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Example – Particle moving in z direction

Example – Particle moving in z direction. Eigenstate of Momentum (Plane wave approximation). Spinor. Spinors. Properties:. Orthogonality. Density increase as volume decreases. Completeness. Proof:. Transform back into rest frame. Projection Operators.

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Example – Particle moving in z direction

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  1. Example – Particle moving in z direction Eigenstate of Momentum (Plane wave approximation) Spinor

  2. Spinors Properties: Orthogonality Density increase as volume decreases

  3. Completeness Proof: Transform back into rest frame

  4. Projection Operators • Break projection operator into two parts: • Electron/positron part; and • Spin up/down part. +=electrons (upper states) - =positrons (lower states) Property for projection operators:

  5. Electron/Positron Projections Since: Projection operators are complete.

  6. Spin Projection Non-relativistic spin Projection operator: Relativistic z-direction (in rest frame): Spin direction is opposite for positrons Conventions:

  7. Z-Axis Spin Projection Operator

  8. Define Electron and Positron States Electron wave functions ε=+1 Positron wave functions ε=-1 Projection Operators

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