Pauli Matrices and orthogonal projections

clumsy9irl
Messages
7
Reaction score
0
Ok, I'm working with the Pauli Matrices, and I've already gone through showing a few bits of information. I've got a good idea how to keep going, but I'm not exactly sure about this one--

say M= 1/2(alphaI + a*sigma)

where alpha E C, a=(ax, ay, az) a complex vector, a*sigma=ax sigmax+ay sigmay+ az sigmz, and I is the identity matrix.


So, an orthogonal projection means that for a matrix P, P^2 and P dagger are both equal to P, right?

Supposedly alpha and a can beonstrained so that M is an orthogonal projection.


How would I go about doing that? :confused:


Thanks muchly!
 
Physics news on Phys.org
Some hints

Hey there !

Here are some hints on a was to do it :

You will only use the fact that M^2 = M...

1) Compute M^2 :

M^2 = \cfrac{1}{4}\left(\alpha I + \sum_i a_i \sigma_i \right)^2 = ...

You will find a term like \sum_i \sum_j a_i a_j \sigma_i \sigma

In order to reduce this term, use some common identities :

\sigma_i \sigma_j = I \delta_{ij} + i \epsilon_{ijk} \sigma_k

and

\sigma_i \sigma_j + \sigma_j \sigma_i = 2 \delta_{ij} I

If you applied them correctly, you should get something quite simple. You then use the fact that M^2 = M and you will find by identification :
\alpha = 1 \text{ and } ||a|| = 1, which is your final answer (Hopefully, I didn't mess up).

If you're not confident with the use of the Levi-Civita symbol, another (more tedious) way is to write all in matrix notation (2x2), compute M^2 and put this equal to M... You will get 4 equations, which will reduce to the answer given above.

Hope this helps,
Cheers,
Florian
 
i must've screwed something up, for I'm getting a sqrt(2) for my alpha?

maybe i screwed up something with the identities...
 
oh, and i almost forgot, THANK YOU VERY MUCH!
 
can we solve the particle in a box problem using schrodingers equation?
 
mayriluseeya said:
can we solve the particle in a box problem using schrodingers equation?

Sure, it's easy; they do it in Halliday and Resnick.

(But not with the Pauli spin matrices.)
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In her YouTube video Bell’s Theorem Experiments on Entangled Photons, Dr. Fugate shows how polarization-entangled photons violate Bell’s inequality. In this Insight, I will use quantum information theory to explain why such entangled photon-polarization qubits violate the version of Bell’s inequality due to John Clauser, Michael Horne, Abner Shimony, and Richard Holt known as the...
Not an expert in QM. AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order. But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical...
I asked a question related to a table levitating but I am going to try to be specific about my question after one of the forum mentors stated I should make my question more specific (although I'm still not sure why one couldn't have asked if a table levitating is possible according to physics). Specifically, I am interested in knowing how much justification we have for an extreme low probability thermal fluctuation that results in a "miraculous" event compared to, say, a dice roll. Does a...
Back
Top