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alisa
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given that [x,p]=ih, show that if x=x, p has the representation p=-iħ∂/∂x+f(x) where f(x) is an arbitrary function of x
alisa said:given that [x,p]=ih, show that if x=x,
Tom Mattson said:What do you mean " if x=x". x=x by definition.
The commutator relation [x,p]=ih is a fundamental equation in quantum mechanics that describes the relationship between the position operator x and the momentum operator p. It states that the commutator of these two operators is equal to ih, where i is the imaginary unit and h is the reduced Planck's constant. This relation is used to derive various important principles in quantum mechanics, such as the uncertainty principle.
The commutator relation [x,p]=ih is used as a starting point in the proof of the momentum operator p=-iħ∂/∂x+f(x). By applying the commutator relation to the position and momentum operators, we can derive the expression for the momentum operator in terms of the position operator and its derivative.
The symbol ħ, also known as h-bar, represents the reduced Planck's constant. It is a fundamental constant in quantum mechanics that relates the energy of a particle to its frequency.
Yes, the commutator relation can be extended to other pairs of operators in quantum mechanics. For example, the commutator relation [A,B]=iC can be used to describe the relationship between two arbitrary operators A and B, where C is another operator.
The commutator relation is used in various practical applications in quantum mechanics. It is used to derive the uncertainty principle, which is a fundamental principle in quantum mechanics that states that there is a limit to the precision with which certain pairs of physical properties of a particle can be known. It is also used in the calculation of quantum mechanical observables, such as energy and momentum, in quantum systems.