Trace of Position and Momentum Operator

In summary, the conversation discusses the mathematics used in quantum mechanics and specifically the finite and infinite dimensional matrix representations of the momentum and position operators. It is shown that finite dimensional representations are not possible due to the commutator [p,x] = -ih, but this argument fails for infinite dimensional matrices such as Heisenberg's. A helpful link is provided for further understanding.
  • #1
unscientific
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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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  • #2
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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  • #3
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.
 
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1. What is the trace of a position operator?

The trace of a position operator is a mathematical quantity that represents the sum of the diagonal elements of the position operator matrix. It is a measure of the average position of a particle in a given state.

2. How is the trace of a momentum operator related to the uncertainty principle?

The trace of a momentum operator is related to the uncertainty principle through the Heisenberg uncertainty relation. The uncertainty principle states that the product of the uncertainty in position and momentum must be greater than or equal to the reduced Planck's constant divided by 2. The trace of the momentum operator can be used to calculate the momentum uncertainty, which is related to the position uncertainty through the uncertainty principle.

3. Can the trace of a position operator be negative?

No, the trace of a position operator cannot be negative. The trace is a sum of the diagonal elements of the position operator matrix, and since position is a positive quantity, the diagonal elements must also be positive, resulting in a positive trace.

4. How is the trace of a position operator related to the expectation value of position?

The trace of a position operator is equal to the expectation value of position. This means that the average position of a particle in a given state can be calculated by taking the trace of the position operator matrix.

5. Can the trace of a position operator change over time?

Yes, the trace of a position operator can change over time. This is due to the fact that the position operator and its corresponding matrix can change depending on the system and its evolution over time. Therefore, the trace of the position operator can also change as the system evolves.

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