1. (10 points)            The     atomic unit     of         time     is         defined           as        ℏ/𝐸!.            Calculate         its        value   in         SI              Hint:    In         the       Energy            Conversion    Factors           table    below, “au”     means atomic unit     and      corresponds  to            EH.

 

  1. (10 points)            Write  the       electronic       Hamiltonian   operator         for       C            atom   in         atomic unit,     with     the       nucleus           placed at         the       origin  of         the

 

  1. (20 points)            Let’s    say      we       have    a          problem         that     we       don’t            know   how     to         solve   its        Schrödinger   equation                    We      decide to            use      linear  variational      principle         to         find     the       approximate  ground            state    energy.           We      have    chosen a          basis   set       that     contains          only     two            real     functions,       f1         and      f2.         The     two      functions        are      orthonormal:            <f1|f1> =          <f2|f2> =          1;         <f1|f2> =          <f2|f1> =          0.         Their   Hamiltonian            matrix satisfies          H11      =          H22      =          E          and      H12      =          H21      =          V,            and      V          takes   a          negative          value.  Solve   this      linear  variational      problem            and      express           the       approximate  ground           state    energy using   E          and      V.

 

  1. (20 points, bonus question)        Find    the       normalized   approximate  ground            state    wave   function          for       the       previous

 

  1. (20 points)            Let’s    say      an        unperturbed  system            has      only    two            eigen   states, the       ground           state    𝜓        and      the       excited            state    𝜓#,            with     the       eigen   energies         Eg        and      Ee.        The     two      states  are            nondegenerate,         e.,       Eg        ≠          Ee.        Given  a          perturbation  operator            𝐻&′,     without           any      derivation,      just      write   down  the       first     order  and            second            order  corrections    to         the       ground           state    energy,           and      the            first     order  correction      to         the       ground           state    wave   function          using   the            four     symbols.         Hint:    when   there   are      a          finite   number          of            unperturbed  eigen   states, the       infinite            sums   in         those   corrections    become            finite   sums   over    the       finite   number          of         unperturbed  eigen   states.

 

  1. (20 points)            The     selection         rule     for       the       transition       of         a          1-D            harmonic        oscillator        (HO)    is         v          –          v’          =          ±1.       v          and      v’            are      the       vibrational     quantum         numbers         of         two      1-D      HO       eigen            Please derive the       transition       selection         rule     for       a          2-D      HO            system.           The     2-D      HO       is         along   the       x          and      y          directions       and            the       eigen   states  can      be        expressed      as        |vxvy>,  which  is         just      a            product          of         the       respective      eigen   states  along   the       two      directions:            |vxvy>   =          |vx>|vy>.          Hint:    the       dipole moment          operator         is         a            vector and      in         2-D,     it          reads  µ!ˆ = i!qxˆ+!jqyˆ,         where q          is         the            charge of         the       particle           that     is         oscillating.      Calculate         the       transition            dipole moment          between         eigen   states  of         the       2-D      HO:      <vx’vy| µ!ˆ |vxvy>,            and      judge   when   it          is         nonzero.

 

 

Data Sheet

Physical Constants

Constant Symbol Value
Atomic mass unit Amu 1.66056 × 10–27 kg
Avogadro’s number N0 6.02205 × 1023 mol-1
Boltzmann constant kb 1.38066 × 10–23 J K-1
Electron rest mass me 9.10953 × 10–31 kg
Electron charge e 1.602191 × 10–19 C
Bohr radius a0 5.29177 × 10–11 m
Permittivity of vacuum ε0

4πε0

8.854188 × 10–12 C2 J–1 m-1

1.112650 × 10–10 C2 J–1 m-1

Planck constant h 6.62618 × 10–34 J s
Reduced Planck Constant ħ 1.0545718 × 10–34 J s
Speed of light in vacuum c 2.997925 × 108 m s–1

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