6. (McQuarrie 5-26) Show that (x²) = [*° 42(x)*x²+2(x)dx = 5 h = 2 √μk for a harmonic oscillator. The square root of (x²) is the root-mean-square-displacement of the oscillator. 7. (McQuarrie 5-27) Show that (p) = 0 and {p²) = [ ¢2(x)*P²v2(x)dx = {2ħ√/µk for a harmonic oscillator.
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- Calculate the energies of the first four rotational levels of 1H127I free to rotate in three dimensions; use for its moment of inertia I = μR2, with μ = mHmI/(mH + mI) and R = 160 pm. Use integer relative atomic masses for this estimate.The energy levels (in cm-1) of diatomic anharmonic oscillator are described by the following expression: Ey = (v + 1/2 )we - (v + 1/2 )< Using this equation, derive expressions for the energies of transition Aɛ for the fundamental vibration and the 2nd harmonic vibration for a diatomic molecule.N = A + £ If Σ = 1; Λ2; then Ω - What are the state(s)?
- 2. A one-particle, one-dimensional system has the state function y=(sinat) ()* e/² + (cosat) (32/C) 1/4 2 1/4 e-x²/² е πρ >-x2/c2 xe where a is a constant and c = 2.000 Å. If the particle's position is measured at t = 0, estimate the probability that the result will lie between 2.000 Å and 2.001 Å.A diatomic molecule has a rotational constant of 8.0 cm-1, and vibrational frequency of 1200 cm 1. What is the energy of the state with v = 1 and j = 6 relative to the lowest energy state, E = E(v=0,j=0)? 0Find the oscillation amplitude of a diatomic molecule (CO) for n=1,2,3,4 & 10,when k=1926 N\m and reduced mass= 6.86 a
- Consider the diatomic molecule AB modeled as a rigid rotor (two masses separated by a fixed distance equal to the bond length of the molecule). The rotational constant of the diatomic AB is 25.5263 cm-1. (a) What is the difference in energy, expressed in wavenumbers, between the energy levels of AB with J = 10 and J = 6? (b) Consider now a diatomic A'B', for which the atomic masses are ma 0.85 mA and mB' 0.85 mB and for its bond length ra'B' = 0.913 rAB. What is the difference in energy, expressed in wavenumbers, between the energy levels of the A'B' molecule with J = 9 and J = 7?A one-particle, one-dimensional system has the state function Y = (sinat) (2)+(cosat) (32) 14 2-x² / c² xe where a is a constant and c = 2.000 Å. If the particle's position is measured at t = 0, estimate the probability that the result will lie between 2.000 Å and 2.001 Å.Consider the rotational spectrum of a linear molecule at 298 K with a moment of inertia of 1.23×10−461.23\times10^{-46}1.23×10−46 kg m2 . (a) What is the frequency for the transition from J = 2 to J = 3? (b) What is the most populated rotational level for this molecule? Would the transition in (a) give the most intense signal in the rotational spectrum?
- The rotational constant of 1H35CI is 317.8 GHz. What is the separation of the lines in its pure rotational spectrum (a) in gigahertz. (b) in reciprocal centimetres?J.G. Dojahn et al. (J. Phys. Chem. 100, 9649 (1996)) characterized the potential energy curves of the ground and electronic states of homonuclear diatomic halogen anions. These anions have a 2Σu+ ground state and 2Πg, 2Πu, and 2Σg+ excited states. To which of the excited states are electric-dipole transitions allowed from the ground state? Explain your conclusion.4) Calculate the energies E of the first three rotational levels (those giving the minimal energy) of 1H35Cl molecule that is free to rotate in three dimensions. For its moment of inertia I = μeffR2 use the effective mass with μeff = mH * mCl / (mH + mCl) and bond length R = 127 pm.