Understanding Spin States: Theoretical Minimum and Normal Coordinate Systems

[Quarlep]

In The Theoritical Minimum we shown all spins states use just two states up and down. How can we do that.?
I am confused about the directions of states and normal coordinate system Can somebody help me ?
Thanks

 

[mfb]

In The Theoritical Minimum

What does that mean?

If you measure the spin direction of a particle with spin 1/2, the only possible measurements are "up" and "down". If the particle has a different spin, the options are different.
Which normal coordinate system?

 

[M Quack]

The spin value 1/2 and two spin states drop out of the Dirac equation. Spin-up and spin-down are chosen more or less arbitrarily, because they are the eigenstates of energy in a static magnetic field along the z-direction (theoretical magnetic fields are always along z...). However, they form a complete basis for spin-1/2 particles, and all spin directions can be written as a linear combination of these two basis states.

Therefore you can have spin directions in each and any direction. The theory is perfectly well developed and used e.g. in neutron diffraction.

https://www.ill.eu/en/instruments-support/instruments-groups/instruments/d3/how-it-works/spherical-polarimetry-with-cryopad/

 

[Nugatory]

In The Theoritical Minimum we shown all spins states use just two states up and down. How can we do that.?

Think of spin-up as a vector pointing east and spin-down as a vector pointing north. We can write any vector as a linear combinations of those two. For example, north-east would be the vector sum of north and east, southeast would be their east minus north, and so forth.

The confusing thing is that the "directions" these vectors point in their abstract vector space isn't the same as the direction that the spin angular momentum vector points in the real world. Spin-up and spin-down are represented by orthogonal vectors in the abstract vector space, even though up and down are opposite directions.

 

[pmacias]

for your question. The concept of spin states is an important aspect of quantum mechanics, and understanding it can be challenging. The Theoretical Minimum is a term used to describe the fundamental concepts and principles of a particular subject. In the case of spin states, the Theoretical Minimum refers to the minimum knowledge required to understand the basics of spin states.

To answer your question, let's start by defining what spin states are. In quantum mechanics, particles have a property called spin, which is a measure of their angular momentum. Spin states refer to the different possible orientations of a particle's spin. In the simplest case, a particle can have two spin states: up and down. These states are represented by the two possible directions of spin, either pointing up or down.

The Theoretical Minimum states that all particles have only two spin states, and this is a fundamental principle of quantum mechanics. This means that regardless of the type of particle or its properties, it will always have two possible spin states: up and down. This may seem counterintuitive, but it is a fundamental aspect of quantum mechanics.

The normal coordinate system is a mathematical tool used to describe the orientation of a particle's spin. It is a set of coordinates that can be used to determine the direction of a particle's spin in three-dimensional space. Understanding the normal coordinate system is important because it allows us to visualize and manipulate spin states in a more tangible way.

I hope this explanation has helped to clarify the concept of spin states and their relation to the Theoretical Minimum and normal coordinate system. If you still have any questions or need further clarification, please don't hesitate to ask. Quantum mechanics can be a complex subject, and it's important to have a solid understanding of its fundamental principles before delving into more advanced concepts.