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QuantumCosmo
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Hi,
I was wondering: What is the physical reason for only choosing linear operators to represent observables?
I was wondering: What is the physical reason for only choosing linear operators to represent observables?
QuantumCosmo said:Hm,
yeah, it's certainly true that the mathematics is simpler and a lot of things we use - for example the spectral theorem - require operators to be linear. But I can't seem to find a physical reason for why that should be so.
The only thing that comes to mind is the requirement of real eigenvalues. For that you need a self adjoint (and therefore linear, since we are dealing with operators on Hilbert spaces, not star algebras or the like) operator. Since we measure those quantities, the should be real. But I'm not sure if this qualifies as a "physical reason"...
Many nonlinear systems are tractable by functional analytic techniques in a bigger linear space. On the other hand, a linear problem may have a more tractable nonlinear representation; then this may be an important advantage. It depends a lot on what sort of questions one is trying to answer.QuantumCosmo said:Ok, so it is really a practical decision rather than a physical necessity?
A non-linear operator is a mathematical function that does not follow the principles of linearity, meaning that the output is not directly proportional to the input. Non-linear operators can be used to describe complex systems and phenomena that cannot be explained by linear equations.
Some common examples of non-linear operators include logarithmic, exponential, and trigonometric functions. Other examples can be found in physics, such as the Schrödinger equation in quantum mechanics and the Navier-Stokes equation in fluid dynamics.
Non-linear operators are used to describe physical systems that exhibit non-linear behavior, such as chaotic systems or systems with complex interactions. These operators allow us to better understand and predict the behavior of these systems, which may not be possible using linear equations.
Non-linear operators can lead to unexpected and complex behavior in a system, such as bifurcations, oscillations, and chaos. This is because small changes in the input can result in large changes in the output, making the system difficult to predict or control.
One of the main challenges in using non-linear operators is the complexity of the equations and the difficulty in solving them. This often requires advanced mathematical techniques and computational methods. Additionally, non-linear systems can be highly sensitive to initial conditions, making it challenging to accurately model and predict their behavior.