- (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.
- (10 points) Write the electronic Hamiltonian operator for C atom in atomic unit, with the nucleus placed at the origin of the
- (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.
- (20 points, bonus question) Find the normalized approximate ground state wave function for the previous
- (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.
- (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.
|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
|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|