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The first water spectra

The first water spectra. CHEM 991J Lecture 1 8/22/2011: A brief history of spin. The first distance measurement. CHEM 991J Lecture 1 8/22/2011: A brief history of spin. 2. Angular momentum and the uncertainty principle. L = r × p. L is a constant of the motion.

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The first water spectra

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  1. The first water spectra CHEM 991J Lecture 1 8/22/2011: A brief history of spin

  2. The first distance measurement CHEM 991J Lecture 1 8/22/2011: A brief history of spin 2

  3. Angular momentum and the uncertainty principle L = r × p L is a constant of the motion Lz = rxpy – rypxLx = rypz – rzpy Ly = rzpx – rxpz The units of L are m × kg × m/s = kg m2/s = J.s The uncertainty principle [rx, px] = rxpx – pxrx = [ry, py] = [rz,pz] = iℏ [ri, pj] = 0; i ≠ j CHEM 991J Lecture 2 8/24/2011: Angular momentum Let’s extend this to angular momentum. [Lx, Ly] = LxLy – LyLx = (ry pz – rz py)(rz px – rx pz) – (rz px – rx pz)(ry pz – rz py) = rypzrzpx – rzpyrzpx – rypzrxpz + rzpyrxpz –rzpxrypz + rxpzrypz + rzpxrz py – rxpzrzpy = rypxpzrz– rz2pxpy – rxrypz2 + rxpyrzpz –rypxrzpz + rxrypz2+rz2pxpy –rxpypzrz = (rxpy–rypx)(rzpz – pzrz) = Lz × iℏ = iℏLz

  4. Properties of L [Ly, Lz] = iℏLx [Lz, Lx] = iℏLy [Li, Lj] = iεijkℏLk countercyclic permutation cyclic permutation εijk is the so called antisymmetric unit tensor εijk = 1 if ijk are in normal order εijk = –1 if they are in reverse order εijk = 0 if any of the three are equal. CHEM 991J Lecture 1 8/22/2011: A brief history of spin Exercise:provetheelegantexpressionL×L=iℏL L2 = L.L = Lx2 + Ly2+ Lz2 [L2, Lz] = [Lx2, Lz] + [Ly2, Lz] + [Lz2, Lz] = [Lx, Lz]Lx + Lx[Lx, Lz] + [Ly, Lz]Ly + Ly[Ly, Lz] + 0 = iℏ(–LyLx – LxLy + LxLy + LyLx) = 0 By symmetry the same must be true for x and y.

  5. Angular momentum and the magnetic dipole Two noncommuting functions cannot share an eigenstate unless one of their eigenvalues are zero. So we can simultaneously measure eigenvalues of L2 and Li, but not Ljand Lk. Usually we measure eigenvalues of L2 and Lz. For an electron: g is dimensionless g = 2 in regular quantum mechanics g = 2.002319 with quantum electrodynamics corrections μ =(–gμB/ℏ)L For a nucleus: μ =(gNμN/ℏ)L CHEM 991J Lecture 1 8/22/2011: A brief history of spin The energy of a nuclear spin in a magnetic field is given by E = –μ.B = –(gNμN/ℏ)L.B SI units: 1 T = 1 kg C–1 s–1 μN = e/2mPℏ = 5.05078324(13) × 10–27 J T–1

  6. gyromagnetic ratio If we make the assumption that the field is along the z direction, and we pull ℏ over to the left, we get ω = E/ℏ = –gNμNLzBz/ℏ2 2|ω| / Bz is the so-called gyromagnetic ratio CHEM 991J Lecture 1 8/22/2011: A brief history of spin

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