Understanding Direct Product States in Linear Algebra

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Discussion Overview

The discussion revolves around the representation of state vectors for distinguishable particles in linear algebra, specifically focusing on the concept of direct product states and their implications within Hilbert spaces.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • Kevin questions why the state vector for two distinguishable particles is expressed as |psi>=|psi_1>|psi_2> and seeks clarification on the nature of this product.
  • One participant suggests that the state is actually in the tensor product of the individual particle spaces.
  • Kevin expresses interest in exploring the idea further after receiving a response.
  • Reilly Atkinson notes that the state |a>|b> is referred to as a "direct product state" and mentions that this concept is commonly discussed in linear algebra literature.

Areas of Agreement / Disagreement

Participants have not reached a consensus, as there are differing views on the nature of the state vector and its representation in Hilbert space.

Contextual Notes

The discussion does not clarify the assumptions underlying the definitions of the Hilbert space or the tensor product, leaving some aspects unresolved.

homology
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So why is it that if you have a system with two distinguisable particles why is their state vector written as

|psi>=|psi_1>|psi_2>

what's this product? Is this product still "in" the original hilbert space?

Kevin
 
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I think the state is in the tensor product of the particle spaces.
 
Huh, is that so, I'll have to play around with that idea and see where it takes me. Thanks SelfAdjoint.

Kevin
 
The state |a>|b> is also called a "direct product state", and this basic idea is discussed in most books on linear algebra.
Regards,
Reilly Atkinson
 

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