Confused about Bogoliubov transformation

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SUMMARY

The discussion centers on the correct application of the Bogoliubov transformation in quantum mechanics, specifically the requirement for four distinct vectors: A1k, A2k, B1k, and B2k. The user expresses confusion over their teacher's instruction, contrasting it with their previous understanding, which involved only two vectors and their complex counterparts. Clarification is sought regarding the necessity of the additional vectors and the corresponding expressions for A and B.

PREREQUISITES
  • Understanding of Bogoliubov transformation in quantum mechanics
  • Familiarity with vector notation in quantum field theory
  • Knowledge of complex numbers and their applications in physics
  • Basic principles of quantum mechanics and particle physics
NEXT STEPS
  • Research the mathematical derivation of the Bogoliubov transformation
  • Study the role of multiple vectors in quantum field theory
  • Examine relevant academic papers discussing the Bogoliubov transformation
  • Learn about the implications of using complex vectors in quantum mechanics
USEFUL FOR

Students and researchers in quantum mechanics, particularly those studying quantum field theory and the Bogoliubov transformation, will benefit from this discussion.

JVanUW
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So I was told by my teacher today that I am doing the bogoliubov transformation wrong because I'm supposed to have 4 different vectors,

i.e

Instead of Ak and Bk

A1k
A2k

and
B1k
B2k

I'm wondering how this makes sense. I've seen papers with 2 different vectors and their complex counterparts but never 4 different vectors. Any help would be great.
 
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Maybe you can give us the expressions you obtained for A and B.
 

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