AEP110_notes_pt2 - AEP 110 Lecture Notes Part II Alex Gaeta...

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1 Wave Nature of Particles We have seen that light can behave as a particle or as a wave. Interestingly particles of matter can also behave as waves. This wave nature is most evident with fundamental particles that are confined to a small region of space, for example, electrons bound to a nucleus. Earlier we introduced the deBroglie relation which is true not only for photons but also for any particle with a mass m . For a particle of mass m , the momentum p = mv where v is the velocity, and the kinetic energy of the particle is given by Thus, Let us calculate what is the wavelength of a E = 10 eV electron. Note that , which is the approximate size of an atom. In order for the atom to exist it must have some kinetic energy (i.e., be orbiting the nucleus) otherwise it would collapse onto the proton. The total energy E of the electron is the sum of the kinetic energy E k and potential energy E p where angular momentum minus sign attractive force p h = λ L m r = ν E E E E m L mr E e r k p k P o = + = = = - 1 2 2 1 4 2 2 2 2 ν πε ε o = × - 8 85 10 12 2 2 . C Nm E m p m = = 1 2 2 2 2 ν h m 2 2 2 150 = eV Å λ = ( 29 150 1 10 4 eV Å eV Å ~ λ π / ~ . 2 0 6 Å λ = = = h p h mE h m E 2 2 1 2 AEP 110 Lecture Notes: Part II Alex Gaeta 1 2 / 3/07
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2 Neils Bohr applied the deBroglie wave theory to the electron by arguing that the orbits of the electron had to be made up of an integral number of wavelengths, that is, these were the allowed modes of the system. As a result, the circumference is given by where This means that the angular momentum L is given by Let us now consider the simplest confinement is one dimension If we assume that the electron is confined to a potential well of width 2 a , there are only certain modes or states of the electron that can exist in the well. As in the case of the optical cavity, the boundary condition will only permit certain wavelengths to exist, and thus for bound electrons only discrete (i.e., quantized) energy levels exist. 2 π λ r n = λ = h p = = = n h p r n h p n p 2 π h h = h 2 π L m r pr n = = = ν h electron is free E x ground state excited state wavefunctions n =2 n =1 -a a Quantized in units of h !
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3 Hydrogen atom: 3-D potential well Let’s now consider the simplest atom which consists of a nucleus (i.e., a proton) with a charge + e and a negatively charged - e electron which orbits the nucleus. A plot of E k and E p suggests that under certain conditions a minimum exists in the total energy E which would allow the atom to be in a stable configuration. To find the minumum energy, we set dE/dr = 0, so that + proton + e electron - e r r a o energy E E r p α 1 dE dr L mr e r o = - + = 2 3 2 2 1 4 0 πε - + ( 29 = = - 1 2 0 1 2 r E E E E K p K p L mr e r o 2 2 2 2 1 8 min min = πε r me L me n o o min = = ( 29 4 4 2 2 2 2 πε πε h = = 4 2 2 2 2 2 2 πε ε π o o me n h me n h E r k α 1 2 Discrete orbits!
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4 Thus, the radius has only discrete values and accordingly only certain orbits are possible.
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