As quantum mechanics matured during the 1920s attention increasingly turned to quantum field theory
here's a list of a few of the important publications from this period
In the previous video we discussed born Heisenberg in Jordan's
1926 paper presenting the quantization of a vibrating string
The following year Dirac applied these ideas to the electromagnetic field in his
The quantum theory of the emission and absorption of radiation
This paper is often cited as the birth of quantum field theory
In 1928 Jordan and thinners paper on the Pauli exclusion principle
showed how to make creation and destruction operators
compatible with the exclusion principle
Thereby allowing the concept to be extended to electrons
We will explore these ideas in the next video
So in 1928 Dirac published his equation of relativistic quantum mechanics for the electron
We covered this in the last video of the quantum mechanics series
The following year
Heisenberg and paulie's paper on the quantum dynamics of wave fields
Presented the first general theory of quantum fields and a canonical quantization
Formalism that continues to play an important role in quantum field theory
Oppenheimer's 1930 paper note on the theory of the interaction of field and matter
Highlighted a central problem in quantum field theory the interaction of a charged particle with its own electric field
produces infinite energy levels shifts
Taming. These infinities was a preoccupation of quantum field theories for years to come
During the summer of 1930
Enrico Fermi visited the University of Michigan where he delivered a series of lectures on the quantum theory of radiation
These were published as a 46 page article which served as a mini textbook on the subject
for physicists in the 1930s and beyond
As
Evidenced by the quote from richard fineman at the beginning of this video
Fermi was known for presenting difficult concepts in a simple intuitive style
So let's look at Fermi's approach to quantizing the electromagnetic field
We start with the simplest case the field in a one-dimensional electromagnetic cavity
We assume the field varies only along the z-axis and
Is confined by perfectly conducting planes to Z values between 0 and L
In
this region assume the vector potential a has only an X component that varies sinusoidally and Z and
vanishes at z equals 0 and L
We write a equals e^x the unit vector in the X Direction times sine K PI over LZ
times a time varying amplitude Q of T
Here K is a positive integer
For this geometry the wave equation requires that the second derivative and T of a
equals the second derivative in Z of a
performing the derivatives and canceling common factors
We obtain Q double dot equals minus Omega squared Q or Omega equals K PI over L
Field is - the time derivative of a or - e X sine K PI over L Z times Q dot
This vanishes at the perfect conductors Z equals zero and L as it must
The magnetic field is the curl of a which evaluates to ey
Omega cosine K PI over L Z times Q
In addition to satisfying the wave equation the vector potential must have zero divergence
Since field components are seem to vary only with Z. This will be true if the Z component of a is 0 which it is
We can repeat these steps assuming a has only a y component
This gives us a second solution in which all field vectors are rotated by 90 degrees from our first solution
So for any positive integer k we have two solutions
representing two independent field polarizations any
Possible field polarization can be represented as a linear combination of these
General field in our 1d cavity is then the sum over K from one to infinity
The sum over alpha from one to two of ye alpha Q K alpha of T
Sine K PI over L Z
Here the unit vectors e1 and e2 represent e^x and ey
For mode the amplitude satisfies QK Alpha double dot
Equals minus Omega K squared Q K alpha with Omega K equal to K PI over L
The solution is QK alpha equals B K alpha times cosine Omega K
times t minus TK alpha
we can alternately express the cosine argument as
Omega K T plus theta K alpha with theta K alpha equal to minus
Omega K T K alpha
Although the frequency Omega K is fixed the amplitude B K alpha and time offset T K alpha can be any real
numbers
So each mode can be arbitrarily scaled and amplitude and arbitrarily shifted in time
A solution of this form is called a standing wave
Because the sinusoidal variation in space remains fixed and merely oscillates an amplitude
More often we are interested in traveling waves representing the flow of energy through unbounded space due to radiation
Let's consider a traveling wave in one-dimensional unbounded free space
Assume a is polarized in the X direction and the field travels along the z axis at the speed of light
Which is 1 in our units
We write a equals exb cosine KZ minus Omega T plus theta
This solves the wave equation if Omega equals K
B and theta are arbitrary amplitude and phase values
To see that this is a traveling wave we set the cosines argument equal to a constant fee and solve for Z we
Find Z equals T plus a constant
So for each increment of time a given point on the wave moves a corresponding increment along the z axis
We can apply a sum of angles trig identity to write B cosine KZ minus Omega T plus theta as
the sum of two terms
Cosine KZ times B cosine Omega t minus theta plus sine KZ
times B sine Omega t minus theta
Let's denote by Q the amplitude B cosine Omega t minus theta
The time derivative Q dot is minus B Omega sine Omega t minus theta by
Analogy with a unit mass harmonic oscillator. Let's consider Q dot analogous to a momentum P
Then we can write the vector potential a as
unit vector e ax times the quantity Q cosine KZ minus
1 over Omega
P sine KZ
Using this expression for a we can calculate the electric field as - the time derivative of a and
the magnetic field is the curl of a
These operations result in the expression shown here
P equals Q dot so P dot equals Q double dot which equals minus Omega squared K
using these relations and the fact that K equals Omega
We can rewrite the electric field as minus e x times the quantity
P cosine KZ plus Omega Q sine KZ
The magnetic field is minus ey times the same quantity
The field energy is the field energy density
One-half magnitude a squared plus magnitude B squared averaged over space
Since E and B have the same magnitude
This is two times one-half the average of P squared cosine squared KZ plus
Omega squared Q squared sine squared KZ plus
2 Omega
PQ cosine KZ sine KZ
The cosine times sine term has zero average
cosine squared and sine squared both average to one half
So the field energy is one half quantity P squared plus Omega squared Q squared
This is formally identical to the energy of our harmonic oscillator
But in place of X the position of unit mass, we have Q the amplitude of an electromagnetic mode
As
We progress through this series. We'll come to appreciate this statement from a quantum field theory textbook
Quantum field theory is just quantum mechanics with an infinite number of harmonic oscillators
So let's do quantum mechanics
We replaced the dynamical variables Q and P in the energy expression with operators Q hat and P hat
Q hat is multiplication by Q P hat is minus IH bar derivative with respect to Q
This gives us the Hamiltonian operator H hat
Then we look for the stationary states where H had applied to a wave function produces energy e times the wave function
But this is simply the harmonic oscillator problem that we've already solved
And we replace Q and P in our field expression with Q hat and P hat
So the electromagnetic field becomes an operator a hat
Q hat and P hat can be expressed in terms of the creation and destruction operators a hat plus and a hat minus
With these substitutions a hat becomes a X
square root 1 over 2 Omega
times the quantity a hat - plus a hat plus
Times cosine KZ plus I a hat - a hat plus times sine KZ
A hat - and a hat plus terms inside the square brackets
We get a hat - times quantity cosine KZ. Plus I sine KZ
Plus a hat plus times the same quantity with I replace by minus. I
Finally we identify the cosine plus or minus I sine expressions as
e to the ikj and it's complex conjugate e to the minus I K Z so
a hat equals e^x square root 1 over 2 Omega
times the quantity a hat minus e to the ikj
plus a hat plus e to the minus I kz
Now let's extend this to describe a radiation field in three dimensional free space
Solution polarized in the X direction and propagating in the Z direction
And
Rotated in space so that it propagates in any desired direction EK
the rotation transforms e^x and ey into he k1 and e k2
Propagation vector k has x y&z components
KX KY and KZ
its magnitude
square root KX squared plus K y squared plus KZ squared is the propagation constant and
Equal to the parameter K in the one dimensional wave
The propagation vector divided by its magnitude is the unit vector ek and the oscillation frequency Omega K
equals K
The propagation constant equals to PI over the wavelength lambda
Where the wavelength is the distance between wave Peaks or valleys?
In place of K times e and the one-dimensional expression we will have wave vector K dot position vector X
Equals KX X plus KY y plus K Z Z
Then the field operator for this mode is a hat equals square root 1 over 2 Omega k
èk alpha times quantity a hat minus k alpha e to the I
KX
Plus a hat plus K alpha e to the minus I K dot X
Here ek alpha can be EK 1 or EK 2?
Field is then the sum over all propagation vectors K and the two polarizations alpha of this mode expression
The operator a hat plus K alpha creates a photon in mode K alpha
the operator a hat minus K alpha
destroys a photon in mode K alpha
The hamiltonian operator for the entire field is simply the sum of the operators for each mode
1/2 quantity P hat K alpha squared plus Omega K squared
Q hat K alpha squared
Which can be written as the sum over all modes of the mode number operator plus 1/2 times H bar Omega K
Or the number operator and hat K alpha is a hat plus K alpha times a hat minus K alpha
Finally the wavefunction of the radiation field is simply a list of the number of photons in each mode n K alpha
Called the occupation numbers
Destruction operator for a given mode modifies only that modes occupation number
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