Spin up and Spin down states can be written as .... why?

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In summary, spin up and spin down states, also known as electron spin states, can be written as 1/2 and -1/2, respectively, because they represent the two possible orientations of an electron's spin. This notation is used to differentiate between electrons in different energy levels and to describe the magnetic properties of atoms and molecules. The spin up state is associated with a positive magnetic moment, while the spin down state is associated with a negative magnetic moment. This notation is important in understanding the behavior of electrons and their interactions in various scientific fields such as quantum mechanics and materials science.
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Sara Kennedy
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Spin up and spin down states in the x direction can be written as

|Upx> = 1/ √2 ( |Upz> + |Downz> )
and
|Downx> = 1/ √2 ( - |Upz> + |Downz> )

My textbook just stated the above facts without referencing why and I've been going through the spin chapter for a while now and I can't see it. Why is this true?

Is it something to do with permutation rotation and orthogonality? For example I could switch the x to y and the z to x and the expressions would be correct
 
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Find the matrix form of ##S_x## in the basis ##|z;\pm\rangle## and then find its eigenvectors. Alternatively, you can also apply the rotation operator about the y-axis by 90 degree to the ##|z;\pm\rangle## states.
 
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Hm okay I have the Sx matrix and a matrix in the n direction Sn. Do you know of any online notes that explain how to change from one basis to another? I found a few questions but they look like homework solutions and I was after explanations, it doesn't explain change of basis in my book. Just to clarify, |z;±> means spin up and spin down states in z direction right?
 
  • #4
Sara Kennedy said:
Do you know of any online notes that explain how to change from one basis to another? I found a few questions but they look like homework solutions and I was after explanations, it doesn't explain change of basis in my book.
Try searching for "change of coordinates matrix", you should find a number of resources on this matter. In general, to find the change of coordinate matrix, you have to know how the vector in one basis is expressed in terms of the basis vectors in the other basis.
Sara Kennedy said:
Just to clarify, |z;±> means spin up and spin down states in z direction right?
Yes.
 
  • #5
Sara Kennedy said:
I was after explanations
See http://feynmanlectures.caltech.edu/III_06.html for a derivation of these equations (6.32) for "amplitudes". The derivation starts from the fact that a 360 degree rotation changes the sign of both amplitudes (because otherwise a 180 degree rotation would be physically equivalent to a 360 degree rotation).
 

1. What are "spin up" and "spin down" states?

"Spin up" and "spin down" states refer to the two possible orientations of an electron's spin in a magnetic field. The spin of an electron is an intrinsic property that can be thought of as a small magnetic moment, and it can either align with the magnetic field (spin up) or align in the opposite direction (spin down).

2. Why can spin up and spin down states be written as superpositions of other states?

This is because the spin of an electron is a quantum mechanical property and follows the rules of quantum mechanics, which allow for the existence of superposition states. This means that an electron can exist in a combination of both spin up and spin down states at the same time.

3. How do spin up and spin down states relate to the Pauli exclusion principle?

The Pauli exclusion principle states that no two electrons can have the same set of quantum numbers. Since the spin of an electron is one of these quantum numbers, spin up and spin down states allow for two electrons to occupy the same orbital, as they have different spin orientations.

4. Can spin up and spin down states be measured separately?

Yes, the spin of an electron can be measured using a magnetic field. If the electron's spin aligns with the field, it will be in a spin up state, and if it aligns in the opposite direction, it will be in a spin down state.

5. How do spin up and spin down states impact the behavior of atoms and molecules?

The spin of electrons plays a crucial role in determining the electronic structure and chemical properties of atoms and molecules. The interaction between electrons with different spin orientations can lead to various phenomena, such as magnetism and chemical bonding.

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