Magnetism 1

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$diffraction from single powder grains - Max Planck Institute for Solid ...$

In addition to the orbital magnetic momentum there is the magnetic momentum related to the spin of the electron. → derived from a fully relativistic treatment! → electron has an intrinsic angular momentum with quantization, spin ½ . 2 Sˆχs( s 1) χ z = + χ = spin part of the electron wave function or or , ( ) ( ) nlm r S χ + ½ −½ Ψ = Ψ orbital part S χ = m χ m = ± ½ s s 2 2 3 2 S χ = S( S+ 1) χ = χ 4 ⇒ additional Zeeman term para =− µ ( 2ˆ) B L+ s H para Ψ =+ µ ( m + 2m ) Ψ B l s spin part 2 = rel. effect ms = ± 1/2 Actually the spin is not represented by a vector whose components are scalars but rather by 2x2 matrices ⇒ the “Pauli-spin” matrices. ⎛0 ˆ σ x = ⎜ ⎝1 1⎞ ⎟, 0⎠ ⎛0 σ y = ⎜ ⎝i −i ⎞ ⎟, 0 ⎠ ⎛1 σz = ⎜ ⎝0 0 ⎞ ⎟, −1⎠ ˆ σ = ( σx, σ y, σ z) be a a vector ⎛ a3 ˆ σ ⋅ a =⎜ ⎝a1+ ia2 a1−ia2⎞ ⎟ −a3 ⎠ such matrices can be multiplied together leading to 2 ˆ σ ⋅ a σb = a⋅ b + iσ a× b and σa = a show in exercise ( ) ( ) ( ) ( ) 2 The spin angular momentum is determined as 1 sˆ = ˆ σ 2 ⎛0 1⎞ ⎛0 −i ⎞ ⎛1 0 ⎞ Sx = ⎜ ⎟, Sy = ⎜ ⎟, Sz = ⎜ ⎟ 2⎝1 0⎠ 2⎝i0 ⎠ 2⎝0 −1⎠ only Sz is diagonal ↑z ⎛1⎞ = ⎜ ⎟ ⎝0⎠ Sˆ z ↑z 1 = 2 ↑ ↓z ⎛0⎞ = ⎜ ⎟ ⎝1⎠ ˆ 1 Sz ↓ = − z ↓ 2 ms - 6 -
The eigenstates concerning spin parallel to x-axis are ↑x = 1 ⎛1⎞ ⎜ ⎟ 2 ⎝1⎠ ↓x = 1 2 ⎛ 1 ⎞ ⎜ ⎟ ⎝−1⎠ ↑y = 1 ⎛1⎞ ⎜ ⎟ 2 ⎝i⎠ ↓y = 1 ⎛ 1 ⎞ ⎜ ⎟ 2 ⎝−i⎠ ⎛1⎞ ˆ σ x ⎜ ⎟ ⎝1⎠ ⎛0 = ⎜ ⎝1 1⎞ ⎛1⎞ ⎟ ⎜ ⎟ 0⎠ ⎝1⎠ ⎛1⎞ = ⎜ ⎟ ⎝1⎠ ⎛ 1 ⎞ ⎛0 ˆ σ x′ ⎜ ⎟= ⎜ ⎝−1⎠ ⎝1 1⎞ ⎛ 1 ⎞ ⎟ ⎜ ⎟ 0⎠ ⎝− 1⎠ ⎛−1⎞ = ⎜ ⎟ ⎝+ 1⎠ ⎛ 1 ⎞ = ( −1) ⎜ ⎟ ⎝− 1⎠ ⎛1⎞ ˆ σ y ⎜ ⎟ ⎝i⎠ ⎛0 = ⎜ ⎝i −i ⎞ ⎛1⎞ ⎟ ⎜ ⎟ 0 ⎠ ⎝i⎠ ⎛1⎞ = ⎜ ⎟ ⎝i⎠ ⎛ 1 ⎞ ⎛0 σ y ⎜ ⎟ = ⎜ ⎝−i⎠ ⎝i −i⎞ ⎛ 1 ⎞ ⎟ ⎜ ⎟ 0 ⎠ ⎝−i⎠ ⎛−1⎞ = ⎜ ⎟ ⎝ i ⎠ ⎛1⎞ = ( −1) ⎜ ⎟ ⎝i⎠ → This two-component representation of the spin wave functions is known as a spinor representation. The eigenstates are referred to as spinors. ⎛a⎞ A general spin state can be written as χ = ⎜ ⎟= a ↑ + b z ↓z ⎝b⎠ 2 2 normalization → a + b = 1 The total angular spin operator is determined as Sˆ ( Sˆ , ˆ , ˆ x Sy Sz) vector of 2x2 matrices! 2 → Sˆ= S 2 + S 2 + S 2 ( x y z ) ( x y z ) 2 S χ = 2 2 2 S + S + S ⎛1 1 1⎞ χ = ⎜ + + ⎟ ⎝4 4 4⎠ 3 χ = χ 4 2 → S 3 χ = χ = s( s+ 1) χ 4 1⎛1 ⎞ 3 ⎜ + 1⎟= 2⎝2 ⎠ 4 = . This is a Commutation relations ⎡Sˆ , ˆ ˆ ˆ ˆ ˆ ˆ x S ⎤ y = SxSy − SySx = iS you have to watch out which operator is first ⎣ ⎦ z applied to the wave function. ˆ 2 and ⎡⎣S , S ⎤ z ⎦ = 0 The components do not commute with each other; each component commutes with ˆS 2 . - 7 -
Page 1 and 2: Magnetism Magnetism is a relativist
Page 3 and 4: Of course within an atom the orbit
Page 5: Operator of orbital magnetic moment
Page 9 and 10: The wave function now takes four co

Magnetism 1

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