Linear operator and linear vector space?

In summary, in quantum mechanics, operators are linear because they follow the axioms of vector spaces. This means that they are closed under addition and multiplication and have certain properties such as associativity and distributivity. The first axiom of QM states that observables are hermitian linear operators with real eigenvalues, while the second axiom is the Born rule which calculates the expected outcome of an observation. At a more advanced level, Gleason's Theorem is also important to understand. Linear is a key concept in the language of vector spaces used in QM.
  • #1
hi, please tell me what do we mean when we say in quantum mechanics operators are linear and also vector space is also linear ?
 
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  • #2
Its to do with the axioms of QM.

At the starting level we say the states form a vector space - and by definition vector spaces are linear - as hopefully you have learned in a course of linear algebra.

The first axioms of QM is that observables are hermitian linear operators such that the eigenvalues (necessarily real since the operator is hermitian) are the possible outcomes of the observation associated with the observable.

The second axiom is the so called Born rule. If a system is in state u and it is observed with observable O the expected value of the outcome is <u|O|u>.

There is more that can be said at an advanced level - especially the very important Gleason's Theorem - see post 137:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

But basically linear is associated the vector space language QM is expressed in.

Thanks
Bill
 
  • #3
An operator R defined on a set S of functions or vectors over a field F (with + and ×)[ with multiplication * between elements of F and elements of S] is linear if, for all f, g in S and all a in F, R(f⊕g) = R(f) ⊕ R(g), and R(s*f) = s*R(f).

A linear vector space is a set S of vectors closed under addition ⊕ and closed under multiplication ⊗ between scalars [from a field F (with +, ×)] and vectors is defined, such that vector addition is associative and commutative, there is a null vector and every vector has an additive inverse in S, and scalar multiplication is distributive: (a+b)*v = a*v⊕v*b, a(v⊕w) = a*v⊕a*w, and finally (a×b)*v = a*(b*v)
 
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  • #4
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1. What is the difference between a linear operator and a linear vector space?

A linear operator is a mathematical function that maps one vector space to another, while a linear vector space is a set of vectors that can be added and multiplied by scalars in a consistent manner. In other words, a linear operator is a transformation between vector spaces, while a linear vector space is the set of all possible vectors that can be operated on by a linear operator.

2. How do you determine if a function is a linear operator?

A function is considered a linear operator if it satisfies two properties: additivity and homogeneity. Additivity means that the function preserves vector addition, while homogeneity means that the function preserves scalar multiplication. In other words, a function f is a linear operator if f(x + y) = f(x) + f(y) and f(αx) = αf(x) for all vectors x and y, and scalar α.

3. What is the importance of linear operators in mathematics and science?

Linear operators play a crucial role in many areas of mathematics and science, including linear algebra, functional analysis, quantum mechanics, and signal processing. They provide a powerful tool for understanding and solving problems involving linear systems, such as those found in physics, engineering, and economics.

4. How do you determine if a set of vectors forms a linear vector space?

A set of vectors forms a linear vector space if it satisfies three properties: closure under vector addition, closure under scalar multiplication, and the existence of an additive identity (zero vector). In other words, the sum of any two vectors in the set must also be in the set, the product of any scalar and vector in the set must also be in the set, and there must be a vector in the set that behaves as the "zero" element under addition.

5. Can a linear operator have multiple inputs and outputs?

Yes, a linear operator can have multiple inputs and outputs. In fact, many linear operators are defined as transformations between vector spaces of different dimensions, meaning they take in multiple vectors and output multiple vectors. For example, a 3x3 matrix can be thought of as a linear operator that takes in a 3-dimensional vector and outputs another 3-dimensional vector.

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