qft_ex2 - Quantum Field Theory Example Sheet 2 Prof N.S Manton October 2010 1 A string has classical Hamiltonian given by 12 p 2n H= 122 2 n qn(1

# qft_ex2 - Quantum Field Theory Example Sheet 2 Prof N.S...

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Copyright © 2010 University of Cambridge. Not to be quoted or reproduced without permission. Quantum Field Theory: Example Sheet 2 Prof. N.S. Manton, October 2010 1.A string has classical Hamiltonian given byH=summationdisplayn=1(12p2n+12ω2nq2n)(1)whereωnis the frequency of thenth mode. (Compare this Hamiltonian to the La-grangian (3) in Example Sheet1. We have set the mass per unit length in thatquestion toσ= 1 to simplify some of the formulae a little.) After quantization,qnandpnbecome operators satisfying[qn,qm] = [pn,pm] = 0and[qn,pm] =nm.(2)Introduce creation and annihilation operatorsanandan,an=radicalbiggωn2qn+i2ωnpnandan=radicalbiggωn2qni2ωnpn.(3)Show that they satisfy the commutation relations[an,am] = [an,am] = 0and[an,am] =δnm.(4)Show that the Hamiltonian of the system can be written in the formH=summationdisplayn=112ωn(anan+anan).(5)Given the existence of a ground state|0)such thatan|0)= 0, explain how, afterremoving the vacuum energy, the Hamiltonian can be expressed asH=summationdisplayn=1ωnanan.(6)Show further that [H,an] =ωnanand hence calculate the energy of the stateparenleftBig