# Why can’t two electrons in the same orbital have the same spin?

This cannot be answered without using very advanced mathematics and the Pauli exclusion rule. However, I can understand why the Pauli principle needs to be questioned. It’s not an acceptable explanation.

This is the “full” explanation of the explanation. You probably won’t get it all.

Two electrons cannot have the same orbit or spin. This is because, according to the PEP (particles that have half-integer spin), two fermions (particles that have half-integer spin). You can have only one spin up and one spin down in an orbital.

Two identical fermions cannot occupy the same quantum status. This is because for particles with half integer spin, swapping their positions multiplies quantum state by negative sign. The result of two identical fermions in the same place is the same state. However, it also has a negative sign which indicates that it is necessarily 0.

Swapping positions multiply by a positive sign because rotating a half integer-spin state 360° multiplies it with a negative signal (some terms to help you understand this: spinor and orientation entanglement. SO(3) is not simply connected. SU(2) is a double-cover of SO(3).

It can be demonstrated that switching the positions of two particles within a quantum state is equivalent to rotating one by 360 degrees.

We liken the rotation of electrons with paths in the SU(2) or SO(3) group because it corroborates our experiments. This is it. You don’t need to know much about the terminology. SO(3) refers to the group of three-dimensional rotations. Aka all directions that you can point a vector in. SU(2) works the same, except that you can go around 360° in any direction and end up in “part two”, which is a second copy SO(3). You will need to complete *two* full rotations to get to the exact point where you started.

There are a few other reasons why SU(2) has double rotatings. It is obvious that vectors are incorrect. Vectors aren’t the right way to describe the state of a spinor. That’s where the confusion comes in. Problem is that the 360-degree rotated vector version of a vector is identified as the same vector. However, it should be identified with the opposite vector. Every vector has a spiral that wraps around it to either the right or the side. If you rotate the vector 360 degrees, the result will be a vector with its spiral turned. It takes two spins in order to return to the true’same state’. Electrons are described by spinors and not vectors. Instead, they are described as oriented vectors. Our math is terrible at describing such things, which is a failure. Some techniques are available, but they don’t appeal to me. Two-dimensional complex numbers described spinors correctly. This is why they are used in quantum mechanics. However, I think we should ditch all imaginary numbers and use better, intuitive ways to describe things.

Anyway.

ps> ‘>

. It is characterized by a wave function that it receives from its eigenvalues as part of the position operators for particles. x^|ps> =ps(x1,x2) This gives you a probability amplitude to find each particle at points (x1,x2)

. You can also multiply the state position operator by the states to specify the states at a specific time t.

According to the statement, swapping particles is equivalent to a positive sign. This means that there is a lower probability of finding particle 1 at (x1) and particle 2 at (x2) than finding them at (x1, x1).

To avoid being terribly vague, let me try to explain. I imagine that swapping their positions is akin to picking a point between x1-x2 and rotating 180 degrees the world around it. So the points swap places. Relative to x1, x2 rotated 360 degrees. This could also be translated. If you accept that spinors gain negative signs under this, and that wave functions are spinors, then it is reasonable to assume that the wave function gains negative signs.

In quantum mechanics, wave function that would equal zero are not possible to be described as existing. Quantitation is used to determine the energy levels of a potential spherical because an electron at n=1.5 energy would have a wave function that would not equal itself once it was wrapped around a circle. It would also be out of phase because it oscillates at the wrong speeds. This would result in destructive interference. It would cancel out itself and we would have 0 wave functions. Only the integer levels of n=1,2, and so on. Give a wave function which constructively interferes with it. The statement that two electrons cannot have the same spin in an orbital is very similar. You have to see (or convince yourself) that they don’t have the same spin because their wavefunctions will destructively interfere with each other, so that state won’t exist. This is due to the 360 degree SU(2) rotation that gives a negative sign.