Download Document

Relativistic Quantum Mechanics
Lecture 3
Books Recommended:
Lectures on Quantum Field Theory by Ashok Das
Advanced Quantum Mechanics by Schwabl
 Relativistic Quantum Mechanics by Greiner
 Quantum Field Theory by Mark Srednicki
 http://www.physicspages.com/2015/11/12/klein-gordon-equation/
 http://www.quantumfieldtheory.info/
Klein Gordon Equation
For non-relativistic case, we know how to get
Schrodinger equation from eq. using corresponding
Hermitian operators
-----------(1)
-------------(2)
For relativistic case, we start with relativistic energy
momentum relationship
-------(3)
Using operator substitution and operating over scalar
wave function ϕ
Considering
-------(4)
Above equation is Klein Gordon equation.
Putting
and c in above equation will look as
-------(5)
Above eq will be invariant under Lorentz
transformations.
Note that D’Alembertian operator is invariant
Also φ is assumed to be scalar and hence K.G.
Equation is invariant
Klein Gordon equation has plane wave solutions:
Function
is Eigen function of energy-momentum operator
Eigen values
Thus, plane waves will be solution of K.G. Eq if
--------(6)
which gives positive and negative energy solutions.
Concept of probability density and current density
in Klein Gordon equation:
Writing K.G. Eq. and its complex conjugate
-------(7)
------(8)
Multiplying (7) from left by ϕ* and (8) by ϕ
--------(9)
Before proceeding further, we recall continuity
equation derived in non-relativistic quantum
mechanics i.e.
----(10)
Where
Probability density
Current density -----(11)
Multiplying Eq (9) by
and using natural
system of units, we can write
-----(12)
Where
-----(13)
------(14)
Above eq gives current density and probability
density from K.G. Eq.
What will be form of J and ρ in Eqs (13) and (14),
if we do not use natural system of units?
Note the difference in probability density from S.E.
And K.G. eq. i.e. Eq. (11) and (14)
In case of S.E. probability density is always positive
definite. It is time independent and hence probability
is conserved
(see chapter 3, QM by Zettili )
But eq. (14) (ρ from KG eq) can take negative
values as well.
This is because KG eq. is 2nd order in time
derivative and we can use any initial value of ϕ
and
.
e.g. For
(plane wave)
As E can be positive or negative and hence, ρ also.
Klein Gordon eq is discarded as quantum mechanical
wave equation for single relativistic particle.
As a quantized field theory, the Klein–Gordon
equation describes mesons.
The hermitian scalar Klein–Gordon field describes
neutral mesons with spin 0.
The non-hermitian pseudoscalar Klein–Gordon
field describes charged mesons with spin 0 and
their antiparticles.