4. Using the dipole approximation, derive the selection rules for the harmonic oscillator and the...
The vibrational absorption spectrum of a diatomic molecule in the harmonic oscillator approximation consists of just one line whose frequency is given by, ν = 1 k . The bond 2π μ length of 12C14N is 117 pm and the force constant is 1630 N m-1. Predict the vibration-rotation spectrum of 12C14N within the harmonic oscillator rigid rotor approximation
a) Describe and sketch the vibrational energy levels observed for diatomic molecules in the harmonic oscillator approximation, using an appropriate formula to support your answer (4 marks) b) State the selection rules for IR transitions in diatomic molecules. (2 marks) c) Briefly explain the implications of anharmonicity for vibrational spectra, with particular reference to the selection rules for diatomic molecules, and the resultant energy levels and spectra observed. (3 marks) a) Describe and sketch the vibrational energy levels observed for...
I. Assuming that it acts as a rigid rotor, harmonic oscillator and using the molecular constants for ,C-O given below, calculate the wavenumbers of the first transitions in both the P and the R branch of the fundamental vibrational transition, the first overtone and the first hot band B 1.931 cm vo 2169.81 cm1
4 A nonlinear oscillator Consider a perturbed harmonic oscillator. Using x p2 H + ke? 2 + ex4 2m 1. write this Hamiltonian in terms of â and at 2. At what frequency or frequencies could this system absorb radiation if € = 0, i.e. the oscillator is unperturbed 1 3. Qualitatively, what do the states look like for the perturbed Hamiltonian? Write the new states as a sum of the unperturbed states, without worrying too much about the amplitude...
4- FOR a Quartun harmonic oscillator OF MASS M, Show That The FUNCTION f(x)= x ě * 2 is EIGENFUNCTION Of The Hamiltonian. Give The genualue, Alue. x= (mk) Esln+ 1 l hv 2 -- For The 37 Excited STATE of the RiGiD ROTOR calculate the energy, the Angular momentul & Lz .
Derive the selection rules for vibrational transitions that result from the first three terms in the electronic dipole moment operator M(R):
Derive the selection rules for vibrational transitions that result from the first three terms in the electronic dipole moment operator M(R):
4. (20 points) Harmonic Oscillator The ground state wave function of a simple harmonic oscillator is (a) = Ae-42", where a = (a) Using the normalization condition, obtain the constant A. (b) Find (c), (), and Az, using the result of A obtained in (a). Again, A.= V(32) - (2) (c) Find (p) and Ap. For the latter, you need to evaluate (p). Hint: For a harmonic oscillator, the time-averaged kinetic energy is equal to the time-averaged potential energy, and...
Using the analytical technique (brute force method), derive the ground state wavefunctions of harmonic oscillator. Explain physics and mathematics of each and every step of your derivations including all the details in addition to those in the textbook. 6.
1. Consider a harmonic oscillator sitting in the ground state with a given spring constant ko m were is constant). We want to change the system to raise the constant to 4ko. [N.B. You will have to use the equation versions of the eigenstates for this question since the system is changing a) Use the ideal instantaneous sudden approximation to find the probability that the system stays in the ground state. Does this approximation include selection rules? b) Assuming that...