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xdeimos
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What is the significance of operator conventions in quantum mechanics?
- Thread starter xdeimos
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- Expected value Quantum Value
In summary, the quantum expected value is defined as the trace of a positive operator multiplied by the operator itself. This formula is valid for both pure and mixed quantum states. The distinction between "bra" and "ket" wavefunctions is purely notational and the sign of the complex part of the wavefunction is also arbitrary.
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xdeimos
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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
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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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Khashishi
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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.
FAQ: What is the significance of operator conventions in quantum mechanics?
What is expected value in quantum?
Expected value in quantum is a mathematical concept that represents the average outcome of a measurement for a quantum system. It is calculated by taking the sum of all possible outcomes, weighted by their probabilities.
How is expected value calculated in quantum?
To calculate expected value in quantum, you first need to determine all possible outcomes of a measurement for a given quantum system. Then, you multiply each outcome by its respective probability and add them together to get the expected value.
Why is expected value important in quantum?
Expected value is important in quantum because it allows us to predict the most likely outcome of a measurement for a quantum system. It also helps us understand the behavior and characteristics of quantum particles.
Can expected value be negative in quantum?
Yes, expected value can be negative in quantum. This can occur when the possible outcomes of a measurement have both positive and negative values, and the probabilities of these outcomes are not evenly distributed.
How is expected value used in quantum experiments?
Expected value is used in quantum experiments to help researchers make predictions about the outcomes of measurements for a given quantum system. It also helps in designing experiments and analyzing the results to better understand the behavior of quantum particles.