Quantum Mechanics of Moving Bound States
In order to define the mechanics inside the bound state, let us go
back to the old problem in the Newtonian world.
Newton's gravity law is well known between two point particles. It is
proportional to masses of the particles and inversely proportional to
the distance (squared) between the particles.
- Since the distance between the sun and earth is much greater than the radius
of the earth, we have a tendency to use his law for point particles.
However, the story becomes more complicated if we take into account the
nonzero radiii of the sun and the earth. It took Newton 20 years to
figure out his gravity law applicable to the spheres with non-zero radii.
He had to invent a new mathematics to solve this problem. The new
mathematics in question is now known as the Integral Calculus.
- How about Newton's law for a ball inside the well that goes through
the center of the earth shown in this figure?
If you drop the ball to the well, it goes all the way to the opposite side
of the earth, and it will come back to you. This cycle will continue,
and the ball will perform a harmonic oscillation.
What would happen if, instead of Newtonian mechanics, we use quantum
Bohr and Einstein
- One hundred years ago,
Bohr and Einstein met occasionally to discuss physics.
Bohr was worrying about why the energy levels of the hydrogen atom
are discrete, while Einstein was interested in how things look to moving
observers. Did they ever discuss how the hydrogen atom looks to a moving
Bohr and Einstein, photo from the AIP Visual Archives.
- Bohr's worry became the present form of quantum mechanics where the
hydrogen atom is a quantum bound state or a standing wave. Thus, the
problem becomes that of a
moving bound state
in Einstein's world.
This world of Einstein is called the
- In this covariant world, moving objects appear differently according to
Lorentz transformations. Click here
- Click here for my review
article on moving bound states in the Lorentz-covariant world.
- The question then is how to construct Lorentz-covariant
wave functions for bound states. The harmonic oscillator wave
function serves as the standard tool for the bound state in
- Paul A. M. Dirac made his life-long efforts
to construct Lorentz-covariant oscillator wave functions. We can mention
the following four papers.
- P. A. M. Dirac,
The Quantum Theory of the Emission and Absorption of Radiation,
Proc. Roy. Soc. (London) A , 243 - 265 (1927).
- P. A. M. Dirac,
Unitary Repercussions of the Lorentz Group,
Proc. Roy. Soc. (London) A [A183], 284 - 295 (1945).
- P. A. M. Dirac,
Forms of Relativistic Dynamics,
Rev. Mod. Phys.  392 - 399 (1949).
- P. A. M. Dirac,
A Remarkable Representation of the 3 + 2 de Sitter Group,
J. Math. Phys. , 901 - 909 (1963).
- In 1962, I had the privilege of spending time
with Dirac to learn his physics directly from him. I was led to study his papers
Dirac's papers are like poems and enjoyable to read.
- However, they do not
contain figures or illustrations.
- Another problem is that Dirac never quoted
his own papers published earlier on the same subject. Presumably, he thought
he was presenting new ideas when he wrote those papers.
We can thus translate his poems into cartoons and synthesize those cartoons.
The net result is
This ellipse (squeezed circle) can provide the resolution of the quark-parton puzzle
and thus the Bohr-Einstein issue. Click here
for a detailed story.
For a published papers on this subject, go to
Integration of Dirac’s Efforts to Construct a Lorentz-covariant Quantum
with Marilyn E. Noz.
Symmetry [12(8)], 1270 (2020),
Physics of the Lorentz Group, Second Edition,
with Sibel Baskal
and Marilyn Noz,
published by the IOP (British Institute of Physics).
Quantum Mechanics of Moving Bound States,
J. Modern Physics  138-165 (2022).
- Poems are written in one-dimensional language. Cartoons constitute a
two-dimensional language. Physics can be best explained in
this two-dimensional language.
- Click here for the pioneer of this important language.
Physics is an Experimental Science.
- Yes, it is OK for the description of relativity with a squeezed circle.
The essential question is whether this elliptic squeeze can be seen the
While there are no observable hydrogen atoms moving with relativistic
speeds (speed comparable with the light speed), modern accelerators started
producing protons with relativistic speed, after 1950. The question is then what
the proton has to do with bound states like the hydrogen atom.
- According to Gell-Mann (1954), the proton at rest is a bound state of three
quarks, sharing the same quantum mechanics of bound states as the hydrogen
According to Feynman (1969), the proton moving with the velocity close to that of
light appears as a collection of an infinite number partons. Are they talking
about the same proton? This question is illustrated in this figure:
Indeed, this elliptic squeeze of the proton shows the following observable effects
in high-energy labs which produce protons moving with velocities very close that of
- Click here for detailed explanations.
By providing the resolution of the quark-parton puzzle, we can settle the
Bohr-Einstein issue of the moving hydrogen atom.
- In order to provide the answer, we had to construct bound-state wave functions
that can be Lorentz-boosted. This problem has a long history in physics. It is
possible to construct the harmonic oscillator wave functions that can be Lorentz-boosted.
We can call them Covariant Harmonic Oscillators and construct the following
E=(m2 + p2)1/2
In addition, the covariant harmonic oscillator can serve as a representation
of the Poincaré group (inhomogeneous Lorentz group).
Click here for the published paper on this subject.
Quantum field theory is an effective theory (with Feynman diagrams) for quantum scattering
processes in Einstein's Lorentz-covariant world. This theory is also a
representation of the Poincaré group. It is thus possible to combine the oscillator
formalism and quantum field theory into the Poincaré (inhomogeneous Lorentz group) as
specified in the following table.
- The last row in the above table asks whether Einstein's Lorentz covariance
can be derived from Heisenberg's starting equations for quantum mechanics.
This is indeed a crazy question, and Einstein will turn over in his grave.
Einstein did not like Heisenberg, who attempted to develop nuclear bombs for
Hitler before 1945.
Dirac's bust at the Fine Hall Library of Princeton University (2000).
The Fine Hall Library was my study place when I was in Princeton (1958-62)
as a graduate student and a postdoc.
- Let us go to
Dirac's 1963 paper on the two-oscillator system. He constructed a Lie
algebra (closed set of commutation relations for the generators of the group)
for the Lorentz group applicable to three space-like dimensions and two
time-like dimensions. This group is known as the O(3,2) deSitter group.
I heard about this paper directly from Dirac when
I met him in 1962.
The remarkable fact is that this set was constructed solely from Heisenberg's
brackets for his uncertainty relations. How is it possible to derive a set
of equations for Einstein's relativity from those for quantum mechanics?
The remaining question is to transform the second time variable of the O(3,2)
system into a useful variable in the Minkowskian system of three space
coordinates and one time. Indeed, it is possible through the group contraction
technique. I published a number of papers on this issue.
- Poincaré Symmetry from Heisenberg's Uncertainty Relations,
with S. Baskal and M. E. Noz,
Symmetry [11(3)], 236 - 267 (2019),
Einstein's E = mc2 derivable from Heisenberg's
with Sibel Baskal and Marilyn Noz,
Quantum Reports [1(2)], 236 - 251 (2019),
Physics of the Lorentz Group, Second Edition,
with Sibel Baskal and Marilyn Noz,
to be published by the IOP (British Institute of Physics).
Two more fundamental issues
- A massive particle at rest has three rotational degrees of freedom. However,
a massless particle has only one degree of freedom, namely around the
direction of its momentum. What happens to rotations around the two
transverse directions when the particle is Lorentz-boosted?
- You also have been wondering why massless neutrinos are polarized, while
massless photons are not.
The answers to these questions are in the bottom row
of this table:
Click on the colored items for further explanations.
E=(m2 + p2)1/2
- If you are interested in symmetry problems, you should be aware that
Wigner's 1939 paper deals with
the subgroups of the Lorentz group whose transformations leave the momentum
of a given particle invariant. Thus, these subgroups dictate the
space-time symmetry of the particle.
It is generally agreed that Wigner deserved a Nobel prize for this
paper alone, but he did not. He got the prize for other issues.
Click here for
my explanation of where the confusion was. Wigner liked my story.
This is the reason why he had photos with me, and I am regarded as
Wigner's youngest student, even though my thesis advisor at Princeton
was Sam Treiman.
- I did enough work for Wigner to deserve this genealogy:
The scientific content of this table is
- I was then able to
expand my scope of research.
Lorentz Group in Other Branches of Physics
- The Lorentz group is the mathematical language for Einstein's special
relativity. Two-by-two matrices are everywhere in physics, and they are
representations of the Lorentz group. Thus, the mathematical language
from this group serve useful services in other branches of physics.
Modern optics is a case in point.
With my younger colleagues, I published a
book on this subject.
- If you like modern optics, including coherent and squeezed states, beam transfer
matrices, polarization optics, as well as periodic systems,
- If you are interested in entanglement problems, particularly
- If you are interested in entropy problems and Feynman's rest of the
universe, click here, and
- Poincaré and Einstein? click here,
- Poincaré sphere for polarization optics and the Poincaré symmetry
for Einstein's relativity.
- I am now working the Lorentz group in condensed matter physics.
- This page is based on the papers I published since 1973.
I wrote many of those papers in collaboration with a number of co-authors,
especially, Sibel Baskal, Elena Georgieva, Daesoo Han, Marilyn Noz,
Seog Oh, and Dongchul Son. Michael Ruiz and Paul Hussar were my graduate
students. They made key contributions to this program. I would like to
- John S. Toll was the chairman of the physics department at the University
of Maryland when I came to the University as an assistant professor of
physics in 1962.
In the same year, Toll invited Paul A. M. Dirac to the department for
one week, and he assigned me to be a personal assistant to Dirac. This was
indeed a great opportunity to learn Dirac's physics directly from him. Here is
my story about meeting this great physicist.
Toll was the Chancellor of the University of Maryland in 1986. He became
very happy when Wigner visited the University at my invitation.
- I am grateful to Professor Eugene Wigner for
clarifying some critical issues concerning his 1939 paper on the internal
space-time symmetries of particles in the Lorentz-covariant world.
for my webpage dedicated to Eugene Paul Wigner.
- It is a serious matter to talk about Albert Einstein. I came to the
United States in 1954 after high-school graduation in Korea, with a strong
mathematical background. For instance, I did the following mathematical
exercise during my high school years.
Einstein's hyperbola is well known among all physicists. On the
other hand, the circle and ellipse are still strange to them. They
are not strange to me.
- Finally, let us examine Einstein's brain. He started as a Kantianist
who thinks the same thing could appear differently depensing on the obserber's
environment or/and state of mind. However, he became a Hegelianist who
could synthesise two opposing aspects into one. Then this create a gap:
many to two. This gap can be bridged by the oriental philosophy of Taoism.
During the 13-th Century, Koreans picked up Neo-Confucianism formulated by
the Chinese scholar named
Zhu Xi (1130-1200).
Koreans called his teaching "Zooja-Hak."
This Neo-Confucianism is a formulation of Confucian doctrines in the
logical frames of Taoism, which says the universe consists of harmonies
of two opposite elements. Man and women, sun and moon, hot and cold, etc.
Korea's last dynasty, which lasted for more than 500 years (1398-1910),
was based on this ideology. Indeed, Korea's national flag tells this
aspect deep-rooted Neo-Confucianism.
Thus, with the Korean background, it is easy to understand the bridge
between many and two as described in the above figure.
The Capitol Building in Washington, DC, USA, where Americans manage
their democracy using the two-party system.
- Another important case is the two-party system of American democracy.
Zhu Xi, Kant, and Hegel are total strangers on American campuses.
Yet, the above figure is the exact description of how Americans run
their democracy. Thus, the above figure is a description of the
natal evolution of human brains.
From their monarchy and dictatorships, Koreans achieved their
two-party system in a short period of time, because Korean
brains are configured like that of Einstein as described above.
Korea's first Presbyterian church set up in 1884. I attended this
church (now in North Korea) until my family moved to Seoul in 1946.
- The North Korean system is still a monarchy, but they will
achieve the democratic system like that of the South soon,
because their Korean brains are the same as those in
the South. Korea will soon become one country.
- I was born and raised in our North Korea before the country was
divided, and my brain was configured there. My family moved to
Seoul in 1946 before Joseph Stalin established his puppet regime
headed by Kim Il-Sung in 1948.
- Click here
for the history of this church.
- Like my colleagues in science, I am not a religious person. However,
I have a very strong Christian background, much stronger than my American
colleagues. With this background, I constructed the following webpages.
- Moses talked to God. How did I talked to Einstein?
- Nicodemus story. I met Paul A. M.
Dirac. I was like Nicodemus meeting Jesus. I was born again.
- Herod Complex. Academic world is
highly competitive world. How can you allow anyone, other than
yourself, becoming more famous than you?
- Garden of Eden. While travelling
around the world, I meet many interesting people, and have photos
with them. I appear very happy when I have photos with two women.
Why? I have the same number of ribs on both sides of my chest.
That means God pulled out two ribs (instead one in Adam's case)
to create women for this world.
Click here for my home page.
Travel around the World.
Click here for my photo-biography