is it true that when symmetry breaks, that which is invisible becomes
M8R-i...@mailinator.com
I have a question about statement that I read on a website about a
year ago. The statement was about that when symmetry is broken, that
which is invisible becomes visible.
My question is if this is true, and if it is, can you provide an
example (preferably not too complex) of a case in which this can be
seen?
Thanks in advance,
Darwin123
As an example I chose: Invisibility.
Consider a deep body of water which is homogeneous over a large
distance. This is very clean water which does not contain any
particles that scatter light. Both its real index of refraction and
its absorptivity are independent of position. This independence
extends in all directions for miles and miles.
You are in this water, wearing something to protect your eyes.
The sunlight is shining through the surface from above, so every cubic
millimeter of water has sunlight passing through it.
I will ignore the question of whether you the diver can see parts
of your own body, wardrobe, or equipment. You are hunting jellyfish.
There is a jellyfish with the same index of refraction of the
water and the same absorptivity of the water floating in front of you.
This jellyfish may change later in the problem, but right now it is
perfectly matched with the water. You, the diver, can not see this
jellyfish because it matches the complex index of refraction of the
water perfectly. I will state in other words why the jellyfish is
invisible.
The system, water plus jellyfish, satisfies two symmetries. First,
the water and the jellyfish are both homogeneous so far as the complex
index of refraction are concerned. The complex index of refraction
does not vary with the point in space. Any point one chooses, in the
water or in the jellyfish, has the same index of refraction. Second,
the jellyfish plus the water is isotropic. That means that if you turn
the jellyfish upside down, the index of refraction still is the same
all throughout the system consisting of jellyfish plus water.
The fact that the system is homogeneous makes the jellyfish
invisible. The water can't scatter from the jellyfish because the
index of refraction matches perfectly. Light passes through the
jellyfish easily with no scatter. The jellyfish is invisible, at least
while it matches the water.
Now consider what happens if the jellyfish changes its index of
refraction. Homogeneity is broken. The index of refraction is not the
same throughout the system of jellyfish plus water. The index of
refraction is different for a point inside the jellyfish when compared
to the index at a point outside the jellyfish. Thus, homogeneity is
broken.
The jellyfish now appears at least three ways: reflection,
refraction, and extinction.
1) Some of the sunlight reflects off the jellyfish. When light is
incident on an interface between two regions with a different index of
refraction, there is a reflection at the interface. You may be able to
see the jellyfish by the reflection. Thus, the glint (i.e., the
reflection) is caused by breaking the homogeneity.
2) Suppose the jellyfish is between you and the sun. The sunlight will
bend by refraction, causing the sun to look distorted. Again, this
distortion doesn't happen if the jellyfish matched its surroundings.
Thus, refraction (the distortion) is caused by breaking the
homogeneity.
3) The jellyfish will cast a shadow. The reflection of sunlight off
the jellyfish, and the diversion of light by refraction, takes energy
out of the sunlight that would normally go in a straight path. Thus,
the jellyfish casts a shadow on the sea floor. The act of casting a
shadow is called extinction. There is no shadow if the jellyfish
matches the water. Thus, extinction (the shadow) is caused by breaking
the homogeneity.
Thus, breaking the homogeneity made the invisible jellyfish
appear. By changing the index of refraction, the jellyfish broke the
homogeneity that made it invisible. Breaking the homogeneity made the
jellyfish appear.
There is a lot of other good examples provided in studies of
camouflage. Basically, the only way animals detect each other is
through processes that break symmetry. Camouflage, and defenses
against camouflage, are based on the breaking of symmetry.
Darwin123
Maybe the invisible jellyfish is too complex. A simple
spectroscopic example may sound better.
Consider a neutral hydrogen atom. The electron is in a
spherically symmetric potential. Use an electric spark to ionize most
of the hydrogen atoms in a sample of gas. Then let the highlly excited
electrons decay to low energy levels. The emission spectrum is called
a Rydberg spectrum, and comes from an electron in a large n state
decaying into a low n state. So it was easy to conclude in the
beginning of the twentieth century that electrons in hydrogen atoms
are confined to electronic shells designated by the letter n. The
Rydberg spectrum shows lines corresponding to transitions from n' to n
where n'>n. Note both n and n' are positive integers. However, there
is more structure in the hydrogen atom than the electronic n levels.
Each n level contains several sublevels designated by the letter l.
The letter l more or less corresponds to angular momentum states. I
will call the l level by their common names in chemistry: s(l=0), p
(l=1), d (l=3), f (l=4), etc.
The problem with the Rydberg spectrum is this. Although it shows
the existence of n states, the Rydberg spectrum does not show the
existence of l levels. This is because in while the potential of the
electron is spherically symmetric, all electrons in the same n level
have the same energy. The l states are invisible if the potential is
spherically symmetric.
How does one determine the existence of l levels? Try Stark
effect.
Apply a electric field. If one applies an magnetic field to a
hydrogen atom, the spherical symmetry is broken. The atom still has a
cylindrical symmetry. If n=2, the l level can be 0 or 1.
So if we repeat the experiment again where a Rydberg spectrum is
generated. We will find that some of the emission lines have broken in
two due to the electric field. Now, the n=2 level is broken up into an
l=0 level and an l=1 level.
The electric field broke the spherical symmetry of the hydrogen
atom, so that there was a preferred direction. This made the l=0 and
l=1 levels have different energies. So the breaking of spherical
symmetry made the l levels visible.