Prove that (x) = 0 for the ground state of a harmonic oscillator. b) Prove that (2²) 2 uk for the ground state of a harmonic oscillator. c) Use the result in (b) to calculate the root-mean-square amplitude of 1ªN2 in its ground state. Use k = 2260 N m-1 for 14N2. The mass of 14N = 14.003 amu. d) The value you obtained in (c) is a measure of how much the bond length in this molecule varies (about its equilibrium length) due to vibrational motion. Given that the equilibrium bond length of 14N2 is 109.77 pm, by how much does the bond length vary due to vibrational motion, as a percent of its equilibrium value?

Hydrogen gas can be placed inside a strong magnetic field B=12T. The energy of 1s electron in hydrogen atom is 13.6 eV ( 1eV= 1.6*10 J ). a) What is a wavelength of radiation corresponding to a transition between 2p and 1s levels when magnetic field is zero? b) What is a magnetic moment of the atom with its electron initially in s state and in p state? c) What is the wavelength change for the transition from p- to s- if magnetic field is turned on?

24. Consider a modified box potential with V(x) = V₁x, Vi(ar), x a Use the orthogonal trial function = c₁f₁+c₂f₂ with f₁ = √√sin (H) and f2 = √√ √√sin sin (2) to determine the upper bound to ground state energy.