A2. Angular momentum in quantum mechanics is given by Ĺ = Îçi+Îyj+Î₂k with componentsÎx = ÛÎz – ZÎy, Îy = ÊÎx − ÎÔz, Îz = ÎÎy – ÎÎx.(a) Use the known commutation rules for â, ŷ, ê, êr, Ây, and ôz to show that [Îy, Î₂] = iħÎx.1 32√ 2πsin(0) exp(-io), where and o(b) Consider the spherical harmonic Y₁,-1(0, 0) =are the polar and azimuthal angles, respectively.(i) Express Y₁,-1 in terms of cartesian coordinates.(ii) Using this expression in cartesian coordinates, show that Y₁,-1 is an eigenfunctionof Î₂.(c) A wavefunction of a particle moving in three spatial dimensions is given as(x, y, z) = cz exp[-√√x² + y² + 2²/(2a)],where c and ao are constants.Determine the constant ao so that this wavefunction solves the stationary Schrödingerequation for the potential energyV(x, y, z)e²4πεονη2 + y? +

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c) The Bohr model of the atom postulated electrons orbiting around the nucleus in stable orbits. De Broglie explained what orbits could exist by postulating that electrons (and any- thing else) with momentum p have an associated wavelength λ, given by λ=h/p where h is Planck's constant. i) For an electron orbiting around a proton (the Bohr model), equating the centripetal force with the Coulomb force gives the expression v² = e²/(4πεmer). Calculate the speed of an electron orbiting at the Bohr radius, ˜Â = 0.053 nm. ii) Calculate the momenta and the de Broglie wavelengths of the electron of part (i) and of a bird (a racing pigeon) that weighs 0.350 kg and flies at 100 km per hour. iii) Compare the wavelength for the electron that you obtain in (ii) with the circumference of the orbit. Comment on this comparison. Explain briefly what it implies about the other possible orbits of the Bohr model and how the higher orbits might be predicted.
i know the answer is NOT 9.42(10^-21)
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