LANDSCAPE ECOLOGY ESM 215
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This 4 page Class Notes was uploaded by Weston Batz on Thursday October 22, 2015. The Class Notes belongs to ESM 215 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 43 views. For similar materials see /class/226965/esm-215-university-of-california-santa-barbara in Environmental Science and Resource Management at University of California Santa Barbara.
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Date Created: 10/22/15
ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 ll Quick Review 0fAn2ular in 0mmme For a central potential energy ie one that depends only on the radial variable r in spherical coordinates the solution to the Schrodinger equation can always be written as a product w R r Y 94 The function Y is the angular momentum eigenfunction requiring two quantum numbers per eigenvalue corresponding to the two degrees of angular freedom at a given r The first one is defined by L2Y 1h2Y 1 where L is the angular momentum operator and i 0 l 2 and the 2quotd quantum number is defined by 4th 2 where A z l and 52 0 l 2 up to a maximum magnitude of In other words 5 defines the total angular momentum and z defines its projection on the z axis Clearly the number of z values for each value of i is NI 2 1 3 Hence it is convenient to think of the orbital angular momentum as the vector lying on the cone shown in Fig1 It has a fixed length of l h and a xed projection on the z axis of zh but a random orientation on the cone consistent with the statistical nature of quantum mechanics All of the possible degenerate eigenstates of L are then represented by Z 1 different cones each having a projection on the z axis given by one of the allowed values of z ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 M i E y lh Fig 1 Threedimensional model of angular momentum vector of electron consistent with rules of quantum mehanics It is a fundamental fact of quantum mechanics that the quantization expressions 1 and 2 remain valid in form for any spin angular momentum S if we replace L by S LZ by SZ and i by s so that the operator S2 has the eigenvalue SS 12 and the operator SZ has the eigenvalue szh The possible increments of s and sZ are de ned by As l Asz l and the maximum magnitude of sZ is s so that there are NS 2sl values values of sZ for each s The key difference between spin and angular momentum is that s can be either integral or halfintegral As we saw in 215A this apparently trivial distinction is the means for classifying all know particles in nature not just electrons Integral spin means the particles called bosons can occupy the same spacespin state and must have symmetric with respect to particle label exchange total wave functions and halfintegral spin means that the particles called fermions can not occupy the same spacespin state and must have an antisymmetric total wave function It is remarkable that the quantization expressions 1 to 3 remain valid in form for a total angular momentum J if we replace L by J and LZ by JZ so that J2 has eigenvalue 77l 12 and J2 has eigenvalue 72h Again the increments are given by A7 l Ayz 1 so that the maximum magnitude of yz is y and there are 2yl values values of yz for each y And just like spin angular momentum neither 7 or yz are necessarily integral This makes total angular momentum difficult in general but relatively simple if the particles in question are electrons and there is no interaction between their orbital and ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 spin operators1 In that case we utilize the fact that the spin for each individual electron is strictly binary spin up or down along an arbitrarily chosen axis and has no component on any other axis We are then free and wise to choose this axis to coincide with the z axis of the angular momentum coordinate system for that electron In this case there are just two possible total angular momentum magnitudes for that electron l lJl IL S and 2 lJl IL S In other words the two possible eigenvalues have 7 3 12 and v 5 7 12 Fortunately the same reasoning applies to a system of electrons such as all the electrons occupying a shell in an atom2 It is simply a binary addition process Each electron contributes two possible total momentum eigenvalues corresponding to the two possible magnitudes 71 51 s1 51 12 and 71 51 s1 51 7 12 When combined with the two possible total angular momentum values for the second electron we get by combinatorial reasoning 2quot unique total momentum total angular momentum quantum numbers Stated more generally when a system eg one electron having total momentum quantum number 71 is combined with another noninteracting system having total momentum quantum number 2 the resulting total angular momentum quantum number has a maximum value of 1 y and a minimum value of 71 7 M This generalization is a very important result of quantum mechanics called the angular momentum addition theorem It works for arbitrary particles and integral or halfintegral values of y In this book we will only use it for electrons Energy in Angular Momentum Given the quantum picture of total angular momentum J the magneticdipole potential energy is UPE m 39 310ml quBJ39B yyglLlBBl L al 39 4 1 An approximation known in atomic physics as the LS approximation 2 For an excellent introduction to shell theory and other aspects of atomic physics see R Eisberg and R Resnick Quantum Physics Wiley New York 1974 ECEZlSBMaterials206B Fundamentals of Solids for Electronics ER Brown Spring 2008 In other words for J pointed anywhere in the upper hemisphere of Fig l the polar axis now being de ned parallel to Blow the potential energy is positive And for J pointed anywhere in the lower hemisphere the potential energy is negative According to quantum mechanics jg must have 2 1 equally spaced values corresponding to 2 1 equally spaced energy levels For example if J 32 then 3 l l 3 7j5 5 5 59 5 representing four equally spaced quantum levels Since spins are hidden variables within atoms and atoms are generally distinguishable we can apply the Boltzmann statistics to determine the microscopic magnetic moment and related macroscopic quantities3 The probability of a magnetic moment being aligned along B is 2J1 2J1 Z mjexpUjkBT Z gmejzexpDJygBkBT j j lt m gt 2J1 2J1 Z exp UjkBT exp y uBBkBT J 6 This is a ratio of partial sums that can be shown to be llt m gtl gJygB x 7 where B J is called the Brillouin function and X is given by BJXE COth w icoth i 8 2 2 J 2 where x gJyBBkBT 9 3 A fact that seems peculiar at first until one recalls that there is nothing inherently classical or quantum mechanical about the Boltzmann statistics
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