Eigenvectors, spinors, states, values

• SoggyBottoms
In summary, eigenvectors, eigenstates, and eigenspinors are all related concepts that refer to the vector or spinor that represents the state of a quantum system. Spinors are a specific way to express spin state vectors, with different components depending on the spin of the particle. State vectors are written as |ψ> and can be represented in a specific base using column vector notation. These terms are often used interchangeably, but they have distinct differences in meaning and notation.
SoggyBottoms
For spin-1/2, the eigenvalues of $S_x, S_y$ and $S_z$ are always $\pm \frac{\hbar}{2}$ for spin-up and spin-down, correct?

What is the difference between eigenvectors, eigenstates and eigenspinors? I believe eigenstates = eigenspinors and eigenvectors are something else? I'm just getting confused, because my teachers use the letter $\chi$ for everything.

Hi SoggyBottoms!

I think "eigenvector" is the correct technical term for integral spin, and "eigenspinor" for half-integral spin.

But i wouldn't worry about it.

So all three terms are actually the same? At least as far as my introductory QM is concerned? Eigenvector = eigenspinor = eigenstate?

well, not exactly the same, since a spinor isn't a vector, and a vector isn't a spinor

but your introductory QM probably doesn't go into the difference between vectors and spinors in detail anyway

It doesn't indeed, but they use all the terms interchangeably it seems, so it's confusing. Thanks.

State is short for state vector, so eigenstate and eigenvector are the same. These terms are general and apply to every quantum system.

Spinors are a specific way to express spin state vectors. For spin 1/2 particles, they have two or four components (Pauli spinor vs. Dirac spinor).

State vectors are written as |ψ>. If you want to write them in a specific base, you use column vector notation (<a1|ψ> <a2|ψ> ... )T. The same notation is often used for spinors, although rectangular brackets are arguably better to make it clear you are talking about spinors: χ=[c1 c2]T.

1. What are eigenvectors and why are they important in mathematics?

Eigenvectors are special vectors that, when multiplied by a square matrix, result in the same vector multiplied by a scalar value. They are important because they represent the directions in which a matrix transformation has no effect other than stretching or shrinking the vector.

2. How are spinors related to eigenvectors?

Spinors are a type of mathematical object that can be thought of as generalized eigenvectors. They are commonly used in quantum mechanics to represent the intrinsic angular momentum (spin) of particles, and their properties can be described using the same mathematical principles as eigenvectors.

3. What is the difference between an eigenstate and an eigenvector?

An eigenstate is a quantum state that is a solution to the Schrödinger equation, while an eigenvector is a mathematical object that represents the direction of no change in a transformation. Eigenstates are often represented as linear combinations of eigenvectors.

4. How are eigenvalues and eigenvectors related?

Eigenvalues are the scalar values that result when an eigenvector is multiplied by a matrix. In other words, they represent the amount by which the eigenvector is stretched or shrunk. Each eigenvector has a corresponding eigenvalue, and they are often used together to solve problems in linear algebra.

5. What is the significance of eigenvectors and eigenvalues in quantum mechanics?

In quantum mechanics, eigenvectors and eigenvalues are used to describe the possible states and energy levels of a quantum system. The eigenvectors represent the possible states of the system, while the eigenvalues correspond to the energy levels associated with those states. They are essential in understanding the behavior and properties of particles at the quantum level.

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