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Chapter II Klein Gordan Field Lecture 3 Books Recommended: Lectures on Quantum Field Theory by Ashok Das A First Book of QFT by A Lahiri and P B Pal Energy Eigenstates Consider the normal ordered Hamiltonian ---(1) Consider the energy eigen state We write Assuming . --------(2) is normalized. Expectation of energy will be ----(3) Which shows that the energy has to be Positive in 2nd quantized theory. Recall following commutation relations for ----------(4) We can write ----(5) Which shows annihilation operator lower the Energy Eigen value ---(6) Similarly, creation operator lowers the energy Eigen value ---(7) Also, we can write -----(8) For minimum energy state ----(9) which is the ground state or vacuum state |0>. ----(10) General Eigen state of higher energy ---(11) Above states are Eigen states of number operator and states are denoted as -----(12) State given in (12) is eigen state of total number operator ----(13) Eq (13) can be proved using ---(14) From which we get -----(15) The way we have definition of Hamiltonian --(16) We can define momentum ----(17) Operating H on (13), we get -----(18) Operating P on (13), -----(19) Thus, we have from (18) and (19) ----(20) Physical meaning of energy eigenstates Consider state -----(21) This satisfy ------(22) We can write, (using 22) (33) Which is a one particle state with four momentum In general we can write Consider the operation of field operator on Vacuum ---(35) Also, we can write, when we have vaccum state On both states ----(36) Non-zero matrix element, involving vacuum states, ---(37) Which represent the projection of . This satisfy along ----------(38) is solution of Klein Gordon eq and we can show -----(39) In Quantum mechanics we write wave function As ----(40) Single particle state ---(41) For multiparticle state ------(42) Above states are symmetric under exchange of particle and thus, describe Bose particles.