Heisenberg's Uncertainty Principle using Linear Algebra

In summary, the conversation discusses the uncertainty principle in linear algebra, specifically the equation x^{ T }x=x^{ T }ABx-x^{ T }BAx with the premise AB-BA=I. The question is how x^{ T }x=x^{ T }ABx-x^{ T }BAx. The conversation also mentions the possibility of proving the premise with an elegant method, using a multiplicative constant.
  • #1
rpthomps
182
19
I am working through linear algebra from MITs MOOC online courses. One of the question refers to the uncertainty principle. It states:
AB-BA=I can happen for infinite matrices with A

[tex]A=A^{ T }\\ and\\ B=-B^{ T }\\ Then\\ x^{ T }x=x^{ T }ABx-x^{ T }BAx\le 2\parallel Ax\parallel \parallel Bx\parallel[/tex]

My question is how does
[tex]x^{ T }x=x^{ T }ABx-x^{ T }BAx [/tex]?
 
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  • #2
Like you said:
rpthomps said:
AB-BA=I

Thanks
Bill
 
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Likes rpthomps
  • #3
bhobba said:
Like you said:Thanks
Bill

Okay. I should have got that one. :) Okay, now I want to prove the premise AB-BA=I, is there an elegant way of doing that?
 
  • #4
rpthomps said:
Okay. I should have got that one. :) Okay, now I want to prove the premise AB-BA=I, is there an elegant way of doing that?

That is the premise of the theorem except for a multiplicative constant - the I is replaced by iC - C a real constant.

Thanks
Bill
 
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Likes rpthomps
  • #5
Thanks again I really appreciate your attention.
 

Related to Heisenberg's Uncertainty Principle using Linear Algebra

1. What is Heisenberg's Uncertainty Principle using Linear Algebra?

Heisenberg's Uncertainty Principle using Linear Algebra is a mathematical formulation of the famous principle in quantum mechanics that states that it is impossible to know both the position and momentum of a particle with absolute certainty. This formulation uses linear algebra to describe the uncertainty in terms of the uncertainty in the measurement of certain observables.

2. How is Heisenberg's Uncertainty Principle derived using Linear Algebra?

Heisenberg's Uncertainty Principle can be derived using the commutation relations between position and momentum operators in quantum mechanics. These commutation relations are then represented in matrix form using linear algebra and the uncertainty in the measurement of these operators is calculated using the eigenvalues and eigenvectors of the matrices.

3. Why is Heisenberg's Uncertainty Principle important in quantum mechanics?

Heisenberg's Uncertainty Principle is important in quantum mechanics because it sets a fundamental limit on the precision with which certain observables can be measured. This principle plays a crucial role in understanding the behavior of subatomic particles and has significant implications in many fields, including quantum computing and cryptography.

4. Can Heisenberg's Uncertainty Principle using Linear Algebra be applied to macroscopic systems?

No, Heisenberg's Uncertainty Principle using Linear Algebra is only applicable to microscopic systems, such as subatomic particles. This is because at the macroscopic level, the uncertainties in measurements of observables are negligible compared to the scales at which they are measured.

5. How does Heisenberg's Uncertainty Principle using Linear Algebra relate to the concept of wave-particle duality?

The wave-particle duality of quantum mechanics states that particles can exhibit both wave-like and particle-like behaviors. Heisenberg's Uncertainty Principle using Linear Algebra is a mathematical representation of this concept, as it shows that the position and momentum of a particle cannot be known at the same time, similar to how a wave's position and momentum cannot be precisely determined.

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