I’ve been reading a lot about massive stellar objects, degenerate matter, and how the Pauli exclusion principle works at that scale. One thing I don’t understand is what it means for two particles to occupy the same quantum state, or what a quantum state really is.
My background in computers probably isn’t helping either. When I think of what “state” means, I imagine a class or a structure. It has a spin field, an energy_level field, and whatever else is required by the model. Two such instances would be indistinguishable if all of their properties were equal. Is this in any way relevant to what a quantum state is, or should I completely abandon this idea?
How many properties does it take to describe, for example, an electron? What kind of precision does it take to tell whether the two states are identical?
Is it even possible to explain it in an intuitive manner?
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ELI5: Imagine a ring on a table. They putting some marbles in that ring. They can roll anywhere, right? Well, quantum mechanics makes you put those marbles in a neat little grid. The marble can still ‘roll’ like a piece moves on a chess board: to unoccupied spots around it. If there’s something already in it, though, it won’t be able to move there. Each of those spots is a ‘quantum state.’
(Importantly, the ring on the table typically represents spots in phase space—something that involves both position and momentum—rather than real space)
ELI15: A quantum state is a state that a quantum system can be in. That isn’t very helpful, but you are asking about something very fundamental and thus something very difficult to describe without getting into the math.
It is important to note that a ‘state’ means very little without knowing what ‘system’ it describes. A system is made up of all the interacting particles that pull on each other and is what we are interested in describing.
All of these interactions collectively generate the ‘potential’ at every point in space and time, and knowing this potential allows us to write down the specific Schrödinger equation that we want to solve. This equation will have a limited1 number of solutions.
Consider your computer science class-instance again. Each instance of the class represents a state, but there is a larger class called System that has an array-type property that holds the States that solve it. Additionally, you should add another property to the State class: occupation. For bosons, this property is an Int, but for fermions, it is a Bool.2 Now, in order for a boson to join the system, it needs to increment the occupation counter on one of the States, but for a fermion to join, it needs to find a State with a False occupation and flip it to True.
Allowing for multiple fermions in the same state would result in a loss of information; nothing changes if a True state is flipped to True. This is thus not allowed, forming the basis of the Pauli exclusion principle.
ELI25: A quantum state is a linear combination of eigenstates for an observable quantum operator—typically the Hamiltonian of a system. Saying that the PEP disallows fermions from entering the same quantum state is a crude way of saying that their combined wavefunction must be antisymmetric:
Ψ(𝛙1, 𝛙2) = -Ψ(𝛙2, 𝛙1)
But then if
𝛙1 = 𝛙2
We get
Ψ(𝛙1, 𝛙2) = -Ψ(𝛙1, 𝛙2)
Which means
Ψ = 0
And is thus disallowed
1 limited means ‘quantized’ here—not ‘finite’
2 notably, you get a different system and thus a different set of States for every particle type. So a clump of neutrons would be unable to ‘exclude’ an electron
I think my answer here does a disservice to your last two questions, but they’re pretty interesting, so I’d like to try answering them here. That being said, I believe a proper explanation would require a greater understanding of quantum field than I currently have.
Regardless, let’s start with
How many properties does it take to describe, for example, an electron?
Wavefunctions are typically described using just two parameters: position and time. That being said, the specifics of a theory will often add two more ‘parameters:’ spin and charge.
For every position in space and time, there are four possible electrons:
Negatively charged with up spin,
Negatively charged with down spin,
Positively charged with up spin, and
Positively charged with down spin
Now, there are a couple conservation laws that prevent the charge of an electron from changing, so we split these four into two categories based on charge, and call members of the latter (positively charged) category ‘positrons.’
There are no such conservation laws preventing an electron from changing its spin, however, so it’s not worth calling the up and down spin electrons separate particles;1 we instead keep track of an electron’s up-ness and down-ness by sticking a two-component vector into the wavefunction.
So, to clarify, you should only need to know the electron’s position and spin at a given time to know its state.
Or, rather, knowing these will give you the local value of the wavefunction of the electron, which is defined at all points in space, at all points in time, and for both possible spins.
This wavefunction describes the state that the particle is in, and it is the solution to our Schrödinger equation.
The magic of quantum mechanics is that there are much fewer possible wavefunctions than you might expect for a given system.2
In many cases, we can actually label each of the allowed wavefunctions with one (or a handful of) quantum number(s),3 which will be integers. The energy (n) and angular momentum (j) are common quantum numbers.
So we can describe states with quantum numbers while the actual wavefunctions associated with these states describe the electrons that occupy them via spin, position, and time.
This means that when our Pauli exclusion principle requires different wavefunctions for different electrons, it is also enforcing that they occupy different states i.e. have different quantum numbers.
But then how does spin let you have two electrons in the same state? It doesn’t
If two electrons have ‘different spins,’ it actually means they have different wavefunctions (remember! Wave-funcs are functions of spin) and are thus in different states. In the absence of a magnetic field, however, up-spin and down-spin are effectively identical in all other aspects, so you will often see that which is described as one state is actually two.
Now for your other question:
What kind of precision does it take to tell whether the two states are identical?
None!
States are quantized, meaning they are described by a (typically finite) set of integer-valued quantum numbers, and if any of the qn’s are different, so too are the states.
It is worth noting that two different quantum states will have zero expected positional overlap, so you can say that they are in different places even if both wavefunctions are defined everywhere.
That being said, the mathematically-enforced inability of electrons to occupy the same state will certainly look like an equal and opposite to any force that energetically favors an already-occupied state… which is why you can pick up your pencil.
Now, if you start to apply a whole lot of pressure on the system, you start to change the states that solve the Schrödinger equation in the first place. So your neutron star isn’t infinitely stable.
1 spin can also mix while charge can’t, so an electron can be partially up and partially down, but not partially negative and partially positive. This is really just a consequence of the (in)ability to switch between states, but I figured I would mention this for clarity.
2 if you let some infinities be ‘fewer than others
3 technically, any linear combination of states can constitute an allowed wavefunction (this is superposition), but, this will constrain the set of allowable wavefunctions for other electrons, meaning the total number of allowed wavefunctions remains the same.
Thanks, that helped a lot, mainly by pointing out some of my misconceptions. I’m basically a tourist in quantum physics with no more than approximate understanding of several concepts, and I don’t think I’ll ever fully understand a field that took dozens of Nobel prize winners and multiple lifetimes to formulate, but I’m a bit closer.
position
I always thought of “position” as simply a point in Euclidean space described by a vector, but I’m guessing that doesn’t translate directly to quantum mechanics because the uncertainty principle gets introduced with having to account for momentum. Does that mean that two electrons can, at the instant they are observed, have no meaningful distance between them, only different momentum?
Not a physicist, but I’ve listened to some podcasts. You’re close enough for a hobbyist.
First off, there are 2 types of particles: Bosons and Fermions. Bosons don’t follow the Pauli Exclusion principal. You can cram more and more photons into a tiny space without reasonable limits. There’s an “unreasonable” limit where eventually things break, but that’s not related to Pauli.
Fermions follow Pauli. Electrons are Fermions, and won’t let you cram them too closely together. Electrons bound to an atom settle into energy levels called “shells”. The first shell can hold two electrons, but only if they’re in different quantum states. Since they’re at the same energy level, we’re left with spin. To fit two electrons into the lowest shell (energy level), one has to be spin-up, and the other must be spin-down. As you add more electrons to the atom, they begin to fill higher shells. Some of these shells can hold more than 2 electrons, but the electrons always get added in pairs. There aren’t any shells with an odd-number limit of electrons because you fill the shells with spin-up and spin-down electrons.
Now, what you’re talking about, massive stellar objects, and degenerate matter, is an atom taken to 11. A star is an equilibrium of outward pressure from the fusion reactions, and inward pressure from gravity. Once a star stops fusing elements, there’s nothing to counteract gravity, and the star collapses catastrophically. Giant stars can leave behind a core so heavy that it keeps compressing until if forms a black hole. But if it’s not quite big enough to do that, it will just squeeze all the leftover matter into some weird shit. First, at the surface all the protons and electrons will be squeezed together to form neutrons. I suppose if there are leftover electrons they just kind of swim around on the surface
Underneath all these neutrons are neutrons that are squeezed so close together they’ve been broken apart into raw quarks. Quarks are fermions, but they have 3 different charges (called color) instead of just 2. And their electrical charge is a fraction of an electron charge, either 1/3 or 2/3 a fundamental charge. A neutron is 3 quarks, and a proton is 3 different quarks (and those are all gone at this depth anyway). Once things are broken down into quarks, it’s called degenerate matter. Conditions are so extreme that matter as we know it can’t form. It’s degenerate.
And the rest is just theory. There’s a spaghetti layer, and a lasagna layer, named for the theoretical shape the degenerate matter takes at that pressure. And below that is…I guess the next pasta type would be some kind of hard, solid block of gluten.
So I think where popsci messaging is bad is because they get all mixed up between particles and waves
A superposition isn’t a particle, it’s a wave. It only has the mass/energy associated with the particle, but it is everywhere it can be all at once
The particle is like… The minimum size the wave can fit into. One photon of light can jump one electron up once energy state, and it has to pick one electron to actually do this
You can channel the wave into a particle long enough to measure it, but to measure any aspect of the wave you have to consume it. Now you’ve converted the energy from one form into another, so any other information that could have been gained is destroyed
That’s my understanding at least, it is very much not digital. There’s no either or, it’s even more analog than analog
One example would be the electrons in an atom. The shell, subshell, orbital and spin together make up one quantum state.
In a more general sense quantum states are solutions to the Schrödinger equation for a given system.
It’s a micronation.
At the subatomic scale, things are less particule-like and more wave-like.
The most famous visualization of this is the double-slit experiment: there’s a source of light, two slits and a wall. There should be two lines right? Nope, you get a wave interference pattern. So which slit did the electron take? Both at the same time, it seems. You can know which path it likely took, but in reality the photon could have taken a detour Taco Bell faster than the speed of light for all we know, as long as the end result doesn’t it’s physically totally fine.
The crazy part of the experiment is that in order measure which slit the photon actually went through, it would have to interfere with your detector. And because it interacted with your detector, the uncertainty collapses and the whole interference pattern disappears. The measurement causes side effects that affect where it possibly could have gone through. You thus only see paths where it did go through your detector.
The universe seems to prefer the path of least action. All possible paths are evaluated at the same time, including ones that would violate the speed of light. You won’t catch the universe doing it, but you can observe that photons and electrons make it places they physically shouldn’t be able to, but mathematically, they can and do in the real world. Do they even actually travel any given path? We don’t know, we know it went from A to B with no idea where it was in-between or how fast it went.
To circle back to your coding example: the particule is a class with getters, but the getters don’t read a property, it makes up the value on the fly. So
particule.spin,particule.locationandparticule.speedwould return you the values, but they would be inconsistent. It only materializes on demand when probed, and you can’t get two of them at the same time. When you check you only get one possible value it can have, but you check again and it’s a different value. In C that would be avolatilevariable.That’s why in atoms you end up with a blurry electron cloud. At this scale, it’s a wave of probable positions, it’s everywhere and nowhere at the same time.
A quantum state is basically that. It’s not a defined state, it’s an equation of all possible states and how probable it is to be in a given state. The only guarantee you have is that all the state will physically make sense if you measure it, so if you measure the spin of an entangled particule, to stay consistent, the other one will take the opposite state because you can’t catch the universe in a lie. But until you observe that state, it’s both at the same time.
PBS Space Time is a great channel on YouTube for this.
In very simple terms, it means that something is changing state so impossibly fast (“on a quantum level”) that we can’t tell what exactly that state is besides at the instant we check it. Exactly is doing a lot of heavy lifting here, because we can have an idea or an area, but not exactly. What that means in turn though is that by checking or measuring that state, we have interacted with it, therefore making the state we measured no longer valid for what it currently is now, or at rest.
Think of it like taking a measure of a water droplet, in the middle of a lake. You can say “there it is, those atoms are in that droplet and theyre this hot”. But the drop you measured is constantly mixing with the water around it. Sure, you measured the temperature of those atoms in that droplet, but if you try to measure it again you could get a different result. (It’s not a perfect example, but it gets the idea through)
Using your programming model, think of it like reading memory in memory that is shorting out. You can read it once, but there’s no guarantee that it will be the same value again next time you read that bit, because it’s in constant flux.
If you put a cat into a box, then leave and come back sometime later and you can’t see or hear the cat (just the outside of the box), you don’t actually know if the cat is still alive or if it died while you were away.
Because of the uncertainty, the cat is considered to be in a quantum state of BOTH dead and alive.
You open the box to observe, and can see it breathing: it’s just asleep. It is no longer in a quantum state, as you can be certain that it is alive.
I’m familiar with that specific interpretation of quantum mechanics. It’s not relevant to the question, and it is a complete misinterpretation of Schrödinger’s thought experiment, both its concept and its purpose. He thought that the indeterministic view of quantum mechanics was wrong, and he used the thought experiment to demonstrate how ridiculous it is by bringing it into a macroscopic scale. It’s become a lie for children that omits many factors, like what it means to “observe or measure” a system.
It’s become a lie for children that omits many factors
Which is perfect for a ELI5!
You’re missing the essential element of this thought experiment - the poison gas canister in the box which releases the gas when an atom decays. That atomic decay is a quantum event that cannot be predicted even if one has perfect knowledge of the atom concerned, and in fact whether it has actually decayed or not only becomes real when it is measured ie observed. Thus, according to this experiment, the cat is simultaneously alive and dead at once, up until the box is opened and an observation made.
If you don’t have that atomic decay, then the cat’s health is merely unknown - it is either alive or dead, not both at once, but the scientist simply doesn’t know which.
Username checks out. Cat is definitely dead.
