Quantum numbers are a set of values that identify the state of an electron. There are two major types of quantum numbers: orbital and magnetic.
Orbital quantum numbers describe the shape and orientation of an electron cloud. Magnetic quantum numbers describe the orientation of the electron relative to an external magnetic field.
The values that make up these two types can be either integer or fractional values. For example, there are three orbital quantum numbers, and they can be labeled as 1, 2, or 3. There are also two magnetic quantum numbers, and they can be labeled as +1 or -1.
The order in which these quantum numbers are listed identifies which orbital shape or magnetic orientation the electron has. For example, an electron with the (n=1,l=0) designation has a linear (1) orbit (n). This article will discuss how to identify which sets of quantum numbers are valid for an electron.
N,ℓ,mℓ
Now that you know how to identify the different quantum numbers, determine which combinations are valid and which are not.
The first quantum number, or set, is N. This set represents the shell an electron belongs in. There are only two possible values for this: 1 or 2.
The second quantum number, or set, is ℓ. This set represents the type of orbital the electron belongs in. There are three possible values for this: 0, 1, or 2.
The third quantum number, or set, is mℓ. This set represents the spin of the electron. There are two possible values for this: -1 or +1/2.
N,ms
The final set of quantum numbers to identify is the N,ms set. This set identifies the orbital energy level, the magnetic quantum number, whether the electron can have a spin or not, and if it does have a spin, what direction it spins.
Orbital energy levels describe where the electron is located in relation to the nucleus. Values range from 1 to infinity, with higher numbers indicating a farther away location.
A magnetic quantum number determines whether or not the electron can have a spin and what direction it spins. A value of 1 means that it can have a spin, while a value of 2 means that it cannot.
If an electron can have a spin, then it must be in an orbital with an inner quantum number of 1 or greater. Spins can only occur in these orbitals due to their higher inner quantum numbers.
N
Next comes the principal quantum number, or N. This value determines the shape of the electron’s wave function.
The value of N can be any integer between 1 and infinity. The higher the value of N, the larger and more elongated the electron’s shape becomes.
Atomic number is determined by which element you are examining. For example, if you are examining a gold atom, then the atomic number is 79. This means that there are 79 electrons in that atom, and any that are added or subtracted will change that number by -1 or +1, respectively.
Valid values of N range from 1 to 7 for any given element due to these being the possible integers for atomic number.
ℓ,mℓ,ms
The fourth, fifth, and sixth quantum numbers are called the ℓ,mℓ,ms quantum numbers. These refer to the angular momentum (spinning), magnetic moment (interaction with a magnet), and the spin-orbit interaction (the interaction between the electron’s spin and its motion).
The ℓ quantum number determines how the electron spins. If an electron has an ℓ value of 1, then it spins on its axis. If it has an ℓ value of 2, then it is a different particle: a photon. The mℓ value determines how strong the electron’s magnetic moment is, and ms determines how strong the spin-orbit interaction is.
Identifying which sets of quantum numbers are valid for an electron can be tricky. There are many resources available to help you do this, however.
ℓ,mℓ
Next, you must determine the spatial orbital and spin angular momentum quantum numbers. These are denoted by the letters ℓ and mℓ.
Spin is a fundamental property of a particle. It is related to the way a particle behaves when interacting with other particles. For example, when a particle spins, it may interact with other particles in a different way depending on the direction of its spin.
The value of ℓ represents the number of waves in an orbital, or how many times the electron goes around the nucleus before returning to its original position. The value of mℓ represents how fast or slow the electron moves in its orbit.
To determine these values, you must use mathematical formulas based on experiments that identify how an electron orbits and spins.
ℓ,ms
The last set of quantum numbers describes the electron’s angular momentum and magnetic moment. Angular momentum is how fast the electron rotates around its axis, along with how large of a radius it has.
Unlike the other two sets of quantum numbers, ℓ cannot be zero. An electron must have some amount of angular momentum, which can be determined by the other sets of quantum numbers.
The magnetic moment is a measurement of how strong an electron’s field is. How strong the field is depends on the size and rotation of the electron, which are determined by ℓ and ms.
These three quantum numbers are not equal in value, and can be difficult to determine for an electron. Due to this, researchers have developed methods to identify which sets of quantum numbers are valid for an electron.
ℓ
The ℓ quantum number represents the electron’s orbital angular momentum, or how it orbits the nucleus. This is similar to how the earth orbits the sun in an elliptical orbit with a tilted axis, which results in seasons.
Orbital angular momentum can be visualized as how a planet spins on its axis while orbiting the sun. The planet takes a certain amount of time to make one complete spin, which is its orbital period.
The ℓ quantum number can have values from 0 to ∞, with 0 representing an electron in a fundamental 1s shell and ∞ representing an electron with infinite orbital angular momentum.
Validation of this quantum number set is important because it determines what type of atomic orbit the electron is in and if it can transition to other types of atomic orbits.
mℓ,ms
So far, we have discussed the first two quantum numbers. Now we will discuss the last two: mℓ and ms.
The mℓ quantum number refers to the electron magnetic spin. This is either +1/2 or -1/2, with no other possible values. The electron can only have one of these two values for its magnetic spin.
The ms quantum number refers to the spin statistics of the electron. This quantum number can be either + or -, with no other possible values. The difference between these two is how they interact with other particles.
So, in general, there are two ways to order an electron’s set of four quantum numbers: (n, ℓ ,mℓ ,ms) or (n, ℓ ,mℓ ,+ or -).
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