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 --------(2) Assuming 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 along . This satisfy ----------(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.