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Jaycurious
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When orthogonal states of a quantum system is projected into subspaces A and B are A and B real spaces?
By "real" do you mean a vector space over the field of real numbers? If so, then no. A subspace of a vector space is, by definition, a vector space over the same field of scalars: in this case the complex numbers.Jaycurious said:When orthogonal states of a quantum system is projected into subspaces A and B are A and B real spaces?
PeroK said:By "real" do you mean a vector space over the field of real numbers? If so, then no. A subspace of a vector space is, by definition, a vector space over the same field of scalars: in this case the complex numbers.
I'm sorry, but I can make no sense of that except your last question.Jaycurious said:Thanks, this is what I was asking. Another question. Are observers associated with subspaces A and B?
The way I understand it is that the states of an electron in Hilbert space like spin up and spin down are projected into subspaces A and B.
These subspaces can be in a pure state with zero entropy or if subspace B is in a mixed state then the entropy of entanglement increases.
Look up the defintion of vector space and subspace. The short answer is that a subspace is only part of a vector space.Jaycurious said:So if subspaces are complex valued like Hilbert space then what's the difference between subspaces and Hilbert space?
That hotel is impossible to sleep in! First, they overbook the rooms. Then they keep waking you up all night, asking you to move one room over in order to make room for the new arrivals. It feels like never-ending exhaustion.sysprog said:Yeah, @PeroK, and who knows how many spaces Dr. Hilbert could show you if you were to check into his hotel . . .
Yup, I agree with you because I experienced this.Bruzote said:That hotel is impossible to sleep in! First, they overbook the rooms. Then they keep waking you up all night, asking you to move one room over in order to make room for the new arrivals. It feels like never-ending exhaustion.
A subspace of Hilbert space is a subset of the larger Hilbert space that is closed under addition and scalar multiplication. It is a vector space that shares the same inner product as the larger Hilbert space and is therefore a subset of the larger space.
Subspaces of Hilbert space are used in a variety of scientific fields, including physics, engineering, and mathematics. They are particularly useful in quantum mechanics, where they are used to represent the states of quantum systems and the operators that act on them.
Subspaces of Hilbert space are both real and mathematical constructs. They are real in the sense that they represent physical systems and their properties, but they are also mathematical constructs that are used to describe and analyze these systems.
Subspaces of Hilbert space are closely related to other mathematical concepts, such as vector spaces, inner product spaces, and Banach spaces. They share many properties and can often be transformed into each other through mathematical operations.
While it is not always possible to visualize subspaces of Hilbert space directly, they can often be represented graphically or through mathematical models. For example, in quantum mechanics, the wave function of a system can be represented as a vector in Hilbert space, which can then be visualized using graphs or other mathematical tools.