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on . Using Planck’s radiation formula show that the maximum of energy density occurs at Amaz = b/T with T being the temperature and 64 is a constant. . According to the classical model of the hydrogen atom (Rutherford model), an electron moving in a circular orbit of radius 0.053 nm around a proton fixed at the center is unstable, and the electron should eventually collapse into the proton. From classical electromagnetic theory, an accelerating charge e radiates energy E given by dE 22 4 dad 3 dmege? where a is the magnitude of the acceleration of the charge and c is the speed of light. Estimate how long it. would take for the electron in the classical model of the hydrogen atom to collapse to the proton. . In Rutherford scattering, the distance of closest approach ryyin is given by 2Ze? Thin = 7—p 7 4regE where F is the incident energy of the a particles and Z is the atomic number of the target. Taking E = 5.3 MeV and Z = 29 for copper, find rmin- . (a) Consider a quantum oscillator inside a cavity wall (blackbody radiation). If it is vibrating with a frequency of 5.0 x 10'4 Hz then calculate the spacing between the energy levels of the oscillator. Note that the spacing is uniform for the quantum oscillator. What about the spacing of energy levels of hydrogen atom? (b) Next consider the vibration of a classical oscillator consisting of mass m = | kg and a spring of spring constant & = 1000 N/m with amplitude 0.1 m. Find the energy of the classical oscillator and compare it with the energy spacing obtained for the classical oscillator assuming energy quantization as in (a). . Consider a simple one-dimensional harmonic oscillator of frequency w. Classically, the energy of the oscillator can be equal to zero. Using uncertainty principle estimate the minimum energy of the quantum mechanical oscillator. . Inthe Bohr model of atom the electron is moving in a circle of radius r around the nucleus. (i) Use the uncertainty principle to estimate (a) the radius r and (b) the ground state energy of the hydrogen atom. . Consider a free particle with a Gaussian wave packet 4)(2,¢) given by 1 ee ‘i (x,t) = —— h)etlk—“0 dk ven= = | dhe where @(k) is the amplitude of the wave packet and is given by ok) = Aelia" (k-ko)?*1/4 with A being a normalization constant. (a) Find the normalization constant A. (b) Show that the width of the Gaussian wave packet A(t) grows with time as a Ane? A(t) = Sy1+ 5 Does it satisfy the uncertainty principle? 
