An eigenstates, eigenvectors and eigenvalues question

Click For Summary

Discussion Overview

The discussion revolves around eigenstates, eigenvectors, and eigenvalues in the context of quantum mechanics, specifically related to the operators associated with spin states. Participants are addressing a problem from past exam papers, exploring the mathematical relationships between state vectors and their corresponding operators.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes the process of showing that specific state vectors are eigenvectors of a given operator using the characteristic equation and simultaneous equations.
  • Another participant suggests an alternative approach to demonstrate that the state vectors are eigenvectors by directly applying the operator to the states, arguing it is a more efficient method.
  • A later reply proposes that the second part of the question can be approached similarly by applying the operator to the combined state and setting up an eigenvalue equation.
  • There is a discussion about the relative difficulty of solving the characteristic equation versus applying the operator to the vectors, with differing opinions on which method is easier or faster.
  • Some participants express uncertainty about the correctness of their suggestions, indicating a lack of confidence in their proposed solutions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to solve the problem. There are competing views on the efficiency of different methods for demonstrating the eigenvector relationships.

Contextual Notes

Some participants note that the complexity of the matrix or operator could affect the ease of solving the eigenvalue equation, suggesting that the discussion is context-dependent.

pigletbear
Messages
2
Reaction score
0
Good evening :-)

I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could point me in the right direction:

S2) Show that the state vectors |Sx+> = \frac{1}{\sqrt{}2} times a 2x1 matrix (1,1) and |Sy+> = \frac{1}{\sqrt{}2} times a 2x1 matrix (1,-1) are eigenvectors of Sx = h/2 times a 2x2 matrix (0 1, 1 0) with respective eigenvalues plus and minus h/2...

This I can do by using |A - λI| = 0, finding the eigenvalues, then using A.v=λv and setting up simutaneous quations to find the eigenvalues

Part two... Of what operator is the state \frac{1}{sqrt{}2}[/itex]/(|S<sub>x</sub>+&gt; + |S<sub>y</sub>+&gt;) and eigenstate, and with what eigenvalue...<br /> <br /> Any help would be great and much appreciated
 
Physics news on Phys.org
pigletbear said:
Good evening :-)

I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could point me in the right direction:

S2) Show that the state vectors |Sx+> = \frac{1}{\sqrt{}2} times a 2x1 matrix (1,1) and |Sy+> = \frac{1}{\sqrt{}2} times a 2x1 matrix (1,-1) are eigenvectors of Sx = h/2 times a 2x2 matrix (0 1, 1 0) with respective eigenvalues plus and minus h/2...

This I can do by using |A - λI| = 0, finding the eigenvalues, then using A.v=λv and setting up simutaneous quations to find the eigenvalues

Part two... Of what operator is the state \frac{1}{sqrt{}2}[/itex]/(|S<sub>x</sub>+&gt; + |S<sub>y</sub>+&gt;) and eigenstate, and with what eigenvalue...<br /> <br /> Any help would be great and much appreciated
<br /> <br /> Hello!<br /> My suggestion is to try to explicitly add up the two states in matrix notation... <br /> The answer should then be obvious to you :)<br /> <br /> Edit:<br /> The answer you gave to the first question is right. Nevertheless it is more time consuming then it was necessary and time is precious during exams :) <br /> You have to <i>show</i> that those are egenvectors with given eigenvalue, so you could simply show that <br /> <br /> S_x |S_x ± \rangle = ± \frac{h}{2}|S_x ± \rangle, <br /> <br /> without solving the egenvalue equation <br /> <br /> |S_x-\lambda I|=0.<br /> <br /> Ilm
 
Last edited:
Ilmrak said:
Hello!
My suggestion is to try to explicitly add up the two states in matrix notation...
The answer should then be obvious to you :)

Edit:
The answer you gave to the first question is right. Nevertheless it is more time consuming then it was necessary and time is precious during exams :)
You have to show that those are egenvectors with given eigenvalue, so you could simply show that

S_x |S_x ± \rangle = ± \frac{h}{2}|S_x ± \rangle,

without solving the egenvalue equation
Ilm

|S_x-\lambda I|=0. is a lot easier and faster to solve ;)

about the second question, it works in the same way:

A(|Sx+> + |Sy+>) = λ (|Sx+> + |Sy+>)

=> (A-λI)(|Sx+> + |Sy+>) = 0

A-λI = 0; Fill in λ = 1/sqrt(2) and the matrix will be (1/sqrt(2) 0; 0 1/sqrt(2) )

It's been a while since I did this, I could be wrong ofcourse.. (and it looks a bit to easy, but hey?)edit: I think I'm wrong...
 
Dreak said:
|S_x-\lambda I|=0. is a lot easier and faster to solve ;)[...]

Yes it is easy but no, it isn't faster.
And if it were a bigger matrix (or worst a differential operator) it wouldn't be so easy to solve that equation while it would still be easy to let a matrix (or a differential operator) act on a vector (or a function).
To solve an equation is (almost) always more difficult than checking one solution ^^

Ilm
 

Similar threads

Undergrad Multiple questions about eigenstates and eigenvalues
  • · Replies 3 ·
Replies
3
Views
2K
Graduate The eigenvalue power method for quantum problems
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Why is this 3D operator with degeneracies only giving me 2 eigenstates
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Finding ##S_x## eigenstate using experiments
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Changing Hamiltonian with some eigenvalues constant
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad How are good quantum numbers related to perturbation theory?
  • · Replies 18 ·
Replies
18
Views
4K
Graduate Degenerate Perturbation Theory: Correction to the eigenstates
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Eigenvalues of Hyperfine Hamiltonian
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
3K
Undergrad Find the Eigenvalues and eigenvectors of 3x3 matrix
  • · Replies 12 ·
Replies
12
Views
3K