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Lecture 13 Space quantization and spin (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies. z-component angular momentum The z-component of the angular momentum operator depends only on φ ˆl = -i ¶ z ¶j The “particle on a sphere” wave function is the eigenfunction of the lz operator. lˆz ( , ) Nlml lml ( )lˆz ml ( ) ml ( , ) imlj e Angular momentum Total angular momentum The total angular momentum operator is 2 æ 1 ¶ 1 ¶ ¶ö ˆl 2 = - 2 total çè sin 2 q ¶j 2 + sinq ¶q sinq ¶q ÷ø 2 æ 1 ¶ 1 ¶ ¶ö ˆ H =+ sin q ÷ 2 ç 2 2 2mr è sin q ¶j sinq ¶q ¶q ø 2 2 l(l + 1) Hˆ Y = Y 2 2mr Total angular momentum The “particle on a sphere” wave function is also the eigenfunction of the l2total operator. ˆl 2 Y(q ,j ) = l(l +1) 2 Y(q ,j ) total Angular momentum squared 2 æ 1 ¶ 1 ¶ ¶ö ˆ H =+ sin q ÷ 2 ç 2 2 2mr è sin q ¶j sinq ¶q ¶q ø 2 2 l(l + 1) Hˆ Y = Y 2 2mr z-component and total angular momenta The “particle on a sphere” wave function, therefore, has well-defined total energy and total angular momentum: 2 l (l 1) El ltotal l (l 1) 2 2m r … and well-defined z-component energy and z-component angular momentum: 2 l 2 m Eml l z ml 2 2m r x- and y-component angular momenta This is because lz and ltotal commute. 2 élˆz , lˆtotal ù=0 ë û and ly operators depend on both θ and φ. They do not commute with lz or ltotal or H. lx ˆl = i æ sin j ¶ + cosj ¶ ö x çè ¶q tanq ¶j ÷ø ˆl = i æ - cosj ¶ + sin j ¶ ö y çè ¶q tanq ¶j ÷ø élˆz , lˆx ù = i ë û élˆy , lˆz ù = i ë û lˆy ¹ 0 lˆx ¹ 0 Uncertainty principle Observable Determined Simultaneously? Energy Yes x angular momentum y angular momentum Yes only for z No for x and y z angular momentum total angular momentum Yes Space quantization If all of x, y, and z components were known, we knew the angular momentum vector (length and direction) exactly and hence the circular trajectory perpendicular to it. The uncertainty in x and y components indicates the precise trajectory cannot be known. Space quantization We can only know the total ( l (l 1) ) and z component ( ml ). x and y components remain undetermined, so we do not know the precise trajectory. Space quantization Even when ml = l, zcomponent does not exhaust the total angular momentum because l (l 1) l . If it were not for “+1”, ml = l would leave nothing for x and y components and precisely determine all x, y, and z components simultaneously, violating uncertainty principle! Angular momentum as a magnet Rotational motion of a charged particle (such as an electron) gives rise to a magnetic field. Angular momentum is proportional to the magnetic moment. Applying an external magnetic field (along z axis) and measuring the interaction, one can determine the (z-component) angular momentum of an electron. Stern-Gerlach experiment The trajectories of electrons in an inhomogeneous magnetic field are bent. The trajectories are “quantized” – the proof of the quantization of angular momentum orientation, namely, space quantization. Summary The total angular momentum and only one of the three Cartesian components (z-component) can be determined exactly simultaneously. One component cannot exhaust the total momentum (because of “+1” in l(l +1) ). The angular momentum is quantized in both its length and orientation – it cannot point at any arbitrary direction (space quantization). Stern-Gerlach experiment Just two trajectories observed for electrons. This suggests l = ½ and m = ½ and –½. This cannot exactly be a particle on a sphere (where l and m must be full integers)? Spin It is the “spin” angular momentum of an electron. It has been discovered that the particle has intrinsic magnetic momentum. Its precise derivation is beyond quantum chemistrty. We can imagine the particle spinning and its associated angular momentum acts like a magnetic moment. Spin An electron has the spin quantum number s = ½ (corresponding to l of particle on a sphere). The total spin angular momentum is s(s 1) 3 . 4 The spin magnetic quantum number ms (corresponding to ml) can take s,…,–s (unit interval). For an electron ms = ½ (spin up or α spin) and –½ (spin down or β spin). A proton or neutron also has s = ½. A photon has s = 1. Indistinguishable particles Because of the uncertainty in the position and momentum of a particle, in a microscopic scale, two particles of the same kind (such as two electrons) nearby are indistinguishable. Indistinguishable particles The probability of finding particle 1 at position r1 and particle 2 at r2 is the same as that of finding particle 2 at r1 and particle 1 at r2 (otherwise we can distinguish the two). (r1 , r2 ) (r2 , r1 ) 2 2 There are at least two immediate possibilities (r1 , r2 ) (r2 , r1 ) (r1 , r2 ) (r2 , r1 ) Fermions Possibility 1: (r1 , r2 ) (r2 , r1 ) This is the case with particles having half integer spin quantum numbers (such as electrons). They are called fermions. Two fermions cannot occupy the same position in space (Pauli exclusion principle). They form matter. Y(r,r) = -Y(r,r) Y(r,r) = 0 Bosons Possibility 2: (r1 , r2 ) (r2 , r1 ) This is the case with particles having full integer spin quantum numbers (such as photons). They are called bosons. Two bosons can occupy the same position in space (e.g., photons can be superimposed to become more intense). They tend to mediate fundamental interactions. Y(r,r) = Y(r,r) Caveat The derivation of the concepts of spin, fermions, and bosons is beyond quantum chemistry. However, spin and the Pauli exclusion principle for electrons (as fermions) are a critical element of chemistry (whenever we have more than one electrons). We treat these as external postulates to quantum chemistry. Homework Challenge #3 Find the origin of spins. Explain why the spin quantum number of an electron, proton, and neutron is ½, whereas that of a photon is 1. Explain why particles with half integer spin quantum numbers are fermions and have wave functions that are anti-symmetric with respect to particle interchange. Explain why particles with full integer spin quantum numbers are bosons and their wave functions are symmetric with respect to interchange.