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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- 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.N = A + £ If Σ = 1; Λ2; then Ω - What are the state(s)?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.In the language of Problem 12C.5, show that if G(t) = (a cos ω1t + b cos ω2t)e−t/τ, then the spectrum consists of two lines with intensities proportional to a and b and located at ω = ω1 and ω2, respectively.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?
- Find 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 aA 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.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?
- 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?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.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.