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Consider a free real scalar field $(x„), where r, = r, y, z for u = 1,2,3 and r4 = ict, satisfying the Klein-Gordon cquation. %3D (a) Write down the Lagrangian density for the system. (b) Using Euler's equations of motion, verify that o does satisfy the Klein-Gordon equation. (c) Derive the Hamiltonian density for the system. Write down Hamil- ton's equations and show that they are consistent with the equation derived in (b)

Consider a free real scalar field $(x„), where r, = r, y, z for u = 1,2,3 and r4 = ict, satisfying the Klein-Gordon cquation. %3D (a) Write down the Lagrangian density for the system. (b) Using Euler's equations of motion, verify that o does satisfy the Klein-Gordon equation. (c) Derive the Hamiltonian density for the system. Write down Hamil- ton's equations and show that they are consistent with the equation derived in (b)

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Consider a free real scalar field $(x„), where r,„ = :r, y, z for µ = 1,2,3 and r4 = ict, satisfying the Klein-Gordon cquation. (a) Write down the Lagrangian density for the system. (b) Using Euler's equations of motion, verify that o does satisfy the Klein-Gordon equation. (c) Derive the Hamiltonian density for the system. Write down Hamil- ton's equations and show that they are consistent with the equation derived in (b)
Transcribed Image Text:Consider a free real scalar field $(x„), where r,„ = :r, y, z for µ = 1,2,3 and r4 = ict, satisfying the Klein-Gordon cquation. (a) Write down the Lagrangian density for the system. (b) Using Euler's equations of motion, verify that o does satisfy the Klein-Gordon equation. (c) Derive the Hamiltonian density for the system. Write down Hamil- ton's equations and show that they are consistent with the equation derived in (b)
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