Ask Question. Asked 6 years, 6 months ago. Active 6 years, 6 months ago. Viewed 3k times. The usual treatment of the Hamiltonian factor is another source of approximations. So naively one would expect operator ordering issues, and further analysis show that they yield in fact non-trivial effects.
References: F. Sakurai, Modern Quantum Mechanics, , Section 2. Because the Wey-ordering prescription is connected to the definition of the measure, right? Do you know any reference? A link to some question in this site that partly answers would be welcome as well. Thank you.
But where is the operator ordering issue? And that this correspondence made the path integral unambiguous. But I do not know in what cases this is right and how the ordering-prescription and the measure's ambiguities cancel each other.
Integrals and Operators
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Year International Collaboration Show this widget in your own website. Mathematics Algebra and Number Theory Analysis. Birkhauser Verlag. How to add Component Coupling Operators for later use. For our example, we first want to calculate the spatial integral over the stationary temperature, which is given by. In the next step, we demonstrate how an Integration operator can also be used within the model. We could, for example, ask what heating power we need to apply to obtain an average temperature of First, we need to compute the difference between the desired and the actual average temperature.
The average is calculated by the integral over T , divided by the integral over the constant function 1 , which gives the area of the domain. By default, such an operator is named aveop1. Note that the average over the domain is the same as the integral for our example. That is because the domain has unit area. The corresponding difference is given by. Next, we need to find the General heat flux on the left and lower boundary, so that the desired average temperature is satisfied.
How to add an additional degree of freedom and a global equation, which forces the average temperature to This value has to be prescribed as a General inward heat flux boundary condition to achieve an average temperature of A frequently asked question we receive in Support is: How can one obtain the spatial antiderivative?
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The following application of integration coupling answers this question. The antiderivative is the counterpart of the derivative, and geometrically, it enables the calculation of arbitrary areas bounded by function graphs. One important application is the calculation of probabilities in statistical analyses. A representation of the antiderivative is the following integral.
In contrast to the integrals above, we here have a function as a result, rather than a scalar quantity. First, a logical expression can be used to reformulate the integral as. Second, we need an integration operator that acts on the lower boundary of our example domain.
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Third, we need to include the distinction of integration and output variable. When using an integration coupling operator, the built-in operator dest is available, which indicates that the corresponding expression does not belong to the integration variable. How to plot the antiderivative by Integration coupling, the dest operator, and a logical expression.
COMSOL provides two other integration coupling operators, namely general projection and linear projection. These can be used to obtain a set of path integrals in any direction of the domain. In other words, integration is performed only with respect to one dimension.
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The result is a function of one dimension less than the domain. For a 2D example the result is a 1D function, which can be evaluated on any boundary. Some more details on how to use these operators are subject to a forthcoming blog post on component couplings. The most flexible way of spatial integration is to add an additional PDE interface.
Pseudodifferential and Singular Integral Operators
The task can be formulated in terms of the PDE. The easiest interface to implement this equation is the Coefficient Form PDE interface, which only needs the following few settings:. How to use an additional physics interface for spatial integration. The dependent variable u represents the antiderivative with respect to x and is available during calculation and postprocessing. Besides flexibility, a further advantage of this method is accuracy, because the integral is not obtained as a derived value, but is part of the calculation and internal error estimation.
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