Quantum State

In classical Physics, we can define the state of a system (ie a particle or something) using it's position and velocity (or momentum), and this completely describes where it is, and where it will be in a given amount of time.

However in quantum mechanics, we think about particles in terms of wave functions. This means that instead of having a (predetermined) fixed property (defined using 'quantum numbers') at a certain time, it could have a number of different possible properties. It is only when we observe it, that the wave function 'collapses' down to give us one particular set of measurements of its state. Furthermore each state can be at a number of different energies.

So basically, in a nutshell, a state is a just a complete description of the properties of a system. It contains information that allow us to determine the state of a system at a later time. A quantum state just means the state describes a quantum system (one obeying the laws of quantum mechanics) rather than a classical system and classical mechanics.

And with the Pauli exclusion principle - protons don't exist in the same state as other protons, and similarly with neutrons (they do obey the Pauli exclusion principle). Basically in an atom each electron, proton and neutron have a different set of quantum numbers to the other particles of the same type.

If you want me to try and explain it in a different way just ask

Time for my Oxford Dictionary for Physics!
Quantum State: "The state of a quantised (discrete rather than continuous values used) system as described by its quantum numbers. For instance the state of a hydrogen atom is described by the four quantum numbers:
$n, l, m, m_s.$
In the ground state they have values 1, 0, 0 and 1/2 respectively."

Cheers for the reply! Following on what pianofluteFTW explained, I assume that the quantum numbers represent specific properties of the atom? For example, the 1/2 for m_s would maybe be the spin? (Whatever that is...)

Hey thanks for the reply! I think I understand. Is it like with Feynman diagrams for the probability of a process occurring? So the electron, proton or neutron has a specific possibility of having a particular set of values, corresponding to a quantum number, depending on the number of ways that set of values can occur?
Also, with quantum states, is position a property? If it is, when an electron and a neutron (for example) share a quantum number and so can exist in the same quantum state, does that mean they can exist at the same point in space-time?

You're correct, but as far as my knowledge extends, the quantum numbers describe electrons inside an atom - I don't know how we use them to describe other quantum systems.

n = energy level of the electron, it describes the shell that an electron is in and can only be a natural number (any number $\geq 1$), so in the ground state, an electron is in the first shell and therefore has n = 1.

l = angular quantum number, and this describes the subshell that an electron is in. Each subshell has small energetic differences. The value of l can range between 0 and (n - 1). The number it is describes what subshell it is a part of, i.e. l = 0 is a s-orbital, l = 1 is a p orbital.

$m_l$ describes the orbital within a subshell - each orbital can only hold two electrons (this is where the Pauli exclusion principle comes in, in a second). The value of $m_l$ varies between $-l$ and [texl

Well as I understand it (but it all gets fairly complicated :P) most quantum properties are expressed in terms of probability - there is a chance of measuring several different states (superposition). So we then look at quantised variables, they have a restricted set of values they can be. This just means that instead of having a variable where the magnitude can be any numerical value, there are fixed values and it has to be one of them.

So for example, for electrons in an atom, with the four quantum numbers n,l,m and m_s we describe the state. n refers to what energy level the electron is on (electron shell), l refers to the shape or type of orbital (each state could be in a few different shapes), m is the magnetic quantum number and therefore can kind of be seen to describe the electrons position in 3D space, and m_s describes the spin of the electron (which is just a property - it's not like the electron spinning as we would imagine lets say a planet spinning, it's an intrinsic property related to angular momentum and is basically really hard to imagine :P).

So yeah, there is a certain probability an electron might be in these states, but once we have made a measurement the electron is fixed to a particular state.

Hope this makes sort of sense - you wont have to know all of this for A level Physics I don't think (but maybe in chemistry?)

Thank you so much! That does make sense. One last question; if the electrons cannot have the same quantum state, can they at least have some of the same quantum numbers? As in, if there are multiple possible states, the electrons must have multiple possible positions; but there are already electrons in the other orbitals and shells of the atom (most of the time). So how do all the electrons manage to have multiple, different quantum states?

Thanks that's brilliant! But I'm still a bit confused about the quantum numbers. If, when unobserved, a particle can exist in a number of different quantum states at the same time, due to quantum mechanics, and each state means different quantum numbers, does that mean that, for example, each electron has a probability of being in each shell and orbital, with different spin? (Sorry for the long sentence )

Thank you so much! That does make sense. One last question; if the electrons cannot have the same quantum state, can they at least have some of the same quantum numbers? As in, if there are multiple possible states, the electrons must have multiple possible positions; but there are already electrons in the other orbitals and shells of the atom (most of the time). So how do all the electrons manage to have multiple, different quantum states?

Sorry, I screwed up the LaTeX, but I just fixed it there!

I don't really understand quantum mechanics past a super, duper basic level so I can't answer your questions here!



They can have the same quantum numbers, as long as ALL of them aren't the same. e.g.

3,1,0, 1/2 and 3,1,0, -1/2 describe two electrons that are in the same p_y orbital. They have opposite spins - this is what prevents them being in the same quantum state.

They have lots of different quantum states simply because there's a lot of variability. I gave you the ranges there (which, I imagine, have been derived mathematically) that describes the permissive numbers depending on other variables.

n = energy level
l can be between 0 and (n - 1)
m_l = between -l and l in integer steps.
m_s = positive or negative one half.

Suppose n = 1; like in a hydrogen atom in ground state. Then the only possibly value for l is zero, therefore the only possible value for m_l is zero, but m_s is independent of the other states. Therefore a hydrogen atom can hold 2 electrons in its first shell/energy level.

Let's scale up to n = 2. Suddenly we have a big change:

n = 2
l = 0, 1
m_l = -1, 0, 1 for the l = 1 subshell.

Therefore 6 electrons.

Let's say we're looking at the electrons in n = 4!

n = 4
l = 0, 1, 2, 3
m_l = -3, -2, -1, 0, 1, 2, 3 in an l = 3 subshell

Therefore 14 electrons in the f orbitals (this is what you call the l = 3 subshell).

As you can see, you can have a large amount of variation within the properties of an electron in an atom.

How about the total number of electrons in a certain energy level? Well, you just have to compute all the values of l for a given n, compute all the possible ranges of m_l given the possible l's, and double that for spin.

So for n = 4, plug all the numbers through, and you have 32 electrons that can be in that one energy level. It's easy to see, then, how we can have lots and lots of electrons in an atom and not have any of them have the same quantum state.

Yep they can share quantum numbers - we know more than one electron can be in the same shell. The quantities above are related, as we kind of see it as there are the main electron shells (n), then each has a certain number of sub shells (l) and these two numbers are related - for a certain value of n, there are only a certain number of values for l. But within each sub shell, the electrons can differ in terms of orbital shape, and in terms of spin.

Looking at the most simple case, the ground state (lowest energy level), where n=1, we have to say l=0. So therefore m=0, but we can have two electrons in the ground state, one spin up, one spin down. So to answer your question, yes they can share the same quantum numbers, but have to differ on at least one - no two electrons can have exactly the same set)

(And this is why electrons fill different shells - it is as a result of quantum Physics and the Pauli exclusion principle)