Pauli Spin Matrix Problem

In summary, based only upon the fact that Pauli matrices are somehow related to rotation, I guessed that the eigenvalues for (σ^.P^) might be something like (p_x)±i(p_y)+|p|. However, after trying out this guess and finding that it does not give a hint of rotation in a complex plane, I am not sure if this is the right answer. I am trying to find a clever trick to solve this equation.
  • #1
neelakash
511
1

Homework Statement



If P^ is the momentum operator, and σ^ are the three Pauli spin matrices, the eigenvalues
of (σ^.P^) are

(a) (p_x) and (p_z) (b) (p_x)±i(p_y) (c) ± |p| (d) ± (p_x + p_y +p_z)

Homework Equations


The Attempt at a Solution



Pauli matrices are related to rotation.So, (b) looks correct to me.

[I am a Bachelor level student and this problem belongs to Masters level.I am trying to do this to see if any tricky method, known to me can be used to solve this.]
 
Physics news on Phys.org
  • #2
Tell us how you did it instead of just showing the answers:)

the eigenvalues
of (σ^3.P^3) are not something that we all here know by heart ;)
 
  • #3
Tell us how you did it instead of just showing the answers:)

the eigenvalues
of (σ^.P^) are not something that we all here know by heart ;)

It was merely a guesswork based only upon the fact that Pauli matrices are somehow related to rotation...and as I told you,possibly it was not a problem from my course.I am trying this to see if any clever trick can solve it.

And we can see none of (a) (c) and (d) give a hint of rotation in a complex plane.This was my basis of guesswork.
 
  • #4
Well if you call "guesswork" a trick then ok ;)

I would construct a matrix eq, and find the eigen values for that matrix.

i.e you get the matrix:

[tex] Q = \vec{p}\cdot \vec{\sigma }= p_x\sigma _x + p_y\sigma _y + p_z\sigma _z [/tex]

Find the matrix-representations for the pauli matrices, evaluate the total matrix Q, and then find Q's eigenvalues. I would do this.
 
  • #5
Yes,I know that.But when you write Q, you need to know matrix representations of momentum operator;then multiply with each Pauli matrix.Then add...and then you make your task of solving an eigenvalue problem.That is time consuming and may be difficult in an MCQ exam hall.

Therefore,I was searching for a nice trick.My guesswork may not be a right one and I do not know if this answer is at all correct.
 

What is the Pauli Spin Matrix Problem?

The Pauli Spin Matrix Problem refers to a mathematical problem in quantum mechanics that involves finding the eigenvalues and eigenvectors of the Pauli spin matrices. These matrices are used to describe the spin states of particles, and solving this problem allows for the calculation of various physical quantities related to spin.

Why is the Pauli Spin Matrix Problem important?

The Pauli Spin Matrix Problem is important because it is used to understand the behavior of particles with spin, which is a fundamental property of matter. It also has practical applications in areas such as quantum computing and magnetic resonance imaging.

What are the solutions to the Pauli Spin Matrix Problem?

The solutions to the Pauli Spin Matrix Problem are the eigenvalues and eigenvectors of the Pauli spin matrices. These values correspond to the possible spin states of a particle and can be used to calculate various physical quantities related to spin.

How is the Pauli Spin Matrix Problem solved?

The Pauli Spin Matrix Problem is typically solved using mathematical techniques such as diagonalization or matrix algebra. These methods allow for the calculation of the eigenvalues and eigenvectors of the Pauli spin matrices, which in turn provide the solutions to the problem.

Are there any real-life applications of the Pauli Spin Matrix Problem?

Yes, there are several real-life applications of the Pauli Spin Matrix Problem. For example, it is used in quantum computing to understand the behavior of qubits, which are the basic units of information in quantum computers. It is also used in magnetic resonance imaging (MRI) to produce images of the inside of the human body.

Similar threads

Replies
15
Views
3K
  • Advanced Physics Homework Help
Replies
10
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
905
  • Advanced Physics Homework Help
Replies
1
Views
5K
Replies
5
Views
2K
  • Advanced Physics Homework Help
Replies
10
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Special and General Relativity
Replies
3
Views
825
  • Advanced Physics Homework Help
Replies
4
Views
8K
  • Advanced Physics Homework Help
Replies
7
Views
1K
Back
Top