdMdEDtehNxf+8bqbDKz9nIs00CDpfFGYc6wRP88B9JoFqPjaEUMnMrZgOiCRUm9QqJgR38eVl0jqvuU7Nvb+o1q+LOMroGJ2gM+SiS1RHd6iBmoiiR/SMXtGb9WS9WO/Wx7y1ZBUzh+gPrM8fIa6XQw==AAACAnicbVBNS8NAEN3Ur1q/op7Ey2IRPJVEBD0W9eCxgv2AJpTNdpIu3WzC7qZQQvHiX/HiQRGv/gpv/hu3bQ7a+mDg8d4MM/OClDOlHefbKq2srq1vlDcrW9s7u3v2/kFLJZmk0KQJT2QnIAo4E9DUTHPopBJIHHBoB8Obqd8egVQsEQ96nIIfk0iwkFGijdSzj7wowt4tcE2wB6li3Kj5iAWTnl11as4MeJm4BamiAo2e/eX1E5rFIDTlRKmu66Taz4nUjHKYVLxMQUrokETQNVSQGJSfz16Y4FOj9HGYSFNC45n6eyInsVLjODCdMdEDtehNxf+8bqbDKz9nIs00CDpfFGYc6wRP88B9JoFqPjaEUMnMrZgOiCRUm9QqJgR38eVl0jqvuU7Nvb+o1q+LOMroGJ2gM+SiS1RHd6iBmoiiR/SMXtGb9WS9WO/Wx7y1ZBUzh+gPrM8fIa6XQw==AAACAnicbVBNS8NAEN3Ur1q/op7Ey2IRPJVEBD0W9eCxgv2AJpTNdpIu3WzC7qZQQvHiX/HiQRGv/gpv/hu3bQ7a+mDg8d4MM/OClDOlHefbKq2srq1vlDcrW9s7u3v2/kFLJZmk0KQJT2QnIAo4E9DUTHPopBJIHHBoB8Obqd8egVQsEQ96nIIfk0iwkFGijdSzj7wowt4tcE2wB6li3Kj5iAWTnl11as4MeJm4BamiAo2e/eX1E5rFIDTlRKmu66Taz4nUjHKYVLxMQUrokETQNVSQGJSfz16Y4FOj9HGYSFNC45n6eyInsVLjODCdMdEDtehNxf+8bqbDKz9nIs00CDpfFGYc6wRP88B9JoFqPjaEUMnMrZgOiCRUm9QqJgR38eVl0jqvuU7Nvb+o1q+LOMroGJ2gM+SiS1RHd6iBmoiiR/SMXtGb9WS9WO/Wx7y1ZBUzh+gPrM8fIa6XQw==AAACAnicbVBNS8NAEN3Ur1q/op7Ey2IRPJVEBD0W9eCxgv2AJpTNdpIu3WzC7qZQQvHiX/HiQRGv/gpv/hu3bQ7a+mDg8d4MM/OClDOlHefbKq2srq1vlDcrW9s7u3v2/kFLJZmk0KQJT2QnIAo4E9DUTHPopBJIHHBoB8Obqd8egVQsEQ96nIIfk0iwkFGijdSzj7wowt4tcE2wB6li3Kj5iAWTnl11as4MeJm4BamiAo2e/eX1E5rFIDTlRKmu66Taz4nUjHKYVLxMQUrokETQNVSQGJSfz16Y4FOj9HGYSFNC45n6eyInsVLjODCdMdEDtehNxf+8bqbDKz9nIs00CDpfFGYc6wRP88B9JoFqPjaEUMnMrZgOiCRUm9QqJgR38eVl0jqvuU7Nvb+o1q+LOMroGJ2gM+SiS1RHd6iBmoiiR/SMXtGb9WS9WO/Wx7y1ZBUzh+gPrM8fIa6XQw==

  • Quantum SystemsMany-electron systems

    Consider a system of independent particles

    Thus, the microstate for a quantum system can be specified by specifying these quantum numbers (n1, n2, n3, …).

    Ĥ = ĥ1 + ĥ2 + ĥ3 + ...AAACGnicbZDLSsNAFIYn9VbrLerSzWARBCFkqqAboeimywr2Am0Ik+mkHTq5MDMRSshzuPFV3LhQxJ248W2cpkFq6w8DH/85hzPn92LOpLLtb6O0srq2vlHerGxt7+zumfsHbRklgtAWiXgkuh6WlLOQthRTnHZjQXHgcdrxxrfTeueBCsmi8F5NYuoEeBgynxGstOWaqD/CKm1k8BrmNMpcBM9+uTbH55oty3LNqm3ZueAyoAKqoFDTNT/7g4gkAQ0V4VjKHrJj5aRYKEY4zSr9RNIYkzEe0p7GEAdUOml+WgZPtDOAfiT0CxXM3fmJFAdSTgJPdwZYjeRibWr+V+slyr9yUhbGiaIhmS3yEw5VBKc5wQETlCg+0YCJYPqvkIywwETpNCs6BLR48jK0axayLXR3Ua3fFHGUwRE4BqcAgUtQBw3QBC1AwCN4Bq/gzXgyXox342PWWjKKmUPwR8bXDxJynlQ=AAACGnicbZDLSsNAFIYn9VbrLerSzWARBCFkqqAboeimywr2Am0Ik+mkHTq5MDMRSshzuPFV3LhQxJ248W2cpkFq6w8DH/85hzPn92LOpLLtb6O0srq2vlHerGxt7+zumfsHbRklgtAWiXgkuh6WlLOQthRTnHZjQXHgcdrxxrfTeueBCsmi8F5NYuoEeBgynxGstOWaqD/CKm1k8BrmNMpcBM9+uTbH55oty3LNqm3ZueAyoAKqoFDTNT/7g4gkAQ0V4VjKHrJj5aRYKEY4zSr9RNIYkzEe0p7GEAdUOml+WgZPtDOAfiT0CxXM3fmJFAdSTgJPdwZYjeRibWr+V+slyr9yUhbGiaIhmS3yEw5VBKc5wQETlCg+0YCJYPqvkIywwETpNCs6BLR48jK0axayLXR3Ua3fFHGUwRE4BqcAgUtQBw3QBC1AwCN4Bq/gzXgyXox342PWWjKKmUPwR8bXDxJynlQ=AAACGnicbZDLSsNAFIYn9VbrLerSzWARBCFkqqAboeimywr2Am0Ik+mkHTq5MDMRSshzuPFV3LhQxJ248W2cpkFq6w8DH/85hzPn92LOpLLtb6O0srq2vlHerGxt7+zumfsHbRklgtAWiXgkuh6WlLOQthRTnHZjQXHgcdrxxrfTeueBCsmi8F5NYuoEeBgynxGstOWaqD/CKm1k8BrmNMpcBM9+uTbH55oty3LNqm3ZueAyoAKqoFDTNT/7g4gkAQ0V4VjKHrJj5aRYKEY4zSr9RNIYkzEe0p7GEAdUOml+WgZPtDOAfiT0CxXM3fmJFAdSTgJPdwZYjeRibWr+V+slyr9yUhbGiaIhmS3yEw5VBKc5wQETlCg+0YCJYPqvkIywwETpNCs6BLR48jK0axayLXR3Ua3fFHGUwRE4BqcAgUtQBw3QBC1AwCN4Bq/gzXgyXox342PWWjKKmUPwR8bXDxJynlQ=AAACGnicbZDLSsNAFIYn9VbrLerSzWARBCFkqqAboeimywr2Am0Ik+mkHTq5MDMRSshzuPFV3LhQxJ248W2cpkFq6w8DH/85hzPn92LOpLLtb6O0srq2vlHerGxt7+zumfsHbRklgtAWiXgkuh6WlLOQthRTnHZjQXHgcdrxxrfTeueBCsmi8F5NYuoEeBgynxGstOWaqD/CKm1k8BrmNMpcBM9+uTbH55oty3LNqm3ZueAyoAKqoFDTNT/7g4gkAQ0V4VjKHrJj5aRYKEY4zSr9RNIYkzEe0p7GEAdUOml+WgZPtDOAfiT0CxXM3fmJFAdSTgJPdwZYjeRibWr+V+slyr9yUhbGiaIhmS3yEw5VBKc5wQETlCg+0YCJYPqvkIywwETpNCs6BLR48jK0axayLXR3Ua3fFHGUwRE4BqcAgUtQBw3QBC1AwCN4Bq/gzXgyXox342PWWjKKmUPwR8bXDxJynlQ=

    E(n1, n2, n3, ...) = ✏n1 + ✏n2 + ✏n3 + ...AAACNnicbZBNS8MwGMdTX+d8q3r0EhzCxFHaTdCLMBTBizDBvcBWSpplW1ialiQVRtmn8uLn8LaLB0W8+hFMtx7c5gMJf37/5yF5/n7EqFS2PTFWVtfWNzZzW/ntnd29ffPgsCHDWGBSxyELRctHkjDKSV1RxUgrEgQFPiNNf3ib+s1nIiQN+ZMaRcQNUJ/THsVIaeSZD3dF7jklyL1yelVKlmWdwWvYIZGkTHck2h7D8zlQXgSVFOhJzyzYlj0tuCycTBRAVjXPfOt0QxwHhCvMkJRtx46UmyChKGZknO/EkkQID1GftLXkKCDSTaZrj+GpJl3YC4U+XMEp/TuRoEDKUeDrzgCpgVz0Uvif145V78pNKI9iRTiePdSLGVQhTDOEXSoIVmykBcKC6r9CPEACYaWTzusQnMWVl0WjbDm25TxeFKo3WRw5cAxOQBE44BJUwT2ogTrA4AVMwAf4NF6Nd+PL+J61rhjZzBGYK+PnF9XBp9o=AAACNnicbZBNS8MwGMdTX+d8q3r0EhzCxFHaTdCLMBTBizDBvcBWSpplW1ialiQVRtmn8uLn8LaLB0W8+hFMtx7c5gMJf37/5yF5/n7EqFS2PTFWVtfWNzZzW/ntnd29ffPgsCHDWGBSxyELRctHkjDKSV1RxUgrEgQFPiNNf3ib+s1nIiQN+ZMaRcQNUJ/THsVIaeSZD3dF7jklyL1yelVKlmWdwWvYIZGkTHck2h7D8zlQXgSVFOhJzyzYlj0tuCycTBRAVjXPfOt0QxwHhCvMkJRtx46UmyChKGZknO/EkkQID1GftLXkKCDSTaZrj+GpJl3YC4U+XMEp/TuRoEDKUeDrzgCpgVz0Uvif145V78pNKI9iRTiePdSLGVQhTDOEXSoIVmykBcKC6r9CPEACYaWTzusQnMWVl0WjbDm25TxeFKo3WRw5cAxOQBE44BJUwT2ogTrA4AVMwAf4NF6Nd+PL+J61rhjZzBGYK+PnF9XBp9o=AAACNnicbZBNS8MwGMdTX+d8q3r0EhzCxFHaTdCLMBTBizDBvcBWSpplW1ialiQVRtmn8uLn8LaLB0W8+hFMtx7c5gMJf37/5yF5/n7EqFS2PTFWVtfWNzZzW/ntnd29ffPgsCHDWGBSxyELRctHkjDKSV1RxUgrEgQFPiNNf3ib+s1nIiQN+ZMaRcQNUJ/THsVIaeSZD3dF7jklyL1yelVKlmWdwWvYIZGkTHck2h7D8zlQXgSVFOhJzyzYlj0tuCycTBRAVjXPfOt0QxwHhCvMkJRtx46UmyChKGZknO/EkkQID1GftLXkKCDSTaZrj+GpJl3YC4U+XMEp/TuRoEDKUeDrzgCpgVz0Uvif145V78pNKI9iRTiePdSLGVQhTDOEXSoIVmykBcKC6r9CPEACYaWTzusQnMWVl0WjbDm25TxeFKo3WRw5cAxOQBE44BJUwT2ogTrA4AVMwAf4NF6Nd+PL+J61rhjZzBGYK+PnF9XBp9o=AAACNnicbZBNS8MwGMdTX+d8q3r0EhzCxFHaTdCLMBTBizDBvcBWSpplW1ialiQVRtmn8uLn8LaLB0W8+hFMtx7c5gMJf37/5yF5/n7EqFS2PTFWVtfWNzZzW/ntnd29ffPgsCHDWGBSxyELRctHkjDKSV1RxUgrEgQFPiNNf3ib+s1nIiQN+ZMaRcQNUJ/THsVIaeSZD3dF7jklyL1yelVKlmWdwWvYIZGkTHck2h7D8zlQXgSVFOhJzyzYlj0tuCycTBRAVjXPfOt0QxwHhCvMkJRtx46UmyChKGZknO/EkkQID1GftLXkKCDSTaZrj+GpJl3YC4U+XMEp/TuRoEDKUeDrzgCpgVz0Uvif145V78pNKI9iRTiePdSLGVQhTDOEXSoIVmykBcKC6r9CPEACYaWTzusQnMWVl0WjbDm25TxeFKo3WRw5cAxOQBE44BJUwT2ogTrA4AVMwAf4NF6Nd+PL+J61rhjZzBGYK+PnF9XBp9o=

    Separable Hamiltonian implies

    Where n’s are quantum numbers. In the present case, each quantum number corresponds to the level occupied by one of the N particles.