Prove Matrix Representations of p & x Don't Satisfy [-ih/2pi]

In summary, the conversation discusses the issue of using trace to prove that there are no finite dimensional matrix representations of the momentum and position operators that satisfy their commutation relation. It is noted that in the finite dimensional case, the trace operation leads to a contradiction, but in the infinite dimensional case, the trace may not be well-defined, making it difficult to determine the validity of the argument.
  • #1
bon
559
0

Homework Statement



By taking the trace of both sides prove that there are no finite dimensional matrix representations of the momentum operator p and the position operator x which satisfy [p,x] = -ih/2pi

Why does this argument fail if the matrices are infinite dimensional?


Homework Equations





The Attempt at a Solution



No idea really..

So I am guessing it'll be something like

[p,x] = px-xp

Let px-xp= some matrix C

Take trace of both sides,

Tr(px-xp) = Tr (C)

0=Tr(C)..

But don't see if/how this answers the question/what would be different in infinite case?

Thanks!
 
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  • #2
You also need to note that in this finite dim representation

[tex] C = - i\hbar I[/tex]

where [tex]I[/tex] is the identity matrix.
 
  • #3
Oh i see. Thanks

so i have Tr(-ih/2pi I)=0 so assuming p and x are nxn, we have (-ih)^n Tr(I) = n(-ih)^n = 0 which implies n = 0?

But how do things change if it is infinite?
 
  • #4
bon said:
Oh i see. Thanks

so i have Tr(-ih/2pi I)=0 so assuming p and x are nxn, we have (-ih)^n Tr(I) = n(-ih)^n = 0 which implies n = 0?

Is n=0 consistent with actually having a matrix representation?

But how do things change if it is infinite?

Is the trace operation well-defined for infinite-dimensional matrices?
 
  • #5
umm well I am not sure..can't it just be the sum to infinity rather than to some finite limit?
 
  • #6
bon said:
umm well I am not sure..can't it just be the sum to infinity rather than to some finite limit?

What about Tr I? Is that well-defined?
 
  • #7
it's infinity? oh i see..we run into problems? is that all that needs to be said?

Thanks!
 
  • #8
Well you have similar problems trying to define Tr(xp) and Tr(px), so it's hard to define the difference between those traces as well.
 

1. How do you prove that matrix representations of p and x do not satisfy [-ih/2pi]?

This can be proven by showing that the commutator [p,x] is not equal to [-ih/2pi], as expected for the canonical commutation relation.

2. What is the significance of [-ih/2pi] in this context?

[-ih/2pi] is the expected value for the commutator of p and x, based on the canonical commutation relation. If the matrix representations of p and x do not satisfy this value, it indicates a violation of the canonical commutation relation.

3. Can you provide an example of a matrix representation of p and x that does not satisfy [-ih/2pi]?

Yes, one example is the matrix representation of p = -i * d/dx and x = x, where d/dx is the derivative operator. This representation does not satisfy [-ih/2pi] since the commutator is equal to -i instead of [-ih/2pi].

4. How does this result impact our understanding of quantum mechanics?

This result highlights the importance of the canonical commutation relation in quantum mechanics and the need for consistency between mathematical representations and physical observables. It also suggests that there may be limitations to our current understanding of the relationship between position and momentum in quantum systems.

5. What further research can be done to explore this topic?

Further research can investigate alternative representations of p and x that may satisfy the canonical commutation relation, as well as the implications of this result for other fundamental concepts in quantum mechanics such as uncertainty and measurement. Additionally, experimental studies could be conducted to test the validity of these representations in real-world systems.

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