Monday, December 23, 2019

quantum mechanics - Fock Subspaces and Weight Vectors


This is my first time taking a physics course (I'm a mathematics major), so I'm encountering a lot of new things, which I'm kind of expected to know. In particular, how to work with Bosons.


I've got the following question to work out;



Let F2 denote the Fock subspace of all 2-boson states, spanned by homogeneous polynomials of degree 2 in the boson creation operators b1, b2, b3 acting on the physical vacuum state |0. From lectures, we know that F2 is a six dimensional gl(3)-module. Explicitly give an orthonomal basis of F2 that is symmetry adapted to the subalgebra chain o(2)o(3)gl(3), and express the highest weight vector of F2 as a linear combination of these basis vectors.




Now, just going through lecture notes, I tried to work out a few things step by step.



  • The gl(3) highest weight vector of F2 must be (2,0,0)

  • F2=c2ΔF0

  • dim(c2) = 5

  • c2={ΨF2|ΔΨ=0}

  • Define Ψ2=k(b1+ib2)2|0


Now, I'm not even sure where to go from there. I've been told to normalise that Ψ2 term, but I'm not even sure how to go about that. I've also been told to act L3 on the Ψ terms, where L3=i(b1b2b2b1)



But I'm not even sure how to even start going about this. Of course I'd love some help on this question, but, links to easy-ish resources that might point me in the right direction, or even give some better background into this would also be much appreciated.




No comments:

Post a Comment

classical mechanics - Moment of a force about a given axis (Torque) - Scalar or vectorial?

I am studying Statics and saw that: The moment of a force about a given axis (or Torque) is defined by the equation: $M_X = (\vec r \times \...