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Chemist@
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How is a dimension defined in quantum mechanics?
I'm not sure I understand the question. The positive integers are the numbers 1,2,3,... and so on. Does that answer your question?Chemist@ said:What do you mean by the positive integer?
That refers to the dimension of spacetime. Spacetime is defined as a smooth manifold, not as a vector space. The simplest way to explain the dimension of smooth manifold is to say that it's the number of coordinates that a coordinate system assigns to each point in its domain.Chemist@ said:What are dimensions in the string theory then? I think there are 11.
I suppose so, but this notion is never given a precise meaning in math books. There's no definition that appears in math books that allows you to pick a specific line, vector or direction and say "this is a dimension". The precise definitions look like the statements I've made above. In the context of vector spaces, it has to do with linear independence. (How many members can a linearly independent set have?) In the context of manifolds, it has to to with how many real numbers a coordinate system associates with a point.ArtistIC said:Isn't a dimension meant to be an independant vector that cannot be described or calculated by another one (i.e. being vertical to another vector(s) ?
That's what Fredrik described in #4.ArtistIC said:Isn't a dimension meant to be an independant vector that cannot be described or calculated by another one (i.e. being vertical to another vector(s) ?
They just get more noticeable the smaller region of space you're looking at, like how the thickness of a wire get more noticeable when you zoom in on it.ttakacs said:In microphysics one can met 11 or even more dimensions while macrospace is of 3(+1) dimensional. Is there any transition between them and is this transition is continuous or discrete?
As far as I know, the number is fixed.ttakacs said:At quantum level we are facing fluctuations of physical variables. What is the official contemporary standpoint: could -in principle- the number of space dimensions itself fluctuate? (I guess the calculation from uncertainty relations is not adequate here.)
Quantum mechanics is a branch of physics that studies the behavior and interactions of subatomic particles. It is based on the principles of quantum theory, which describes the probabilistic nature of particles at the quantum level.
In quantum mechanics, there are three spatial dimensions (length, width, and height) and one temporal dimension (time). However, some theories suggest the existence of additional dimensions beyond the four we can perceive.
In classical physics, dimensions are considered to be fixed and continuous, while in quantum mechanics, dimensions are described as discrete and probabilistic. This means that at the quantum level, particles can exist in multiple places at once and have uncertain properties.
In quantum mechanics, dimensions are defined by the number of possible states a particle can occupy. This is known as the quantum state space, which is a mathematical representation of all the possible configurations a particle can have.
Some theories in quantum mechanics, such as string theory, suggest the existence of additional dimensions beyond the four we can perceive. These higher dimensions are compactified, meaning they are too small for us to detect, but they play a role in governing the behavior of particles at the quantum level.