Trace of Position and Momentum Operator

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SUMMARY

The discussion centers on the impossibility of finite-dimensional matrix representations for the momentum operator \( p \) and the position operator \( x \) in quantum mechanics, specifically under the commutation relation \([p, x] = -ih\). It is established that while finite-dimensional representations fail to satisfy this relation, infinite-dimensional representations, as utilized by Heisenberg, do succeed. The participants highlight the mathematical foundations necessary for understanding these operators, emphasizing the distinction between finite and infinite dimensions.

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  • Quantum Mechanics fundamentals
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  • Commutation relations in quantum mechanics
  • Understanding of differential operators
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Homework Statement



Hi guys, I've not started a course on QM yet, but we are currently learning the maths used in QM.


Show, by taking the trace of both sides show that finite dimensional matrix representations
of the momentum operator p and the position operator x which satisfy [p, x] = −ih are not possible.

Why does this argument fail if the matrices are infinite-dimensional (as Heisenberg’s were)?

15plh50.png


Homework Equations





The Attempt at a Solution



Are the position and momentum operators supposed to be matrices? I've looked up on wikipedia, the position operator is more of like ih(d/dp) while the momentum operator is ih(d/dx).
 
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I believe this link will help you understand.

http://www.asianscientist.com/books/wp-content/uploads/2013/05/7271_chap02.pdf
 
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Bryson said:
I believe this link will help you understand.

http://www.asianscientist.com/books/wp-content/uploads/2013/05/7271_chap02.pdf

It looks difficult but I will give it a go.
 
Last edited by a moderator:

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