- #1
xdeimos
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What is the significance of operator conventions in quantum mechanics?
- Context: Graduate
- Thread starter xdeimos
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- Expected value Quantum Value
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Discussion Overview
The discussion revolves around the significance of operator conventions in quantum mechanics, particularly focusing on the placement of operators in relation to wavefunctions and the implications of these conventions for expected values and physical observables. The scope includes theoretical aspects and conceptual clarifications regarding the mathematical framework of quantum mechanics.
Discussion Character
- Conceptual clarification
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant questions why the expected value
is expressed with the wavefunctions ψ* and ψ, seeking clarification on the reasoning behind this convention. - Another participant notes that in Dirac notation, the expected value is represented as E(A) =
- A different viewpoint suggests that operators can be represented by matrices and that the convention of operators acting on the right or left is arbitrary, emphasizing that physical observables are self-adjoint operators.
- This participant also argues that the distinction between "bra" and "ket" wavefunctions is merely notational and does not have real significance, suggesting that the placement of the operator is flexible.
- Additionally, it is mentioned that the sign of the complex part of the wavefunction does not hold real significance, and the choice of placement for the conjugate is based on convention rather than necessity.
Areas of Agreement / Disagreement
Participants express differing views on the significance of operator placement and the implications of conventions in quantum mechanics. There is no consensus on the necessity or implications of these conventions, indicating an unresolved discussion.
Contextual Notes
The discussion reflects varying interpretations of operator conventions and their implications, with some participants emphasizing the arbitrary nature of these conventions while others point to specific mathematical frameworks that may suggest deeper significance.
- #2
xdeimos
- 8
- 0
quantum expected value
1. why <x> is squeez between ψ* and ψ what we doing this?
2. for <p> [h/i d/dx ] is sqeeze between ψ* and ψ why is that?
3. if you put latter operator between ψ* and ψ what is going to happen?
thank you
1. why <x> is squeez between ψ* and ψ what we doing this?
2. for <p> [h/i d/dx ] is sqeeze between ψ* and ψ why is that?
3. if you put latter operator between ψ* and ψ what is going to happen?
thank you
- #3
bhobba
Mentor
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- 3,861
Its easier to see in the Dirac notation - E(A) = <u|A|u>
But that is only a special case valid for so called pure states - the full rule is E(A) = Trace(PA) where P is a positive operator of unit trace which is the correct definition of a quantum state - pure states |u><u| are a special case. For pure states Trace(|u><u| A) = <u|A|u>
As to why that formula check out Gleason's Theorem:
http://kof.physto.se/theses/helena-master.pdf
Thanks
Bill
But that is only a special case valid for so called pure states - the full rule is E(A) = Trace(PA) where P is a positive operator of unit trace which is the correct definition of a quantum state - pure states |u><u| are a special case. For pure states Trace(|u><u| A) = <u|A|u>
As to why that formula check out Gleason's Theorem:
http://kof.physto.se/theses/helena-master.pdf
Thanks
Bill
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- #4
Khashishi
Science Advisor
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The operators can be represented by matrices. Although usually we think of operators as operating on everything to the right, this is just a matter of convention. The physical observables are self-adjoint operators so it makes no physical difference if they act on the right or the left. This is good because the distinction between "bra" and "ket" wavefunctions is really just an artificial notational part of the model, and can't have real significance. We put the operator in the middle so it can act on the left or the right (but only once).
The sign of the complex part of the wavefunction also has no real significance. Multiplying a number by its complex conjugate gives the absolute value squared. This is a handy mathematical trick, but there's no reason other than convention to put the conjugate on the left factor or the right factor.
The sign of the complex part of the wavefunction also has no real significance. Multiplying a number by its complex conjugate gives the absolute value squared. This is a handy mathematical trick, but there's no reason other than convention to put the conjugate on the left factor or the right factor.
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