# Intro to quantum mechanics - Spin and linear algebra

• Graham87
In summary: The matrix is the representation of the operator ##\hat{H}## in the eigenbasis of ##\hat{s}_z##. You must distinguish between operators, living in the abstract 2D Hilbert space and their components wrt. some basis. Then all this confusion wouldn't occur.
Graham87
Homework Statement
Measure Sz from the following expression see pic.
Relevant Equations
spin=1/2
Operators Sy and Sz represent standard spin vectors and w is a positive constant.

So this expression is apparently in Sz basis? How can you see that?
How would it look in Sy basis for example?
The solution is following. They are putting Sz as a basis, bur how do you know that Sz is the basis here?

Thanks

You know it's the z basis because of the matrix form of the operators.

topsquark and Graham87
PeroK said:
You know it's the z basis because of the matrix form of the operators.
Aha, so what signifies z basis? Is it this:

It looks like z and y basis added together?
How would a y basis look?

Thanks!
I will polish my linear algebra soon.

The matrix representing ##S_z## is a diagonal matrix with ##\pm \frac \hbar 2## on the diagonals. That indicates the z basis.

topsquark and Graham87
PeroK said:
The matrix representing ##S_z## is a diagonal matrix with ##\pm \frac \hbar 2## on the diagonals. That indicates the z basis.
Yes. They are asking for Sz of H, so I thought you first need to make a basis change with Sz basis. I think they did that in the solution. But I can’t tell what basis it is. I thought alpha and beta are the new basis matrix?

The first thing they did was express ##H## in the z basis, using the familiar form of ##S_y## and ##S_z## in that basis.

Next, they find the eigenvalues and eigenvectors of ##H##, with the vectors represented in the z basis.

topsquark and Graham87
PeroK said:
The first thing they did was express ##H## in the z basis, using the familiar form of ##S_y## and ##S_z## in that basis.

Next, they find the eigenvalues and eigenvectors of ##H##, with the vectors represented in the z basis.
Aha, I don’t get how
is expressed in the z basis. It looks like the diagonal of z and diagonal of y.

Graham87 said:
Aha, I don’t get how View attachment 313217is expressed in the z basis. It looks like the diagonal of z and diagonal of y.
I suspect you don't understand what it means for a matrix to be expressed in a particular basis.

You seem to be leaning towards wanting to express ##H## in its own eigenbasis.

Graham87 and topsquark
The matrix is the representation of the operator ##\hat{H}## in the eigenbasis of ##\hat{s}_z##. You must distinguish between operators, living in the abstract 2D Hilbert space and their components wrt. some basis. Then all this confusion wouldn't occur. The matrix elements are by definition ##H_{\sigma_1 \sigma_2} = \langle \sigma_1 | \hat{H} | \sigma_2 \rangle##.

Graham87 and malawi_glenn

## 1. What is quantum mechanics?

Quantum mechanics is a branch of physics that studies the behavior of matter and energy at a very small scale, such as atoms and subatomic particles. It explains how these particles behave and interact with each other.

## 2. What is spin in quantum mechanics?

Spin is an intrinsic property of particles in quantum mechanics. It is a measure of the angular momentum of a particle, and it can have only certain discrete values.

## 3. How is linear algebra used in quantum mechanics?

Linear algebra is used to represent and manipulate the state of a quantum system in mathematical terms. It allows us to describe the properties and behavior of particles in a quantum system, and to make predictions about their behavior.

## 4. What is the significance of spin in quantum computing?

The spin of particles is used as a way to store and manipulate information in quantum computing. By controlling the spin of particles, quantum computers can perform calculations and solve problems that are not possible with classical computers.

## 5. How does quantum mechanics relate to the classical world?

Quantum mechanics is the underlying theory that explains the behavior of matter and energy at a small scale. While classical mechanics describes the behavior of larger objects, it can be seen as an approximation of quantum mechanics for macroscopic objects.

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