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Quantization of the Radiation Field
Physics — Quantum Mechanics · dirac-field-charge-operator
Problem (a) Show for the spinor field that
Q = − e 0 ∑ k ∑ r ( b r k † b r k − d r k † d r k ) Q = -e_0\sum_{\mathbf{k}}\sum_r\left(b^\dagger_{r\mathbf{k}}b_{r\mathbf{k}} - d^\dagger_{r\mathbf{k}}d_{r\mathbf{k}}\right) Q = − e 0 k ∑ r ∑ ( b r k † b r k − d r k † d r k ) Q = − e 0 ∫ d 3 x : ψ ˉ ( x ) γ 0 ψ ( x ) : . Q = -e_0\int d^3x\,:\bar{\psi}(x)\gamma^0\psi(x):\,. Q = − e 0 ∫ d 3 x : ψ ˉ ( x ) γ 0 ψ ( x ) : . (b) Show, furthermore, that
[ Q , b r k † ] = − e 0 b r k † and [ Q , d r k † ] = e 0 d r k † \left[Q, b^\dagger_{r\mathbf{k}}\right] = -e_0 b^\dagger_{r\mathbf{k}} \quad\text{and}\quad \left[Q, d^\dagger_{r\mathbf{k}}\right] = e_0 d^\dagger_{r\mathbf{k}} [ Q , b r k † ] = − e 0 b r k † and [ Q , d r k † ] = e 0 d r k † Browse the problem library