Once in a while, a great scientist comes along who notices a pattern in nature that others
may have completely missed. For example, Isaac Newton realized that the forces acting on
a fired cannonball are the same as that on the moon orbiting the earth.
Ernest Rutherford was another such person. When he realized that atoms have a heavy nucleus,
he hypothesized that the way the moon orbits earth is the same as the way an electron orbits
the nucleus of atoms. Newton was right. Rutherford we now know was
wrong. An atom would not look anything like this. Yet we still use the Rutherford model
to depict the way atoms look. It is the most familiar image we have of the atom.
A real atom would look shockingly different from anything like this. And it would definitely
not be mostly empty space as you’ve been told.
What does an atom really look like? And how did we determine that? Why doesn’t an electron
fall onto the nucleus if they attract each other so strongly? Why can’t you squeeze
an atom? The answer requires us to look deeper into the meaning of quantum mechanics. Why
do we trust quantum mechanics anyway? And did we arrive at this revelation – perhaps
the most accurate and proven theory in science. The answer is coming up right now...
In 1911, Ernest Rutherford proposed a planetary
model of the atom. According to the model, the solar system and the atom were almost
identical. Just like the moon is held in orbit around
the earth due to gravity, the negatively charged electron was held in orbit around a positively
charged nucleus. The attractive force was electromagnetism instead of gravity in the
case of the atom. This seemed to make sense because Isaac Newton’s
law of universal gravitation was almost identical to coulomb’s law for electric force. This
was beautiful. A fantastic symmetry of nature. But alas, it was too good to be true. One
big problem is that when electrons are accelerated, as they would be when constantly changing
direction traveling in circular orbits, they create electromagnetic waves according to
Maxwell’s equations. This means that photons would be constantly radiated from the electron,
causing it to lose energy. It would radiate a rainbow of colors, getting
bluer and more purplish as the electron came closer and closer to the nucleus, until it
collapsed into it. So using Classical laws for the atom just
did not work if you presume the Rutherford model.
A bold new step was needed. But little did Rutherford realize that 11 years earlier,
in 1900, Max Planck had already made this step, by showing that energy of photons was
quantized. Planck’s theory had shown that matter emitted only discrete amounts of radiation,
with energy E proportional to the frequency f – the proportionality constant h, being
Planck’s constant. A young scientist by the name of Neils
Bohr came along and combined Rutherford’s model and Planck’s theory to hypothesize
that the electron could exist in certain special orbits without radiating energy. How did he come up with this?
What Bohr noticed is that Planck’s constant had units of angular momentum. So he hypothesized
that only those orbits would be allowed where the angular momentum of the electron is quantized
based on Planck’s constant. And he guessed that the lowest orbit would have the momentum
h/2pi where the 2pi comes from the geometry of circular orbits. And any orbit could exist
as long as it was an integer multiple of this number. So the next orbit would be 2 x h/2pi.
Then 3X for the next one and so on.
And Bohr predicted that the electrons would
only radiate or absorb energy when these electrons jumped from one orbit to another.
But Bohr could not explain why electrons would not emit photons all the time, or why these
special orbits should exist in the first place. But presuming he was right, you could find
the size of the orbit of the electron using coulomb’s law. It turns out that the radius
of the lowest orbit would be .529 X 10^-10 m, about half an angstrom. So now we knew
the size of the atom, which had been a mystery in the past.
Knowing this, the energy emitted by the electron as it changes orbits could be calculated.
Lo and behold, observations of energy emissions, confirmed these calculations. So Bohr’s
model was right. Since this model predicted values that could
be verified, it was thought that this was it! This must be what atoms really look like.
The problem was nobody knew why Bohr was right. So this was probably not the final answer.
Then along came brilliant French scientist by the name of Louis de Broglie. He said: look, if a particle
has a momentum and it has a wavelength associated with it because of Planck’s constant, then
an electron is probably a wave. This required a huge philosophical leap, because here was
a guy suggesting that solid matter – things that we can see and feel on a macro scale
were composed of waves. Matter was somehow a particle and a wave at the same time.
De Broglie suggested that electrons can only exist in orbits where their waves interfere
constructively. And that can only happen if the circumference of the orbit is equal to
the wavelength, or twice the wavelength or 3 times the wavelength, or any integer times
the wavelength. This made sense, and explained why orbits would be at the radii that they
are, something that Bohr could not do. The Bohr model was looking more and more legit.
But there were still many questions. What is the nature of these waves? How and why
do they exist? A crucial piece of the puzzle was solved by
Austrian physicist, Erwin Schrodinger. He said: look guys, if it’s a wave, it can exist
anywhere in 3 dimensional space. And he formulated the rules to describe the behavior of these
waves. The rules are encompassed in the Schrodinger equation, which became arguably the most important
equation in quantum mechanics. The rules were so spectacularly successful
in making predictions that no one could really dispute it. Schrodinger’s equation could
describe the hydrogen atom with more detail and precision than the Bohr model. But it
could also describe all the other atoms in the periodic table, which the Bohr model could
not. It was truly a revelation. This was not Newtonian mechanics any more,
but a new kind of mechanics based on the quantum revolution started by Max Planck. It was quantum
mechanics! And Schrodinger’s equation containing the
wave functions of atoms is the one critical piece of information that we need to determine
what an atom most likely actually looks like. But why do we even have to guess? Why don’t
we settle the debate by just looking inside any material to see what these pesky little
things actually look like? The problem is that in order to be seen, the
object has to be large enough to reflect light. But the largest atom is 1000 times smaller
than a wavelength of visible light. So visible light just goes right through the atoms – it
can’t really be seen because no light is reflected back.
What about using shorter wavelength light, like X-rays? The problem is that short wavelength
light carries so much energy, that it interacts with the electron, changing it.
This is one consequence of Heisenberg's Uncertainty Principle, which says that the range of an
electron's position times the range of its momentum can not be below a certain constant
– h over 4Pi. We cannot say where the electrons are and how fast they are moving at the same
time, so we can’t draw a picture with a little electron in one place sitting on a
circular orbit. So we have to take our best guess based on
what the wave equation tells us. And what it tells us is that the electron forms a cloud
around the nucleus. The shape of the cloud is governed by the wave function. It is called
a wave function because the quantum wave equation resembles the equation for classical waves.
The cloud represents the probable position of the electron if you were to measure it.
And this cloud exists everywhere from the nucleus to far away from the nucleus. The
volume of the atom is thus definitely not empty. It is filled everywhere with a cloud
of electrons. We can use the Schrodinger wave equation to
find the electron probability. This plot shows the probability of finding the electron at
various distances from the nucleus for a hydrogen atom. The highest probability occurs at, wouldn’t
you know it, 0.529 X 10^-10 meters, which is exactly the radius calculated by Bohr.
So the most probable radius obtained from quantum mechanics is identical to the radius
calculated by classical mechanics.
In Bohr’s model, however, the electron was assumed to be at this distance all the time,
whereas in the Schrödinger model, it is at this distance only some of the time. It has
the highest probability of being at this radius, but it could be elsewhere too. The difference
is due to the Heisenberg uncertainty principle, and the fact that electron acts like a 3D
wave.
And the same wave equation tells us that the nucleus of atoms, which in the case of the
Hydrogen atom is a proton, is also a cloud. But the extent of the proton cloud is much
smaller than the electron cloud because it is much more massive.
So if you look at the uncertainty equation, you can see that the extent of the delta x
would be much smaller, given a large m, in order to satisfy the inequality.
The proton cloud is so small in fact, that if the electron cloud was the size of Michigan
stadium, the largest sports stadium in America, which seats about 110 thousand people, the
proton cloud would be the size of a marble at the 50 yard line. The electron cloud is
about 100,000 times larger than the proton cloud. So the idea of discrete spherical protons
and neutrons in the nucleus, or electrons orbiting them is really fiction. It is good
for showing to kids in class rooms, but not for students of quantum mechanics.
Let me clarify though that you could never actually see a hydrogen atom in this state
because the act of seeing it would necessarily change it. But if by some magic you could
see its state without measuring it, this is a good approximation.
Note that there are other shapes that the cloud of the hydrogen atoms could take as
well depending on the energy level and quantum state of the electron.
But here are actual images of a hydrogen atoms taken by an international team of researchers
in 2013. So this is definitive proof that our 3D model is likely correct.
Now, given what I just said, how can a photograph like this be taken?
Well, this is not a direct image. If you read the paper, you will find that it is a composite
image based on the trajectory of electrons emitted by hundreds of hydrogen atoms after
they are excited by laser pulses. So now we can answer our original 2 questions
First, why doesn’t the electron just fall to the proton in the nucleus if it is attracted
to it? After all if you drop a meteor directly onto the earth with no angular momentum, it
will hit the earth with a colossal amount of energy, enough to kill all the dinosaurs like 65 million
years ago. But when it comes to the atom, even if you drop an electron with no angular momentum
directly onto a proton, the electron will not fall and hit the proton. Why? Because
if it did, it would violate the Heisenberg uncertainty principle, because both delta X
and delta P in the equation would be zero. And that can't be. What happens is that there will always be
a balance between the position and momentum of the electron such that the uncertainty
principle is obeyed. What happens is the electron forms a cloud around the proton even if you dropped it.
You might say ok, I get that, but if it is a cloud, why can’t I squeeze two atoms together?
We have to go to a chart which shows the energy of the electron as a function of distance
from the nucleus. The electron prefers to be in its lowest energy state which is at
a distance .529X10^-10 meters from the nucleus. In order to squeeze atoms to a smaller size,
we have to increase the energy of the electron. This requires the electron to go to a higher
energy state. When you consider that there are quadrillions of atoms in any object we
can see. For example a grain of sand has about 10^18 atoms, the cumulative amount of energy
is humongous. Squeezing two objects together requires energies on the order of a small atomic bomb. And
this is the reason we have solid objects that we can hold in our hand without our hand being
able to squeeze them to nothing, and why we don’t sink if we stand on solid ground.
So while quantum mechanics results in some very mysterious phenomenon that we have a
hard time explaining, like the double slit experiment, or entanglement, it also gives
us a deeper and more complete picture of reality as it probably is. While you might
feel uncomfortable with quantum mechanics because of it unintuitive in nature, remember
that nature has no obligation to be intuitive or understood by us conceited hairless apes,
who think we deserve to know the deepest secrets of the universe.
Quantum mechanics is ultimately at the root of reality. And if you want to get a deeper
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