At the dawn of the 20th century, most scientists thought we had learned almost everything
there was to learn about the world of physics. The commissioner of the US patent office in 1899
is even purported to have said, “Everything that can be invented has been invented.”
The classical world picture which scientists had painstakingly built over hundreds of years
was doing wonders, as the industrial revolution was in full swing. There remained only a few
problems, which it was thought, would soon be solved.
Then came 1905. Einstein published the special theory of relativity. And the classical Newtonian
view of fixed space and time began to crumble. Soon thereafter, when the nucleus was discovered
by Rutherford, it became clear the classical world was not reality.
One problem was that according to classical physics atoms shouldn’t even exist, or at
the very least be unstable. Consider the hydrogen atom. It is made of
a negatively charged electron and a positively charged proton. According to classical electromagnetism,
the electron should orbit the proton, held together due to their electromagnetic attraction,
similar to the way the earth is held in orbit around the sun due to gravity.
But the electron is accelerating as it is orbiting, and according Maxwell’s equations,
this should create electromagnetic waves. In other words the electron should be radiating
photons. This is the concept behind lasers for example, where accelerated electrons radiate
photons. But if the orbiting electron constantly sends
off photons, its orbital energy would continually decrease until eventually it crashed into
the proton. This would be a disaster, of course, because it would mean atoms would not exist,
and consequently we would not either. This problem was solved by Niels Bohr who
postulated that electrons orbit in distinct orbitals, and can only gain or lose energy
in chunks, not continuously. But this by itself would not save the atom.
In order to do that, we have to go farther back in time, to Max Planck, who really started
the quantum revolution with a deceptively simple looking formula, the full implications
of which scientists are still trying to figure out.
It also led to what we call the uncertainty principle. It underlies all of quantum mechanics.
And if you truly want to understand how quantum mechanics works, you have to understand how
this principle works and where it comes from. What was Planck’s formula? How did it lead
to the uncertainty principle? And how does it show that reality was after all, is not
deterministic, continuous or even logical? The answer is coming up right now…
Before we get into the fascinating story of the uncertainty principle, I’d like to acknowledge
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the link in the description. Whether we are aware of it or not, we view
the world with a classical perspective. When we think of something, like a tennis ball,
we think of it as a discrete object with a distinct position in space and time, with
a plainly observable velocity, or speed and direction.
The problem is that to the best of our knowledge, this is not how the universe actually works.
Quantum mechanics says that the position and velocity of a particle does not really exist.
What really exists is a wave function which is spread out in time and space, which can
tell us only the probability of the position and velocity of that particle if we were to
measure it. It’s just that in our macro world, the wave function of large objects
is so localized that we don’t experience the uncertainty.
To understand the real world, which has measurable effects only on very small scales, let’s
start off at that really small scale, and see what happens when we scale up.
If we look at Bohr’s quantum solution to the classical picture of the hydrogen atom,
we see that he theorized that the electron spins around the proton at only certain orbitals
where it would be stable. The stability happens because its angular momentum would be an integer
multiple of Planck’s constant. And the electron would only emit light when it went from one
discrete orbital to another orbital. It would not go anywhere in between.
The lowest energy radius can actually be calculated using both classical and quantum formulas. And that radius
turns out to be 5.29 X 10^-11 meters, which is also called the Bohr radius.
Bohr’s solution rested on the heels of Max Planck, who in 1900 had shown that energy
was quantized in discrete chunks. He introduced Planck’s constant, h, from his work on the
radiation emitted by a black body. This led to the idea that energy is quantized according
to this equation. Where Planck’s constant is the proportionality
factor between the energy, and the frequency of the energy emitted. Bohr noticed
that Planck’s constant had units of angular momentum, so he guessed that this was the
minimum angular momentum that an electron would need to have to remain in a stable orbit. And
it explained how there were spontaneous emissions of photons when the electron in the atom goes
from a high energy state to a lower one, proportional to this constant.
Now, you might ask, how does Planck’s equation lead to the uncertainty principle? So let’s
figure that out. Recall that in the famous double slit experiment,
we see that when monochromatic light goes through two slits, we get an interference pattern
just like we would with a wave. There is a brighter pattern in the center and not as
bright the further away you go. What would this pattern look like with just a single
slit? Well, we get a diffraction pattern where we
see a central very high peak, and very faint peaks elsewhere, much fainter than with the
double slit. For our purposes here, as an approximation, let’s ignore the faint peaks
and analyze only the central peak. We can do the same thing with electrons instead
of light. We are going to shoot electrons through the single slit. Just like light,
as the electrons go through the slit, they spread out. They are waves after all.
Let’s use some basic geometry to figure out the width of the pattern we will get on
the other side. If we draw a line from the slit to the edge of the pattern, and also
draw line from the slit to the center of the pattern, we can see that it will make an angle
theta near the slit. Why is there an angle, and how is the angle
determined? Well, if we do a closeup of the slit, we can see two waves that will destructively
interfere at the edge of the pattern. They destructively interfere, or cancel each other
out, if they are out of phase by the wavelength lambda.
This is because the top wave will cancel the bottom wave at every position point where
the waves are one half wavelength apart. This would be everywhere in between the two waves.
Using basic trigonometry, we see that lambda equals d sine theta.
Using geometry, we also see that Theta is the same as our original angle theta. Notice
that if lambda, the wavelength doesn’t change then as d, the width of the slit gets smaller,
theta gets larger. And this is exactly what we see when we do such an experiment. The
width of the pattern enlarges as D becomes smaller. This is counter intuitive only if you think
of the electron as a discrete particle. It is not so counter intuitive if you think of
it as a wave. Note that D is the width of the slit. It represents
all the spots where the electron could be as it goes through the slit, so D really represents
the uncertainty in the electron’s position. We will call this delta x.
The distance from the slit to the point where the waves cancel is the same as the momentum
p. The uncertainty of this momentum is the distance
between the edge and the center of the pattern. We will call this delta P. Again using trigonometry,
p sine theta equals delta p Louis de Broglie had shown that the wavelength
lambda for a particle with mass, like an electron, is equal to h over p, where h is Planck’s
constant, and p is the momentum. If we substitute h over p for lambda in our
original equation, we get that it equals d sine theta, or delta x sine theta. Now some
rearranging shows that h equals delta x, p sine theta. Substituting delta p for p sine
theta, we get delta x times delta p equals h.
So this is essentially what the uncertainty principle is. The uncertainty in the position
times the uncertainty in the momentum is approximately equal to Planck’s constant. Now this is
slightly different than how the equation is typically written with is greater than or
equal to h/4*pi, because of some imprecisions in this approach to the derivation. But you
get the general idea. A more precise approach requires higher level
math and use of Fourier transforms. I will put a link in the description if you want
to see that rigorous mathematical derivation. Now don’t click away because we are coming
to the best part. We’re going to relate this to the Bohr model, and to the tennis
ball. The important thing to remember is that the
uncertainty in the momentum and the uncertainty in the position are inversely related. As
one gets bigger, the other gets smaller. And critically, this is not a limitation of our
ability to measure the position and momentum. It is a limitation of reality. It is not an
observer effect. Now let’s put all this together to save
our atom. How does this relate to the Bohr model of the atom?
If you do the calculations, this is what we come up with. The Bohr radius, where Niels
Bohr calculated the electron would be at its lowest energy state in the hydrogen atom is
5.29x 10^-11 m. The velocity can be calculated to be 2.18 x 10^6 m/s, for the same
Bohr model electron. And we know the measured mass of the electron is 9.11 x 10^-31 kilograms.
Now we can calculate the momentum of this electron because momentum is just mass times
velocity. This would be equal to about 2 x 10^-24 kg*m/s. But there is always
some uncertainty in the velocity, so for example if there was a 10% uncertainty, then delta
p would be one tenth of this or 2 x 10^-25 kg m/s.
If we plug this into our uncertainty equation, and calculate what the uncertainty in the
position would be, we get the following: Delta X is equal to h which equals to 6.63
X 10^-34 m^2*kg/s divided by 4 pi times 2 x 10^-25 kilograms*meters/second. So delta
x, or the uncertainty in the position of the electron is about 26 x 10^-11 meters.
If we compare this to the Bohr radius, we see that it is almost 5 times the radius. So
the uncertainty in the position is vastly larger than what Bohr calculated to be the
preferred radius of the electron. And this is based on just a 10% uncertainty in momentum.
A 10% uncertainty results in a huge uncertainty in the position. What does this tell us?
It tells us that the Bohr radius is fundamentally wrong, or at least incomplete. The truth is
that the electron doesn’t just sit at a radius of 5.29 x 10^-11 meters. There is a
constant balance between the uncertainty in its momentum and position, such that it forms
a cloud of probability around the proton. This cloud extends far below and far beyond
the Bohr radius. And using the Schrodinger equation, we can calculate the probability
of finding it at certain locations if we were to measure it.
So if this is wrong, why do we keep teaching students about Bohr’s inaccurate model. Well, we use it
because it is useful for approximate calculations, and explains a number of features of atoms
on the periodic table, so it is useful for chemistry.
But the Heisenberg uncertainty principle is the essential concept of quantum mechanics.
And it is difficult for most people to understand because it is not something we experience
in our macro world. But the truth is that the uncertainty equation also applies to big
things like a tennis ball. So for example, if we used the same equation
and put in the mass of a tennis ball which is about 0.06 kg, and the speed of a serve by
Venus Williams, who has one of the fastest serves at 57 meters per second (128 miles
per hour). And since we can calculate the mass and speed fairly accurately, let’s
say there is a 1% uncertainty in the momentum. Delta p would be 0.034 kg * meters/second.
If we then calculate the uncertainty in the position, we get that the uncertainty in the
position, delta x equals 1.55 x 10^-33 meters – this is so small that we would never notice
it. Similarly, we don’t notice it in anything else either that we can see with our eyes.
The big lesson to be learned here is that the central concept in quantum mechanics – which
comes from Planck’s equality and the uncertainty equation is only noticeable at super tiny
scales. We don’t experience it in our everyday visible
world, so many people don’t think it is real. But I want most is for you to walk away
from this video realizing that the way the real world works is counter intuitive, invisible
and even illogical, but it is…the truth! I’d like to thank my generous supporters
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