Ch4 - SJP QM 3220 3D 1 Angular Momentum (warm-up for...

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SJP QM 3220 3D 1 Angular Momentum (warm-up for H-atom) Classically, angular momentum defined as (for a 1-particle system) r Note: L defined w.r.t. an origin of coords. (In QM, the operator corresponding to L x is etc. , , z i p p z p y L z y z x = - = ˆ ˆ ˆ ˆ  according to prescription of Postulate 2, part 3.) Classically, torque defined as , F r × τ and dt L d = (rotational version of a m = F ) If the force is radial (central force), then . const L F r = = × = 0 H-atom: In a multi-particle system, total average momentum: = i i L L ˆ tot is conserved for system isolated from external torques. sum over particles Internal torques can cause exchange of average momentum among particles, but tot L remains constant. In classical and quantum mechanics, only 4 things are conserved: energy linear momentum angular momentum electric charge Page H-1 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 z y x p p p z y x z y x p r L ˆ ˆ ˆ = × O x y m v m p = ) ( ) ( ˆ ) ( ˆ z y z x y z yp xp z xp zp y zp yp x L - + - + - = + - proton at origin electron (Coulomb force) r 2 2 - F r ke r ˆ ˆ =
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SJP QM 3220 3D 1 Back to QM. Define vector operator L ˆ operator unit vector z L y L x L L z y x ˆ ˆ ˆ ˆ ˆ ˆ ˆ + + = Recall [ ] Q i dt Q d ˆ , ˆ H = = L i dt L d ˆ , ˆ ˆ H + < + < y dt L d x dt L d y x ˆ ˆ Claim: for a central force such as in H-atom [ ] 0 then 2 = - = = L r ke r V V ˆ , , ) ( ˆ H (will show this later) This implies 0 = dt L d (just like in classical mechanics) Angular momentum of electron is H-atom is constant, so long as it does not absorb or emit photon. Throughout present discussion, we ignore interaction of H-atom w/photons. Will show that for H-atom or for any atom, molecule, solid – any collection of atoms – the angular momentum is quantized in units of ħ . | | L can only change by integer number of ħ 's. Claim: [ ] z y x L i L L ˆ ˆ , ˆ = and (i, j, k cyclic: x y z or y z x or z x y ) Page H-2 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] = × = = = = = = r r k p r p rp L L L L ) (since Note of Units , [ ] k j i L i L L ˆ ˆ , ˆ =
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SJP QM 3220 3D 1 To prove, need two very useful identities: [ ] [ ] [ ] [ ] [ ] [ ] B C A C B A C AB C B C A C B A , , , , , , + = + = + Proof: [ ] [ ] = - - = z x y z y x xp zp zp yp L L , , z x y L i yp xp i = - + = ) ( (Have used [ ] [ ] [ ] [ ] etc. 0 0 x, 0 , , , , , , , = = = = y x y x p p p y x i p x I'm dropping the ˆ over operators when no danger of confusion. Since [L x , L y ] ≠ 0, cannot have simultaneous eigenstates of . and y x L L ˆ ˆ However, 2 2 2 2 z y x L L L L L L + + = = does commute with L z . Claim: [ ] 0 2 = z L L , , i = x, y, or z Proof: [ ] [ ] [ ] [ ]    0 2 2 2 2 z z z y z x z L L L L L L L L , , , , + + = [ ] [ ] [ ] [ ] y x z y x z y y x y z x y z x x L L i L L L i L L L L L i L L L i L L L + + + + - + - = , , , , = 0 (Note cancellations) [L 2 , L z ] = 0 => can have simultaneous eigenstates of i) any (or 2 2 i z L L L L ˆ , ˆ ˆ , ˆ Page H-3 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 [ ] [ ] [ ] [ ] [ ] [ ] = + + - - -  
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This note was uploaded on 02/27/2012 for the course PHYSICS 3220 taught by Professor Stevepollock during the Fall '08 term at Colorado.

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Ch4 - SJP QM 3220 3D 1 Angular Momentum (warm-up for...

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