Electrons DO NOT Spin
Summary
TLDR量子力学には多くの奇妙な現象がありますが、量子自転という現象は特に理解が難しいとされています。この動画は、電子の量子自転がどのように物質の構造に影響を与え、またどのように量子力学の深遠な洞察につながるかを探求しています。実験と歴史的背景を交えて、量子自転の磁的性質、量子力学での自転の記述方法、そして自転がどのように粒子の相互作用と物質の構造に関わるかを解説しています。
Takeaways
- 量子力学には多くの奇妙な現象がありますが、量子自転という現象は誰も理解していません。
- 物理学の授業で行われる経典的なデモンストレーションは、角動量保存の説明に役立ちます。
- アイアンシリンダーが垂直方向の磁場に置かれ、急に定常速度で回転することを示す実験は、エインシュタイン・デハース効果と呼ばれます。
- 電子は自転していませんが、角動量を持っているようです。これは量子力学的なものであり、電子のような粒子の性質です。
- Stern-Gerlachの実験では、銀原子が磁場を通過し、原子内の電子が磁矩を与えるため、特定の方向に偏向します。
- 量子自転は、電子の性質として量子化されています。測定する方向に依存するため、古典的な回転とは異なります。
- 量子力学では、スピンを記述するために、特別な数学的对象であるスピンオルを使います。
- スピンオルは、360度の回転ではなく、720度の回転が必要とされます。これは、電子のような粒子がどのように振るうかを説明するのに役立ちます。
- 粒子の線形モメンタムは、その位置に根本的に関連しています。同様に、角動量は、粒子の方向性に関連しています。
- スピンは、粒子が持つ回転の自由度から生じるため、角動量は定義できます。
- 粒子はスピン数で区別され、半整数(fermion)または整数(boson)です。この違いは、粒子の相互作用に深い影響を与えます。
- fermionの「反社会的的な」性質は、物質の構造を形成するポアリンエクザル原理に責任があります。
- 量子統計学の定理は、スピンと粒子の性質に関連する現象を説明します。
Q & A
量子スピンとは何ですか?
-量子スピンは、電子などの粒子が持つ非常に奇妙な種類の角動量です。これは、古典的な回転のように考えられがちですが、電子は古典的に回転しているわけではありません。量子スピンは、粒子の基礎的な量子力学的プロパティであり、质量や各種の電荷と同様に重要です。
アインシュタイン・デハアス効果とは何ですか?
-アインシュタイン・デハアス効果は、1915年にアインシュタインとデハアスが行った実験によるものです。この実験では、鉄の円筒をスレッドからぶら下げ、垂直方向の磁場をかけることで、円筒がすぐに一定の速度で回転することを示しました。これは、角動量保存の原理に反するように見える現象です。しかし、実際には、外部の磁場が鉄を磁化し、鉄の外殻の電子が自らのスピンを整列させることで、角動量が保存されます。
Zeeman効果とは何ですか?
-Zeeman効果は、原子が外部の磁場に置かれると、電子がエネルギーレベル間を跳躍させる際に放出されるフォトンの特定の波長が、磁場の存在によって分裂する現象を指します。この効果は、古典的な物理学の考え方で説明できますが、さらに複雑な分裂が観察されたことから、異常Zeeman効果と呼ばれる新たな現象が発見されました。
Stern-Gerlach実験とは何ですか?
-Stern-Gerlach実験は、1921年にオットー・スターンが提案し、翌年ヴァルター・ゲルラークが行った実験です。この実験では、銀の原子が磁場のグラディエントを通過するように射出され、原子の外殻の単一電子が原子に磁矩を与えるため、外部の磁場によって原子に力が働き、それが原子を偏向させます。予想される乱反射の代わりに、銀の原子が2つの地点にのみ当たり、最も極端な偏向に対応する場所に現れます。
量子力学におけるスピンの数学的表現は何ですか?
-量子力学において、スピンはスピンオブジェクトである特別な数学的オブジェクトとして表現されます。これは、一般的なベクトルとは異なり、360度の回転で元の状態に戻る代わりに、720度の回転が必要であるという、非常に奇妙な回転特性を持っています。
fermionsとbosonsの違いは何ですか?
-fermionsは半整数のスピン量子数を持つ粒子で、電子、プロトン、中子などがそれに該当します。一方、bosonsは整数のスピン量子数を持つ粒子で、光子やグルーオンなどの力の媒介者粒子が該当します。fermionsはパウリ排他原理により、同じ量子状態を共有できず、bosonsは同じ量子状態に集積することができます。
スピンが物質の構造にどのように影響を与えるか?
-スピンは物質の構造に根本的な影響を与えます。fermionsのスピンは、パウリ排他原理をもたらし、電子が独自のエネルギーレベルを持つことになり、これが元素の周期表や物質の構造を形成する基礎となります。また、bosonsのスピンは、宇宙の基本的な力の媒介者であり、物質間の相互作用に関与しています。
スピンが現実の構造にどのように関連しているか?
-スピンは、物質の構造を決定するだけでなく、現実の構造にも深く関連している可能性があります。スピンオブジェクトは、空間時間の亜原子的な織物に沿って考えることができます。これらのオブジェクトは、奇妙な結び目のようなものです。スピンは、これらの結び目がどのように機能し、現実が形成されるかを理解する可能性のある手がかりです。
エントロピーと量子纠缠の関係は何ですか?
-エントロピーは、システムの混乱度を測定する指標であり、量子纠缠は、システムの部分間の関連性です。エントロピーは、観察者の視点によって相対的であり、量子纠缠は、システム全体の情報含量を決定する要素の一つです。量子纠缠が低い場合、システムのvon Neumannエントロピーも低くなります。
宇宙の初期状態におけるエントロピーはどのようになっていたか?
-宇宙の初期状態におけるエントロピーは、非常に低いとされています。これは、宇宙が非常に小型で高温で均一であったため、重力的な自由度がほとんど占有されておらず、低いエントロピーを有していたと考えられています。しかし、物質のエントロピーは非常に高かったため、重力エントロピーが物質エントロピーを圧倒していたのです。
電子が回転しているわけではないなぜスピンを如此く表現するのか?
-電子が回転しているわけではないが、「スピン」という言葉を使用するのは、電子が持つ非常に奇妙な種類の角動量を説明するためです。この角動量は、古典的な物理学的な回転とは別の現象であり、量子力学的な性質を持っています。そのため、別の言葉で説明することが難しいため、「スピン」という言葉が使われることがあります。
スピンが電子の磁矩にどのように影響を与えるか?
-スピンは、電子の磁矩に直接影響を与えます。電子はスピン量子数を持っているため、スピンが変化すると、電子の磁矩も変化します。これは、電子が磁场の中で力をを受けたり、銀の原子がStern-Gerlach実験で偏向するようにする現象の原因となります。
アインシュタイン・デハアス効果で観察された角動量保存は、どのようにして解明されたのか?
-アインシュタイン・デハアス効果で観察された角動量保存は、外部の磁場が鉄を磁化し、電子のスピンが整列化することで解明されました。この整列化によって、鉄の円筒が角動量を得、円筒の回転が起こることで角動量が保存されると考えられています。
Outlines
🌀 量子自旋の神秘性と実験的検証
この段落では、量子自旋という量子力学における複雑で神秘的な現象について説明されています。量子自旋は、電子のような粒子が持つ非常に基本的な量子的角運動です。実験的な検証として、アインシュタイン・デ・ハアス効果が紹介されており、鉄の円筒が外部の磁場によって回転することを示しています。また、電子が自旋を持っていることを示すゼーマン効果も触れられています。電子の自旋は、古典的な回転ではなく、量子力学的なものであり、粒子の構造や物質の構成に深く関わることを強調しています。
🔬 スターン・ゲラフ実験と量子自旋の磁的性質
この段落では、スターン・ゲラフ実験について説明されています。この実験は、銀原子が磁場を通過する際に量子自旋が磁的モーメントを帯び、それがどのように影響を与えるかを示しています。実験の結果、原子が画面の2つの地点にしか当たりませんでした。これは、量子自旋が特定の方向を持つことが示されています。また、自旋は量子力学で非常に重要な役割を果たしており、保則と統計の原理に基づく粒子の性質と相互作用を決定します。
📐 量子自旋と角運動の関係性
この段落では、量子自旋と角運動の関係性に焦点を当てています。自旋は、粒子が持つ回転の自由度から生じる保存量であり、古典的な回転とは異なります。自旋は、粒子の位置と関連付けられた角位置の概念です。また、自旋は、粒子がどのように相互作用するかを定めるための鍵となる性質であり、フェルミオンとボーソンという2種類の粒子に分類されます。この違いは、物質の構造形成や宇宙の基本的な法則に影響を与えることになります。
🌌 宇宙の初期状態とエンタングルメント
最後の段落では、宇宙の初期状態とエンタングルメントの関係性について議論されています。宇宙の初期状態における低いエントロピーと量子自旋の役割が説明されています。宇宙の非常に早期段階で、粒子がすでにエンタングルメントであった可能性や、宇宙の膨張がエンタングルメントな地域を永遠に分離した可能性についても触れられています。また、情報理論の創始者であるクロード・シャノン博士にちなんで「The Cloud」という言葉の由来についても言及されています。
Mindmap
Keywords
💡量子スピン
💡角運動量の保存
💡アインシュタイン・デハース効果
Highlights
Quantum spin is a fundamental property of particles that has deep insights into the nature of the quantum world.
The conservation of angular momentum is demonstrated in a classic physics experiment involving a spinning wheel and a rotating professor.
The Einstein de-Haas effect shows that an iron cylinder starts rotating in a magnetic field due to the alignment of electron spins.
Electrons are not spinning like bicycle wheels, but they possess a strange type of angular momentum without classical rotation.
Quantum spin is a manifestation of a deeper property of particles responsible for the structure of all matter.
The Zeeman effect and the anomalous Zeeman effect led to the understanding of electron spin and its magnetic properties.
The Stern-Gerlach experiment revealed the quantized nature of electron spin, showing that it can only take on specific directions.
Electron spin is an intrinsic angular momentum that is a quantum mechanical property, not explainable by classical physics.
Pauli's work on the two-valuedness of electrons led to the development of the concept of spinors in quantum mechanics.
Dirac's equation, which incorporates special relativity, also includes spinors, revealing the importance of spin in quantum theory.
Spinors describe particles with strange rotation properties, requiring a 720-degree rotation to return to their original state.
The concept of spin is related to the orientation of particles and their rotational degree of freedom in quantum mechanics.
Fermions, particles with half-integer spins, and bosons, with integer spins, exhibit different behaviors and interactions.
The Pauli Exclusion Principle, a result of fermion behavior, is responsible for the structure of matter and the periodic table.
Spin statistics theorem explains the fundamental differences between fermions and bosons and their interactions.
The nature of quantum spin and its implications for the structure of reality are still being explored and understood.
Entropy and quantum entanglement are interconnected, with the former being relative and dependent on the context of observation.
The low entropy at the Big Bang suggests a highly ordered early universe, which may have implications for understanding quantum entanglement.
The concept of 'The Cloud' in information theory is mistakenly named and mispronounced, originally referring to Dr. Shannon.
Transcripts
Quantum mechanics has a lot of weird stuff - but there’s one thing that everyone agrees
that no one understands. I’m talking about quantum spin. Let’s find out how chasing
this elusive little behavior of the electron led us to some of the deepest insights into
the nature of the quantum world.
There’s a classic demonstration done in undergraduate physics courses - the physics
professor sits on a swivel stool and holds a spinning bicycle wheel. They flip the wheel
over and suddenly begin to rotate on the chair. It’s a demonstration of the conservation
of angular momentum. The angular momentum of the wheel is changed in one direction,
so the angular momentum of the professor has to increase in the other direction to leave
the total angular momentum the same.
Believe it or not, this is basically the same experiment - suspend a cylinder of iron from
a thread and switch on a vertical magnetic field. The cylinder immediately starts rotating
with a constant speed. At first glance this appears to violate conservation of angular
momentum because there was nothing spinning to start with. Except there was - or at least
there sort of was. The external magnetic field magnetized the iron, causing the electrons
in the iron’s outer shells to align their spins. Those electrons are acting like tiny
bicycle wheels, and their shifted angular momenta is compensated by the rotation of
the cylinder.
That explanation makes sense if we imagine electrons like spinning bicycle wheels - or
spinning anything. Which might sound fine because electrons do have this property that
we call spin. But there’s a huge problem: electrons are definitely NOT spinning like
bicycle wheels. And yet they do seem to possess a very strange type of angular momentum that
somehow exists without classical rotation. In fact the spin of an electron is far more
fundamental than simple rotation - it’s a quantum property of particles, like mass
or the various charges. But it doesn’t just cause magnets to move in funny ways - it turns
out that quantum spin is a manifestation of a much deeper property of particles - a property
that is responsible for the structure of all matter. We’ll unravel all of that over a
couple of episodes - but today we’re going to
Today we’re going to talk about what spin really is and get a little closer to
understanding what this weird property of nature.
The experiment with the iron cylinder is called the Einstein de-Haas effect, first performed
by, well, Einstein and de-Haas in 1915. It wasn’t the first indication of the spin-like
properties of electrons. That came from looking at the specific wavelengths of photons emitted
when electrons jump between energy levels in atoms. Peiter Zeeman, working under the
great Hendrik Lorenz in the Netherlands, found that these energy levels tend to split when
atoms are put in an external magnetic field. This Zeeman effect was explained by Lorentz
himself with the ideas of classical physics. If you think of an electron as a ball of charge
moving in circles around the atom, that motion leads to a magnetic moment - a dipole magnetic
field like a tiny bar magnet. The different alignments of that orbital magnetic field
relative to the external field turns one energy level into three.
Sounds reasonable. But then came the anomalous Zeeman effect. In some cases, the magnetic
field causes energy levels to split even further - for reasons that were, at the time, a complete
mystery. One explanation that sort of works is to say that each electron has its own magnetic
moment - by itself it acts like a tiny bar magnet. So you have the alignment of both
the orbital magnetic moment and the electron’s internal moment contributing new energy levels.
But for that to make sense, we really need to think of electrons as balls of spinning
charge - but that has huge problems. For example, in order to produce the observed
magnetic moment they’d need to be spinning faster than the speed of light. This was first
pointed out by the Austrian physicist Wolfgang Pauli. He showed that, if you assume electrons
have a maximum possible size given by the best measurements of the day, then their surfaces
would have to be moving faster than light to give the required angular momentum. And
that’s assuming that electrons even have a size - as far as we know they are point-like
- they have zero size, which would make the idea of classical angular momentum even more
nonsensical. Pauli rejected the idea of associating such
a classical property like rotation to
the electron, instead insisting on calling it a “classically non-describable two-valuedness”.
OK, so electrons aren’t spinning, but somehow they act like they have angular momentum. And this
is how we think about quantum spin now. It’s an intrinsic angular momentum that plays into
the conservation of angular momentum like in the Einstein de-Haas effect, and it also
gives electrons a magnetic field. An electron’s spin is an entirely quantum mechanical property,
and has all the weirdness you’d expect from the weirdest of theories. But before we dive
into that weirdness, let me give you one more experiment that reveals the magnetic properties
that result from spin.
This is the Stern-Gerlach experiment - proposed by Otto Stern in 1921 and performed by Walther
Gerlach a year later. In it silver atoms are fired through a magnetic field with a gradient
- in this example stronger towards the north pole above and getting weaker going down.
A lone electron in the outer shell of the silver atoms grants the atom a magnetic moment.
That means the external magnetic field induces a force on the atoms that depends on the direction
that these little magnetic moments are pointing relative to that field. Those that are perfectly
aligned with the field will be deflected by the most - either up or down. If these were
classical dipole fields - like actual tiny bar magnets - then the ones that were only
partially aligned with the external field should be deflected by less. So a stream of
silver atoms with randomly aligned magnetic moments is sent through the magnetic field.
You might expect a blur of points where the silver atoms hit the detector screen - some
deflected up or down by the maximum, but most deflected somewhere in between due to all
the random orientations. But that's not what’s observed. Instead, the atoms hit the screen
in only two spots corresponding to the most extreme deflections.
Let’s keep going. What if we remove the screen and bring the beam of atoms back together.
Now we know that the electrons have to be aligned up or down only. Let’s send them
through a second set of Stern-Gerlach magnets, but now they’re oriented horizontally. Classical
dipoles that are at 90 degrees to the field would experience no force whatsoever. But
if we put our detector screen we see that the atoms again land in two spots - now also
oriented horizontally.
So not only do electrons have this magnetic moment without rotation, but the direction
of the underlying magnetic momentum is fundamentally quantum.
The direction of this "spin" property
is quantized - it can only take on specific values. And that direction depends on the
direction in which you choose to measure it. Here we see an example of Pauli's two-valuedness
manifesting as something like the direction of a rotation axis, or the north-south pole
of the magnetic dipole.
But actually this two-valuedness is far deeper than that. To understand why we need to see
how spin is described in quantum mechanics. It was again Pauli who had the first big success
here. By the mid 1920s physicists were very excited about a brand new tool they’d been
given - the Schrodinger equation. This equation describes how quantum objects behave as evolving
distributions of probability - as wavefunctions.It was proving amazingly successful at describing
some aspects of the subatomic world. But the equation as Schrodinger first conceived it
did not include spin. Pauli managed to fix this by forcing the wavefunction to have two
components - motivated by this ambiguous two-valuedness of electrons.
The wavefunction became a very
strange mathematical object called a spinor, which had been invented just a decade prior.
And just one year after Pauli’s discovery, Paul Dirac found his own even more complete fix
of the Schrodinger equation - in this case to make it work with Einstein’s special
theory of relativity - something we’ve discussed before. Dirac wasn’t even trying to incorporate
spin, but the only way the equation could be derived was by using spinors.
Now spinors are exceptionally weird and cool, and really deserve their own episode. But
let me say a couple of things to give you a taste. They describe particles that have
very strange rotation properties. For familiar objects, a rotation of 360 degrees gets it
back to its starting point. That’s also true of vectors - which are just arrows pointing
in some space. But for a spinor you need to rotate it twice - or 720 degrees - to get
back to its starting state.
Here’s an example of spinor-like behavior. If I rotate this mug without letting go my
arm gets a twist. A second rotation untwists me.
We can also visualize this with a cube attached to nearby walls with ribbons. If we rotate
the cube by 360 degrees, the cube itself is back to the starting point, but the ribbons
have a twist compared to how they started. Amazingly, if we rotate another 360 - not
backwards but in the same direction - we get the whole system back to the original state.
Another thing to notice is that the cube can rotate any number of times, with any number
of ribbons attached, and it never gets tangled.
So think of electrons as being connected to all other points in the universe by invisible
strands. One rotation causes a twist, two brings it back to normal. To get a little
more technical - the spinor wavefunction has a phase that changes with orientation angle
- and a 360 rotation pulls it out of phase compared to its starting point.
To get some insight into what spin really is,
think not about angular momentum, but regular or linear momentum.
A particle's momentum is fundamentally connected to its position.
By Noter's theorem, the invariance of the laws of motion to changes in
coordinate location gives us the law of the conservation of momentum. For related reasons
in quantum mechanics position and momentum are conjugate variables.
Meaning you can represent a particle wavefunction in terms of either of these properties.
And by Heisenberg's uncertainty principle
increasing your knowledge of one, means increasing the unknowability of the other.
If position is the companion variable of momentum, what's the companion of angular momentum?
Well it's angular position. In other words the orientation of the particle.
So one way to think about the angular momentum of an electron
is not from classical rotation,
but rather from the fact that they have a rotational degree of freedom
which leads to a conserved quantity associated with that.
They have undefined orientation, but perfectly defined angular momentum.
Some physicists think that spin is more physical than this. Han Ohanian,
author of one of the most used quantum textbooks.
shows that you can derive the right values of the electron spin angular momentum and magnetic moment
by looking at the energy and charge currents in the so called Dirac field.
That's the quantum field surrounding the Dirac spinor aka the electron,
imply that even if the electron
is point like, it's angular momentum can arise from an extended though still tiny region.
However you explain it, we have an excellent working definition of how spin works.
We say that particles described by spinors have spin quantum numbers that are half-integers
- ½, 3/2, 5/2, etc. The electron itself has spin ½ - so does the proton and neutron.
Their intrinsic angular momenta can only be observed as plus or minus a half times
the reduced Planck constant,
projected onto whichever direction you try to measure it. We call these
particles fermions. Particles that have integer spin - 0, 1, 2, etc. are called bosons, and
include the force-carrying particles like the photon, gluons, etc. These are not described
by spinors but instead by vectors, and behave more intuitively - a 360 degree rotation brings
them back to their original state.
This difference in the rotational properties of fermions and bosons
results in profound differences
in their behavior - it defines how they interact with each other. Bosons, for example, are
able to pile up in the same quantum states, while fermions can never occupy the same state.
This anti-social behavior of fermions
manifested as the Pauli Exclusion Principle and is responsible
for us having a periodic table, for electrons living in their own energy levels and for matter
actually having structure. It’s the reason
you don’t fall through the floor right now. But why should this obscure rotational property
lead to such fundamental behavior? Well this is all part of what we call the spin statistics
theorem - which we’ll come back to in an episode very soon.
Electrons aren’t spinning - they’re doing something far more interesting. The thing
we call spin is a clue to the structure of matter - and maybe to the structure of reality
itself through these things we call spinors - strange little knots in the subatomic fabric of
spacetime.
Last time we talked about the connection between quantum entanglement and entropy - this was
a heady topic to say the least, but you guys had such incredibly insightful comments and
questions.
Joseph Paul Duffey asks whether entropy is an illusion created by our observation of
isolated components within a "larger" entangled system? Well the answer
is that entropy is sort of relative. It's high or low depending on context.
The air in a room may be perfectly mixed and
so considered “high” entropy. But if that room is warm compared to a cold environment outside,
then the total room + environment is at a relatively low entropy compared to the maximum
- if you opened the doors and let the temperature equalize.
Von Neumann entropy is different to thermodynamic entropy in that it represents the information
contained in the system and extractable in principle, versus information that’s lost
to the system by entanglement with the environment with the environment. On
the other hand, classical or thermodynamic entropy represents
information that is hidden beneath the crude properties of the system, but may in principle
be extracted. And yet von Neumann entropy has a similar contextual nature. If your system
has no entanglement with the environment then its von Neumann entropy is zero. But if you
consider a subsystem within that system then that entropy rises.
Randomaited asks the following: If entropy only increased over time, which implies it
was at its minimum at the Big Bang, does that mean there was no quantum entanglement at
the Big Bang? To answer this we’d need to know why entropy is so low at the Big Bang - and
that’s one of the central mysteries of the universe. But, I’ll give it a shot anyway.
So we can’t really talk about the t=0 beginning of time, because that moment lost in our ignorance
about quantum gravity and inflation and whatever other crazy theory we haven’t figure out
yet. But what we do know is that at some very, very small amount of time after t=0, the universe
was extremely compact - which meant hot and dense, and it was also extremely smooth. The
compact part is where the low entropy comes from. The “gravitational degrees of freedom”
were almost entirely unoccupied. On the other hand, the extreme smoothness meant that the
entropy associated with matter was extremely high. Energy was as spread out as it could
get between all of the particles and the different ways they could move. The low gravitational
entropy massively outweighed the matter entropy, so entropy was low. That smoothness seems
to suggest the particles of the early universe were already entangled - otherwise how did
they spread out their energy?
Chris Hansen makes the same point, asking if the conditions of the Big Bang meant everything
started out entangled. You’d think so - but that’s not necessarily the case. Remember
that von Neumann entropy is relative to the
system you’re talking about, and so is entanglement.
Let’s say you have a bunch of particles that are not entangled with each other but
are all entangled with another bunch of particles somewhere else. If you ignore those other
particles then it seems like there’s no entanglement in the particles of the first
system.
And yet those particles may have correlated thermodynamic properties due to their mutual
connection to the outside. In the early universe, the extreme expansion of cosmic
inflation may have permanently separated entangled regions, but left those regions with an internal
thermal equilibrium which does NOT require maximal entanglement within the regions themselves.
In other words, the universe - or our patch of it - may have started out unentangled and
at low entropy, even if it was at thermal equilibrium.
Lincoln Mwangi also dropped some knowledge, informing us that “The Cloud” - is actually
named after Dr, Shannon, the founder of the field of information theory. As with many
of these things, the word has been corrupted over time and is now routinely mispronounced.
This is very disrespectful, and I intend to write a series of op-eds to correct the matter.
Right after we upload this video to the Claude.
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