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Example Sheet 1 - Quantum Field Theory Example Sheet 1 Dr...

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Unformatted text preview: Quantum Field Theory: Example Sheet 1 Dr David Tong, October 2007 1. A string of length a , mass per unit length σ and under tension T is fixed at each end. The Lagrangian governing the time evolution of the transverse displacement y ( x, t ) is L = integraldisplay a dx bracketleftBigg σ 2 parenleftbigg ∂y ∂t parenrightbigg 2 − T 2 parenleftbigg ∂y ∂x parenrightbigg 2 bracketrightBigg (1) where x identifies position along the string from one end point. By expressing the displacement as a sine series Fourier expansion in the form y ( x, t ) = radicalbigg 2 a ∞ summationdisplay n =1 sin parenleftBig nπx a parenrightBig q n ( t ) (2) show that the Lagrangian becomes L = ∞ summationdisplay n =1 bracketleftbigg σ 2 ˙ q 2 n − T 2 parenleftBig nπ a parenrightBig 2 q 2 n bracketrightbigg . (3) Derive the equations of motion. Hence show that the string is equivalent to an infinite set of decoupled harmonic oscillators with frequencies ω n = radicalbigg T σ parenleftBig nπ a parenrightBig . (4) 2. Show directly that if φ ( x ) satisfies the Klein-Gordon equation, then φ (Λ − 1 x ) also satisfies this equation for any Lorentz transformation Λ. 3. The motion of a complex field ψ ( x ) is governed by the Lagrangian L = ∂ μ ψ ∗ ∂ μ ψ − m 2 ψ ∗ ψ − λ 2 ( ψ ∗ ψ ) 2 . (5) Write down the Euler-Lagrange field equations for this system. Verify that the La- grangian is invariant under the infinitesimal transformation δψ = iαψ , δψ...
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Example Sheet 1 - Quantum Field Theory Example Sheet 1 Dr...

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