Chapter4_20 - PHY4604 R D Field Rotations in Spin-Space...

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Unformatted text preview: PHY4604 R. D. Field Rotations in Spin-Space Taylor Series Expansion: The Taylor series expansion for the function f(x) is given by ∞ a n d n f ( x) df ( x) a 2 df 2 ( x) a 3 df 3 ( x) = f ( x) + a + + +L f ( x + a) = ∑ dx n dx 2! dx 2 3! dx 3 n = 0 n! but U (a) = e iap x / h =e a d dx ∞ an d n iap x / h f ( x) . n and f ( x + a ) = U ( a ) f ( x ) = e n = 0 n! dx =∑ For a = ε << 1 then f ( x + ε ) = eiεp x / h f ( x) = (1 + iεpx / h) f ( x) = f ( x) + ε df ( x) dx The operator px / h is the generator of spatial translations. The Taylor series expansion for the function f(φ) is given by ∞ ϕ n ∂n f f (φ + ϕ ) = ∑ n n=0 n! ∂ φ and ∂ ∞ ϕ ϕ n ∂n iL ϕ / h ∂φ Rz (ϕ ) = e =e =∑ and n n=0 n! ∂ φ y-axis ϕz φ' φ x-axis z f (φ + ϕ ) = Rz (ϕ ) f (φ ) = eiLzϕ / h f (φ ) , where I used Lz = −ih ∂ . The operator Lz / h is the generator of rotations ∂φ about the z-axis. Rotations in Spin-Space: For spin ½ spinors we have S z = h σ z 2 | χ ' >=| χ (ϕ ) >= Rz (ϕ ) | χ >= e iS zϕ / h | χ >= eiσ zϕ / 2 | χ > and 1 iϕ Rk (ϕ k ) = e 2 kσ k ∞ ∞ 1i 1i ( 2 ϕk ) n + σ k ∑ ( 2 ϕk ) n = cos( 1 ϕk ) + iσ k sin( 1 ϕ k ) (k = 1, 2, 3) 2 2 even n! odd n! =∑ n 1 iϕ Rz (ϕ z ) = e 2 zσ z n 0 cos( 1 ϕ z ) − i sin( 1 ϕ z ) 2 2 = cos( 1 ϕ z ) + iσ z sin( 1 ϕ z ) = 2 2 0 cos( 1 ϕ z ) + i sin( 1 ϕ z ) 2 2 Similarly, 1 iϕ Rx (ϕ x ) = e 2 Ry (ϕ y ) = e Department of Physics xσ x 1 iϕ σ 2 yy cos( 1 ϕ x ) 2 = cos( 1 ϕ x ) + iσ x sin( 1 ϕ x ) = 2 2 i sin( 1 ϕ ) 2x cos( 1 ϕ y ) 2 1 1 = cos( 2 ϕ y ) + iσ y sin( 2 ϕ y ) = − sin( 1 ϕ ) 2y Chapter4_20.doc i sin( 1 ϕ x ) 2 cos( 1 ϕ x ) 2 sin( 1 ϕ y ) 2 cos( 1 ϕ y ) 2 Note that Rx(2π) = Ry(2π) = Rz(2π)=-1! University of Florida ...
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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