Dirac_12 - Particle Moving in the Z-Direction For a...

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PHY4604 R. D. Field Department of Physics Dirac_12.doc University of Florida The Helicity Operator The bonus embodied in the Dirac equation is the extra two-fold degeneracy. For example, the two positive energy solutions + = + + > χ σ 2 ) 1 ( 0 ) ( mc E p c N p u E r r r and + = > 2 ) 2 ( 0 ) ( mc E p c N p u E r r r have the same energy and momentum. This means there must be another observable which commutes with H op and op p r , whose eigenvalues can be used to distinguish the two states. Consider the spin operator = r r h r 0 0 2 S Helicity Operator: The helicity is defined to be the component of the particle’s spin along the direction of motion as follows: = = Λ p p p S ˆ 0 0 ˆ 2 ˆ r r h r where p z p y p x p p z y x / ) ˆ ˆ ˆ ( ˆ + + = is a unit vector in the direction of p r . The eigenvalues of Λ are 2 h ± and Λ commutes with H op and op p r . Hence the component of the spin of the particle along the direction of motion is a “good” quantum number and can be used to label the solutions.
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Unformatted text preview: Particle Moving in the Z-Direction: For a particle with p x = p y = 0 we see from above that ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = Λ z z z z S 2 h and ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + = ± ± > 2 ) 2 , 1 ( ) ( mc E p c N p u z z z E . These two 4-component spinors are the eigenkets of the helicity operator with eigenvalues 2 h + and 2 h − , respectively, since ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ± = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ± ± ± ± ± ± 2 2 2 2 2 2 mc E p c mc E p c mc E p c z z z z z z z z z z h h h , where I used the fact that ± ± ± = z . positive helicity spin momentum negative helicity spin momentum...
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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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