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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.