Question

3 (b) The energy of a Bohr atom in the n-th excited state is given by the formula E--a2mc2 2,7, where α-e2/(4πέρ,10hc)-1 /137, m is the electron mass and e denotes the electron electric charge. i) Why is the total energy negative? Explain briefly your answer. ii) What is the radius of the electron in the n-th excited state in the Bohr atom? To answer that correctly follow the next steps Use Bohr's angular momentum quantization principle to obtain an expression for the velocity of a electron as a function of the orbit radiu:s Write down the energy of the electron as a sum of its kinetic energy and its potential energy in the electric field of the nucleus. Use the fact that kcehac where kc is the Coulomb constant. 21 Substitute in this expression the velocity and solve for the radius to show that it iii) Suppose that you want to observe the electron orbit using light. What would be the energy of the photon in order to make such an observation? Do you think that you can really use this procedure to observe the electron trajectory? Explain briefly why. Suppose that the atoms are in a bottle in the n transition n 2 → n-1 happens 2 excited state and the iv) What is the wavelength of the emitted photons? v) The atoms in the bottle are heated and they start moving in random directions Will that change the observed wavelength of the emitted photon? Explain qual itatively without the use of equations. [2 vi) Assume that the state at the n-2 excited state has a lifetime of 10 ns before the transition to n-1. Calculate the relative change of the observed wavelength of the emitted photon will change (in %) and cornrnent on how easy it is to measure this change 4πεομ。hc = 1 / 137, c = 197 eV nmi You may want to use the fine structure constant, or the electron inass m 0.511 MeV/c21 l α

Answer 1

l37 1Teo ㅻ.hc ue Eno, aqura mn "(우)-(뿌 ) ho 13.6 A he 13.6 3 1- 13-61 큽 ]"뜻 26 = 79.2 X10 2