This video explains how Heisenberg discovered Matrix Mechanics, the original version of Quantum Mechanics.
*https://www.youtube.com/watch?v=BcNHTv7Rw3c
**https://300.ya.ru/v_z7y6RNR6
таймкоды
00:00:00 Введение в квантовую механику
- Каждый элемент во Вселенной обладает уникальными свойствами.
- Вернер Гейзенберг открыл квантовую механику, сформулировав матричную механику.
00:00:17 Отпуск Гейзенберга и его открытие
- В мае 1925 года Гейзенберг попросил отпуск из-за сенной лихорадки.
- На острове Гельголанд он сформулировал первую математически непротиворечивую теорию квантовой механики.
00:01:17 Критика модели Бора
- Модель Бора не учитывала наблюдаемые признаки более сложных атомов.
- Гейзенберг предложил новую теорию, основанную на наблюдаемых характеристиках атомов.
00:02:48 История атомной модели
- Философы Греции и Индии независимо друг от друга рассматривали вопрос о неделимости объектов.
- Джон Дальтон в 1808 году предложил теорию атомов, объясняющую законы химии.
00:04:34 Открытие электрона
- Джей Джей Томпсон открыл электрон в 1897 году, изучая катодные лучи.
- Томпсон разработал модель атома «сливового пудинга».
00:06:00 Эксперимент Резерфорда
- Эрнест Резерфорд и его команда обнаружили ядро атома в 1909 году.
- Ядро состоит из положительно заряженных частиц, вокруг которых вращаются электроны.
00:07:59 Спектральные линии
- Уильям Волластон заметил чёрные линии в спектре солнечного света.
- Густав Кирхгоф объяснил спектральные линии, связанные с поглощением и излучением света газами.
00:09:53 Формула Ридберга
- Иоганн Балмер обнаружил математическую зависимость между длинами волн света и спектральными линиями водорода.
- Йоханнес Ридберг модифицировал формулу, открыв знаменитую формулу Ридберга.
00:11:28 Модель Бора
- Нильс Бор предложил новую модель атома в 1913 году, решив проблему нестабильности атомов согласно теории Максвелла.
- Модель Бора включала четыре постулата, первые два из которых устраняли несоответствие между теориями Максвелла и Резерфорда.
00:12:59 Теория Максвелла и квантование
- Теория Максвелла неприменима к объектам размером с атом.
- Планк предположил, что энергия света может быть только в целых числах, кратных постоянной Планка.
- Эйнштейн показал, что идея Планка даёт представление о природе света.
00:13:55 Постулаты Бора
- Третий постулат Бора: электрон не может вращаться по орбите в любом месте, а только при определённых фиксированных значениях момента импульса.
- Угловой момент электрона должен быть целым числом, кратным уменьшенной постоянной Планка.
00:14:55 Квантовые скачки
- При перемещении между орбитами электрон поглощает или испускает фотон с энергией, равной разнице энергий между орбитами.
- Модель Бора объясняет спектральные линии водорода.
00:15:48 Проблемы модели Бора
- Модель Бора хорошо описывает спектр водорода, но не работает для атомов с более чем одним электроном.
- Макс Борн предложил перестроить систему понятий физики.
00:16:46 Переосмысление Гейзенберга
- Гейзенберг отказался от ненаблюдаемых величин, таких как орбита электрона.
- Он предложил создать квантовую механику, основанную на соотношениях между наблюдаемыми величинами.
00:18:52 Комбинационный принцип Ридберга-Ритца
- При сложении двух частот в спектре водорода получается третья частота, также присутствующая в спектре.
- Этот принцип известен как комбинационный принцип Ридберга-Ритца.
00:22:18 Замена орбит на амплитуды переходов
- Гейзенберг заменил основные частоты наблюдаемыми, а коэффициенты Фурье — амплитудами переходов.
- Амплитуды переходов связаны с интенсивностью наблюдаемых частот.
00:24:23 Матричное умножение
- Гейзенберг обнаружил, что амплитуды должны умножаться по правилам матричного умножения.
- Порядок умножения величин имеет значение, что является основным свойством матриц.
00:25:15 Публикация статьи Гейзенберга
- Гейзенберг поделился своими результатами с Максом Борном.
- Борн понял, что это матричное умножение, и отправил статью для публикации.
- Атомная модель начала переосмысливаться, квантовая революция шла полным ходом.
In this video
Intro
0:00
Every element in the universe has a
0:02
unique signature.
0:04
This simple idea led a young German
0:07
physicist to discover the most
0:09
revolutionary scientific theory of the
0:12
entire 20th century, quantum mechanics.
0:17
In May 1925, Verer Heisenberg asked his
0:21
mentor Max Bourne for a 14-day leave of
0:24
absence. The reason, a severe attack of
0:28
hay fever that left Heisenberg with a
0:30
swollen face, a stuffy nose, and a foggy
0:34
mind. In his own words, Heisenberg
0:36
recounted, «I made straight for
0:38
Helgoland, where I hope to recover
0:40
quickly in the bracing sea air far from
0:44
blossoms and meadows.»
0:46
Little did he know, this short trip
0:48
would bring about a dramatic paradigm
0:51
shift in fundamental physics. It was
0:54
here that Heisenberg would formulate the
0:56
first mathematically consistent theory
0:59
of quantum mechanics, what would later
1:01
be called matrix mechanics. At the
1:04
forefront of his mind was the prevailing
1:06
model of the structure of the atom, the
1:09
bore model, which pictured an atom as
1:11
consisting of electrons orbiting a
1:13
nucleus, just like planets orbit around
1:16
the sun. The Boore model was known to be
1:19
inadequate to many physicists including
1:22
Heisenberg and even Neils Boore himself.
1:26
Although it worked for hydrogen, it
1:28
entirely failed to account for the
1:29
observed signatures of more complicated
1:32
atoms.
1:33
As he pondered the inadequacy of Boore’s
1:36
model, Heisenberg made a brilliant leap
1:39
of abstraction. What if picturing the
1:41
electron as having an orbit was the
1:44
fundamental problem? We never actually
1:46
observe the orbits of electrons. So
1:49
Heisenberg reasoned that we should
1:51
instead formulate a new version of
1:53
physics which made no reference
1:55
whatsoever to electron orbits. The new
1:58
theory should be entirely based on
2:00
quantities that we actually observe and
2:03
nothing else. In this case, Heisenberg
2:06
focused on the observed signatures of
2:08
atoms, what scientists call atomic
2:11
spectra. To truly appreciate
2:13
Heisenberg’s brilliant leap and get a
2:16
sense for his thought process that led
2:18
to quantum mechanics, we’ll now take a
2:20
tour of the history of the atomic model
2:23
and its relation to observed spectra.
2:26
Along the way, we’ll gain a deep
2:28
understanding of Heisenberg’s discovery
2:30
of matrix mechanics, how it
2:32
revolutionized our concept of the atom,
2:35
and the reason why matrices were so
2:37
fundamental to his work.
2:48
As early as 2500 years ago, two
History of Atomic Model
2:51
philosophers, one from Greece and the
2:54
other from India, both independently
2:56
considered whether an object could be
2:59
divided into smaller and smaller pieces
3:01
forever. They both believed that at a
3:04
certain point we would arrive at
3:06
something fundamental, an indivisible
3:08
object that could not be further
3:10
divided. [music] In Greek, this object
3:13
was atos. In Sanskrit, a new. It
3:17
wouldn’t be until more than 2,000 years
3:19
later that a modern theory of the atom
3:22
based on scientific evidence would
3:24
emerge. In 1808, the British chemist
3:28
John Dalton proposed that all matter was
3:30
made up of tiny indivisible objects
3:33
called atoms. [music] Though in
3:35
substance, his idea was very much like
3:38
these earlier thinkers, Dalton’s theory
3:40
was significantly more developed.
3:43
According to him, every atom was a
3:45
sphere of a certain mass and each
3:48
element consisted purely of identical
3:50
atoms of that element. So hydrogen was
3:54
made of a collection of one type of
3:56
atom, helium another type of atom,
3:59
oxygen [music] yet another and so on. It
4:02
was the characteristics of the atoms
4:04
that gave the elements their properties.
4:07
Moreover, these different types of atoms
4:09
could also rearrange and combine with
4:12
other atoms to form new compounds.
4:15
Dalton’s theory could explain all the
4:17
laws of chemistry at the time and
4:19
remained the standard model of the atom
4:21
for almost 100 years. But [music] in
4:23
1897, this model was shattered by a
4:26
British physicist named JJ Thompson.
Electron Discovered
4:29
Thompson was interested in studying the
4:31
behavior of something called cathode
4:33
rays. When two plates, one positively
4:36
charged and the other negatively
4:38
charged, were separated by a small
4:40
distance, a stream of particles moved
4:42
from the negative plate to the positive
4:44
plate. It was unclear what these
4:46
particles were exactly, so physicists
4:49
just called it a cathode ray. Now,
4:51
Thompson had a clever idea. He added
4:54
another two plates within the path of
4:57
the cathode ray. Once he did this, he
4:59
observed that the cathode ray was
5:01
deflected up toward the positive plate.
5:04
So the rays must be made up of
5:06
negatively charged particles. Particles
5:10
known today as electrons. That’s right.
5:13
It wasn’t until this experiment in 1897
5:16
that the electron was discovered. A
5:19
truly remarkable discovery that earned
5:20
Thompson a Nobel Prize 10 years later.
5:24
After conducting this experiment with
5:26
various types of materials, Thompson
5:29
synthesized his results and came up with
5:32
a new model of the atom, the plum
5:35
pudding model. Inspired by Dalton’s
5:37
earlier model, atoms were spheres that
5:40
had no net charge. Since electrons had a
5:43
negative charge, according to Thompson,
5:46
they must be embedded in some sort of
5:48
positively charged region of the sphere,
5:51
just like raisins embedded in [music]
5:53
pudding. This model would last for only
5:55
about 10 years until the groundbreaking
5:57
work of one of Thompson’s students,
Nucleus Discovered
6:00
Ernest Rutherford.
6:02
In 1909, a researcher named Hans Guyger
6:06
and an undergraduate named Ernest
6:08
Marsden worked with their adviser Ernest
6:11
Rutherford to perform an experiment that
6:13
would put Thompson’s plump pudding model
6:15
to the test. They began with a sample of
6:18
radium stored in a box with a very small
6:21
hole. It was known at the time that
6:23
radium naturally decays and emits
6:25
positively charged particles called
6:27
alpha particles. The stream of alpha
6:30
particles were then aimed at a piece of
6:32
gold foil which was surrounded by a
6:34
circular-shaped screen made of zinc
6:37
sulfide.
6:38
As the alpha particles hit the gold
6:40
foil, they would scatter at different
6:42
angles and produce light flashes on the
6:45
screen.
6:46
If the plum pudding model was correct,
6:49
Rutherford expected to see most of the
6:51
alpha particles pass straight through
6:53
the gold foil with some getting slightly
6:56
deflected due to an encounter with an
6:58
electron.
7:00
For most of the experiment, this is
7:02
indeed what they saw.
7:05
But to their utter shock, for every one
7:07
in around 8,000 detections, they saw
7:10
that the alpha particle essentially
7:12
bounced right back at an angle very
7:14
close to 80°.
7:17
As Rutherford would describe it, it was
7:20
quite the most incredible event that has
7:22
ever happened to me in my life. It was
7:25
almost as incredible as if you fired a
7:27
15-in shell at a piece of tissue paper
7:30
and it came back and hit you. Rutherford
7:33
concluded that in order for the alpha
7:35
particles to bounce back, there must be
7:38
something small, dense, [music] and
7:40
positively charged that it interacted
7:42
with. The nucleus of the atom was
7:46
discovered. In Rutherford’s updated
7:48
model, the atom now consisted of a
7:50
massive positively charged nucleus at
7:53
the center with negatively charged
7:55
electrons orbiting like planets around
7:57
the sun. Fittingly, the next development
8:01
in the model of the atom that would pave
8:02
the way for Heisenberg came from
History of Atomic Spectra
8:04
observations of the sun.
8:07
You see, in the early 1800s, when the
8:10
physicist William Wallist used a prism
8:13
to examine light from the sun, he
8:15
observed an interesting pattern. The
8:17
light not only split into the various
8:20
colors of the rainbow, producing a
8:22
beautiful spectrum, but there were also
8:24
certain black lines scattered
8:26
throughout. It was as if some of the
8:28
colors were just missing. It wasn’t
8:31
until the 1850s that Gustaf Kirkoff
8:34
developed a limited understanding of
8:36
these spectral lines. What he observed
8:39
was that when he shined a light in the
8:41
laboratory, it produced a continuous
8:43
spectrum with no gaps at all. However,
8:46
when the light was passed through a cool
8:48
lowdensity gas, the gaps in the spectrum
8:51
started [music] to appear. Additionally,
8:53
if he removed the light source and
8:55
instead slowly heated up the same gas
8:58
until [music] it started glowing, he
9:00
discovered that he would see the
9:02
reverse. The continuous spectrum would
9:05
not be present, but there would be clear
9:08
visible signs that corresponded exactly
9:10
to where the missing lines were earlier.
9:13
So, the gas must be somehow absorbing
9:15
some of the light in this case,
9:17
resulting in an absorption spectrum.
9:21
In the case of the gas heating up, it
9:23
emitted a very specific color, producing
9:26
an emission spectrum. While certainly a
9:29
step forward in understanding spectra,
9:32
Kirkoff still didn’t know why this
9:34
occurred. Another important step
9:36
occurred in 1855
9:39
due to the work of a mathematician named
9:41
Johan Balmer. After carefully examining
9:44
the spectral lines of hydrogen, he
9:47
discovered a mathematical relationship
9:49
between wavelengths of light [music] and
9:52
these lines. The relationship he found
9:54
was given by this formula where h is a
9:57
constant, n= 2 and m can be an integer
10:01
greater than 3. Just a few years later,
10:04
the Swedish physicist Johannes Ryberg
10:07
modified it and discovered the famous
10:09
Ryberg formula where again n= 2, m= an
10:13
integer greater than 3 and r is a
10:16
constant. This formula worked incredibly
10:18
well for calculating the observed
10:20
spectral lines of hydrogen. Another
10:23
thing that works incredibly well is this
10:26
video sponsor, brilliant.org. If you’re
10:29
looking for a new way to hone your
10:31
skills in math, science, or computer
10:33
science, Brilliant is a great place to
10:36
begin. Lately, I’ve had a lot of fun
10:38
going through their new coordinate
10:39
transformations course, which you can
10:41
see takes a beautiful visual approach to
10:44
learning. They’ve also added tons of new
10:46
courses recently and even offer
10:48
personalized practice problems now, so
10:50
you can get instant feedback and test
10:52
your mastery of any topic. If you’d like
10:55
to try it out, you can begin learning
10:57
for free on Brilliant for a full 30 days
11:00
by either going to
11:01
brilliant.org/abidebyreason,
11:03
scanning the QR code on screen, or
11:06
clicking on the link in the description.
11:08
Brilliant’s also given my viewers 20%
11:10
off an annual premium subscription,
11:13
which gives you unlimited daily access
11:15
to everything on Brilliant. Though
11:17
Ryberg’s formula accurately predicted
Bohr Model
11:19
the hydrogen spectrum, a deeper
11:22
understanding of why would not occur
11:24
until the arrival of the Danish
11:26
physicist Neils Boore. In 1913, when
11:30
Boore proposed a new model of the atom,
11:33
he was well aware of the observed
11:36
hydrogen spectrum. More importantly, he
11:39
was also aware of a fundamental flaw in
11:42
Rutherford’s model. In the late 1800s,
11:45
James Clark Maxwell discovered a set of
11:48
four equations that described the
11:50
behavior of electricity and magnetism.
11:52
According to Maxwell’s theory, any
11:55
accelerating charged particle must emit
11:57
energy in the form of electromagnetic
12:00
radiation. And since the electrons in
12:03
Rutherford’s model orbited the nucleus,
12:05
they were continuously changing
12:07
direction and thus accelerating.
12:10
Consequently, they must radiate energy.
12:13
So a huge problem emerged. If you
12:16
combined Maxwell’s theory with
12:18
Rutherford’s, the electrons should lose
12:20
energy so quickly that they would spiral
12:23
into the nucleus in a fraction of a
12:25
second. In other words, stable atoms
12:28
should not exist. Bore’s updated model
12:31
of the atom sought a solution to this
12:34
dilemma. It consisted of four postulates
12:38
with the first two directly addressing
12:40
the inconsistency between Maxwell’s and
12:43
Rutherford’s theory. First, he
12:45
postulated in agreement with Rutherford
12:48
that the electron moves in a circular
12:50
orbit around the nucleus with the
12:52
acceleration caused by the positive and
12:54
negative electrical attraction between
12:56
the two.
12:59
Second, he postulated that Maxwell’s
13:01
theory just simply doesn’t apply to
13:04
objects as small as an atom. Even though
13:07
the electron is constantly accelerating,
13:09
it does not emit any electromagnetic
13:12
radiation.
13:14
The next two postulates were inspired by
13:16
the recent work of Max Plank and Albert
13:19
Einstein. As a result of his work trying
13:21
to solve the famous UV catastrophe,
13:24
Plank proposed that the energy of light
13:27
can only come in integer multiples of
13:29
some discrete amount. E= HF, where H is
13:34
Plank’s constant and F is the frequency
13:37
of light. This assumption, which Plank
13:40
referred to as an act of desperation on
13:42
his part, was simply a mathematical
13:44
trick. He had no clue why it worked, but
13:48
it led to an incredibly accurate
13:50
prediction that matched the experimental
13:52
observations of black body radiation.
13:55
Then in 1905, Albert Einstein took this
13:59
further and showed that Plank’s idea
14:01
wasn’t [music] just a wild guess, but
14:03
was actually giving us insight into the
14:06
nature of light. There really were
14:08
photons, or discrete packets of light,
14:10
that had energies equal to HF. Einstein
14:14
used this formula to explain the
14:16
photoelectric effect, a phenomenon where
14:19
light ejects electrons from a metal
14:22
which only occurs when a certain minimum
14:24
frequency is reached. Influenced by
14:26
planks and Einstein’s quantization idea
14:30
or’s third postulate was that the
14:32
electron cannot orbit at just any
14:34
location but rather could only orbit at
14:37
certain fixed values values for which
14:40
the angular momentum of the electron is
14:42
equal to certain multiples of plank’s
14:44
constant or rather in this case a factor
14:47
of 2 pi came in and the angular momentum
14:50
was integer multiples of h bar the
14:53
reduced plank’s constant The final
14:56
postulate was similarly influenced by
14:58
quantization. As the electron moved
15:01
between these fixed orbits, it did so by
15:04
either absorbing or emitting a photon
15:07
where the energy of the photon was equal
15:09
to the energy difference between orbits.
15:12
This movement came to be known as a
15:14
quantum jump since it was believed to
15:16
happen instantaneously.
15:18
Amazingly, with this model, Bor was able
15:21
to find the correct energy levels of the
15:23
hydrogen atom and finally explain why
15:26
hydrogen produced those spectral lines
15:28
that Ryberg’s formula predicted. As the
15:31
electron underwent these so-called
15:33
quantum jumps, it released a photon with
15:36
a specific frequency of light, exactly
15:39
coinciding with what was observed in the
15:42
hydrogen spectrum.
15:48
It is here that we can finally
15:50
understand the model that was at the
15:52
forefront of Heisenberg’s thinking
15:54
during his retreat in Heloland. [music]
15:58
Heisenberg already knew that Bor’s model
16:00
had to be modified in some way.
16:04
It worked incredibly well for
16:06
reproducing the spectrum for hydrogen
16:08
and other atoms with only one electron.
16:12
But it completely failed for every atom
16:14
that [music] had more than one electron.
16:18
In fact, in 1923, the situation was so
16:21
dire that Heisenberg’s mentor Max Bourne
16:24
stated the whole system of concepts of
16:27
physics must be reconstructed from the
16:30
ground up. The next year in 1924,
16:33
Heisenberg decided to visit Boore
16:35
himself at the Institute for Theoretical
16:38
Physics in Denmark. Together, they
16:40
puzzled over the problems with Boore’s
16:42
model and what could be done about
16:44
formulating a better theory. They worked
16:46
together for over a year with little
16:49
success. A breakthrough would eventually
16:52
come in the summer of 1925 when
16:55
Heisenberg worked in isolation during
16:58
his Helgoland retreat. The
Heisenberg’s Reinterpretation
17:00
groundbreaking paper that Heisenberg
17:02
worked on during this trip is known as
17:04
the umung paper, German for
17:07
reinterpretation
17:08
and reinterpret he did. Heisenberg
17:12
begins this paper with a radical
17:14
overhaul of previous physics ideas.
17:17
According to him, it seems more
17:20
reasonable to completely discard
17:22
unobserved quantities like the electrons
17:24
orbit and period and instead try to
17:27
establish a theoretical quantum
17:29
mechanics analogous to classical
17:32
mechanics but in which only relations
17:34
between observable quantities occur. It
17:37
is this opening statement from
17:39
Heisenberg that quantum theory is only
17:42
about observable quantities and nothing
17:44
else that would spark countless debates
17:47
over the next decade over the real
17:49
meaning of quantum mechanics. Debates
17:52
between figures like Einstein,
17:54
Schroinger, Paulie, Bourne, Boore and
17:58
Heisenberg himself amongst many others.
18:01
So, Heisenberg begins in a totally
18:03
different way than how previous
18:05
modifications to the atomic model
18:07
occurred. Instead of adding an object to
18:10
the model and slightly changing it, as
18:12
many of his predecessors did, Heisenberg
18:15
immediately discards the very concept of
18:18
electron orbits altogether. But if Bor’s
18:21
electron orbits are not in the theory,
18:24
what are the observable quantities
18:26
Heisenberg allows? Well, we never
18:28
observe the actual orbit, but we do
18:31
observe the transitions or rather the
18:34
effects of the transitions,
18:36
the spectral lines. These are truly
18:39
empirical quantities, things we can
18:42
actually observe in a laboratory. And
18:45
these lines occur at certain
18:47
frequencies, which for the hydrogen atom
18:49
are given by Ryberg’s formula.
18:53
If we now try to add two frequencies
18:55
that appear in the hydrogen spectrum,
18:58
something very interesting happens.
19:00
First, we include indices for each
19:02
frequency to keep track of the integer
19:05
values. Then we add
19:08
the middle indices cancel out and we get
19:11
a third frequency that also appears
19:13
somewhere in the spectrum. So for
19:16
example, if we took the frequency
19:17
corresponding to the quantum jump from
19:20
n= 6 to k= 3 and added it to the jump
19:24
from k= 3 to m= 1, this would equal the
19:28
frequency corresponding to the jump from
19:31
6 straight to 1. So whenever you add two
19:34
observed frequencies, you don’t just get
19:37
an arbitrary frequency, but always
19:40
another frequency in the spectrum. This
19:44
amazing fact wasn’t just a nice
19:46
mathematical tool, but a wellestablished
19:49
empirical fact. It worked for all the
19:52
frequencies of hydrogen, even beyond the
19:54
visible spectrum into the infrared and
19:57
ultraviolet region. This law of addition
20:00
was known as the Ryberg Ritz combination
20:03
principle and it’s one of the central
20:05
ideas to grasp in order to understand
20:07
what Heisenberg was doing in his
20:10
reinterpretation paper. So, how did
20:12
Heisenberg use this to radically remodel
20:15
the atom? In Bour’s model, the electron
Heisenberg’s Use of Rydberg-Ritz Combination Principle
20:18
orbits around the nucleus. And a
20:20
convenient way physicists at the time
20:22
studied this was to consider it as a
20:25
one-dimensional system where the
20:27
electron was undergoing simple periodic
20:30
motion.
20:35
In classical mechanics, the trajectory
20:37
of the electron here would then be
20:39
described by some function x of t and
20:42
the rate at which it repeats itself,
20:45
which is referred to as the fundamental
20:48
frequency omega.
20:51
Since this motion is periodic, it’s
20:53
possible to represent it using a forier
20:56
series which consists of an infinite sum
21:00
where each of the terms is called a
21:02
harmonic. So each harmonic is written in
21:04
terms of time of frequency and something
21:08
called a forier coefficient.
21:10
Now every energy level of the hydrogen
21:13
atom will have a different fundamental
21:15
frequency.
21:17
So omega has a dependency on what energy
21:20
level n the electron is at. We can then
21:23
encode this as follows.
21:28
One of Bor’s insights was that every
21:30
possible quantum jump corresponded to
21:33
one of these harmonics.
21:35
So the quantum jump from the nth level
21:37
to n minus one corresponds to the first
21:41
harmonic. The jump from n to n minus2
21:44
corresponds to the second harmonic and
21:46
so on.
21:48
But remember bor’s model only worked for
21:51
atoms with one electron and failed for
21:54
everything else. This is where
21:56
Heisenberg’s brilliant leap of insight
21:59
comes to the rescue. Heisenberg wanted a
22:01
theory that only relied on observable
22:04
quantities. And we never actually
22:06
observe this motion nor any of these
22:09
fundamental frequencies.
22:12
But there are frequencies we do observe
22:15
the frequencies in the spectra. So
22:18
Heisenberg replaced the fundamental
22:20
frequencies with the observed
22:23
frequencies
22:25
and the forier coefficients with
22:27
corresponding transition amplitudes
22:30
which are directly related to the
22:32
intensity of the observed frequencies.
22:36
The general form for all transitions
22:38
will then be written as follows.
22:42
And the electron’s orbit has now been
22:44
entirely replaced by a whole set of
22:47
purely observable quantities. [music] Or
22:50
rather to be more precise in the case of
22:52
the transition amplitudes something that
22:55
is directly connected to the observable
22:57
intensity. As Heisenberg would say, one
23:00
may readily regard the ensemble of these
23:02
quantities as a representation of the
23:05
quantity X. Now since each of these
23:08
terms uses two states in its
23:10
formulation, what Heisenberg has done
23:13
here has been to take the classical
23:15
forier harmonics of the electrons orbit
23:18
and turn them into elements of a matrix.
23:21
This becomes readily apparent when you
23:23
calculate x^2. For the classical
Why Matrices Appear
23:26
calculation, we have this
23:29
relabeling results in this expression
23:34
and the frequencies just add according
23:36
to normal addition rules.
23:40
If we now [music] attempt to apply
23:42
Heisenberg scheme, we get this
23:44
expression. And the crucial question is
23:47
how [music] do we add frequencies here?
23:50
Remember these are observed frequencies.
23:53
So they must add according to the Ryberg
23:55
Ritz combination principle. So any
23:58
observed frequencies should add together
24:00
to produce a third observed frequency
24:03
and the indices must match accordingly.
24:06
By staying grounded in observations,
24:09
Heisenberg knew that the frequencies
24:11
must add this way. Consequently, the
24:13
transition amplitudes and frequency
24:16
labels must be given by this instead.
24:19
Reabeling again, we can now correctly
24:21
add the frequencies.
24:23
And we also find that the amplitudes now
24:26
must multiply in [music] this manner,
24:29
which is just matrix multiplication.
24:31
Though Heisenberg had no clue what it
24:34
was, he was even more puzzled when he
24:36
realized that if you consider two
24:38
quantities X and Y, the order in which
24:41
you multiply them matters. So X * Y in
24:45
general does not equal Y * X. a basic
24:48
property of matrices but a total mystery
24:51
to Heisenberg. Reflecting on this
24:54
important discovery, he said, «At first,
24:56
I was deeply alarmed. I had the feeling
24:59
that through the surface of atomic
25:01
phenomena, I was looking at a strangely
25:04
beautiful [music] interior and felt
25:06
almost giddy at the thought that I now
25:08
had to probe this wealth of mathematical
25:10
structures nature had so generously
25:13
spread out before me.» Upon leaving his
25:16
Hellooland retreat, Heisenberg finished
25:19
up writing his paper and returned to
25:21
work with his adviser, Max Bourne. He
25:24
shared his new results and asked Borne
25:26
[music] whether it was worth publishing.
25:28
Bourne was also puzzled at first by
25:30
seeing this strange [music]
25:31
multiplication rule, but soon it
25:34
reminded him of something he had seen in
25:36
a linear algebra course many years ago.
25:38
He realized it was matrix
25:40
multiplication, of course. He sent
25:42
Heisenberg’s paper off for publication
25:45
himself and then proceeded to develop
25:47
matrix mechanics in a mathematically
25:49
rigorous way. [music] The atomic model
25:51
was just beginning to be reimagined and
25:54
the quantum revolution was well
25:56
underway. If you enjoyed this video,
25:59
please consider supporting the channel
26:00
by becoming a member. And if you want an
26:04
even more indepth technical explanation
26:06
of Heisenberg’s matrix mechanics, go
26:10
check out this beautiful video by Dr.
26:12
Jorge Diaz, which I learned an
26:14
incredible amount