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kakarukeys
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sometimes I see [\hat{q},\hat{p}] = i\hbar\widehat{\{q,p\}} + O(\hbar^2)
what does the last term O(\hbar^2) mean?
[tex]x=y[/tex]
what does the last term O(\hbar^2) mean?
[tex]x=y[/tex]
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Hurkyl said:It's asymptotic notation. Since you're probably in a context where everything's analytic... O(x^2) is essentially just shorthand for "possibly plus some more terms involving powers of x with exponent at least 2".
kakarukeys said:finally latex is back:
here's the equation that's puzzling me:
[tex][\hat{q},\hat{p}] = i\hbar\widehat{\{q,p\}} + O(\hbar^2)[/tex]
without [tex]O(\hbar^2)[/tex], the equation is just the usual canonical quantization recipe. what is that term for?
kakarukeys said:
Read post#7 in the thread "transition from poisson brackets to ..." if you follow my derivation, you will be able to see where the [tex]O(\hbar^{2})[/tex] term comes from.
O(something) stands for "Of Order Of" that "something".
sam
The symbol \hbar, also known as "h-bar", is the reduced Planck's constant in quantum mechanics. It is a fundamental constant that relates to the discrete nature of quantum systems and is used to determine the uncertainty in measuring a particle's position and momentum simultaneously.
In quantum mechanics, commutation relations describe how two physical quantities, such as position and momentum, behave when measured simultaneously. \hbar^2 is often seen in these equations and represents the square of the reduced Planck's constant. It is used to calculate the uncertainty in the measurement of these quantities.
O(\hbar^2) is known as the "order of magnitude" of \hbar^2 in commutation relations. This means that \hbar^2 is being used as a small quantity compared to other terms in the equation. It represents the level of precision in calculating the uncertainty between two quantities.
Since \hbar^2 is a small quantity, it is often used in commutation relations to represent the uncertainty in measuring two quantities simultaneously. This is because, in quantum mechanics, the more precisely one quantity is measured, the less precisely the other can be measured. \hbar^2 helps to quantify this uncertainty.
The value of \hbar^2 can affect the behavior of quantum systems in terms of the uncertainty principle. The smaller the value of \hbar^2, the more precise the measurements of two quantities can be, but this also means that the uncertainty in the third quantity will increase. This is a fundamental concept in quantum mechanics that helps to explain the probabilistic nature of quantum systems.