Discrete Models for Arguments and Continuous Variables in Quantum Mechanics

In summary: McCulloughIn summary, the conversation discusses the use of discrete models in presenting arguments and the potential issue with taking the limit as the resolution is increased. The speaker raises concerns about the applicability of this approach in quantum mechanics due to the existence of irrational numbers. The possibility of using different operators for different types of numbers is also mentioned, but doubts are expressed about its physical meaning. The conversation ends with a request for hints on how to improve the formulation of the question without having to pursue a degree in mathematics.
  • #1
Derek P
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Arguments can often be presented using a discrete model on the assumption that continuous variables can be accommodated by taking the limit as the resolution is increased.

I would have thought that this would be just fine in QM where functions are continuous. But maybe mathematicians here can say where it breaks down. For instance there might be a problem with irrational numbers. We could specify an operator which works differently with different classes of numbers but could it have any physical meaning?

Any hints, other than "take a degree in mathematics and come back when you can formulate your question in better language" :mad: would be appreciated.
 
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  • #2
Derek P said:
Arguments can often be presented using a discrete model on the assumption that continuous variables can be accommodated by taking the limit as the resolution is increased.

I would have thought that this would be just fine in QM where functions are continuous. But maybe mathematicians here can say where it breaks down. For instance there might be a problem with irrational numbers. We could specify an operator which works differently with different classes of numbers but could it have any physical meaning?

Any hints, other than "take a degree in mathematics and come back when you can formulate your question in better language" :mad: would be appreciated.

Taking the continuum limit of discrete theories is a little problematic. An example is the "Fermion doubling" problem that comes up when you try to do quantum field theory on a lattice. I'm not qualified to talk about that, but it's described in Wikipedia:

https://en.wikipedia.org/wiki/Fermion_doubling

--
Daryl
 

1. What is the difference between discrete and continuous variables in quantum mechanics?

Discrete variables refer to quantities that can only take on specific, separate values, such as energy levels in an atom. Continuous variables, on the other hand, can take on any value within a given range, such as position or momentum of a particle.

2. How are discrete models used in quantum mechanics?

Discrete models are often used to simplify complex systems and make them more manageable for calculations. They can also help in understanding the behavior of a system by breaking it down into smaller, discrete components.

3. What are some examples of discrete models used in quantum mechanics?

Some common examples of discrete models in quantum mechanics include the particle in a box, the quantum harmonic oscillator, and the hydrogen atom. These models are used to describe the behavior of particles in specific scenarios and provide insights into the overall principles of quantum mechanics.

4. Can a continuous variable also be treated as a discrete variable in quantum mechanics?

Yes, in certain cases, a continuous variable can be discretized for the purposes of analysis. This is often done in numerical calculations where it is more efficient to work with discrete values than with a continuous range. However, it is important to note that this discretization is an approximation and may not accurately represent the true nature of the system.

5. How do discrete models and continuous variables relate to each other in quantum mechanics?

Discrete models can be used to understand the behavior of continuous variables in quantum mechanics. By breaking down a continuous system into discrete components, we can gain a better understanding of how the system behaves as a whole. Additionally, continuous variables can also be used to describe the behavior of discrete systems, as long as we account for the discrete nature of the system in our calculations.

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