Completeness of basis in quantum mechanics

In summary, In a QM course, we learn that a complete set of basis vectors can represent an operator, and there exist coefficients that can be used to express the identity matrix. Additionally, in linear algebra, a set of basis vectors is considered complete if any vector can be expressed as a linear combination of those vectors. Both definitions are equivalent.
  • #1
wowowo2006
13
0
In a QM course,
I learn that an operator can be represented by basis vectors
If the basis vector is complete, the following relation holds
There exist coefficient Mij such that
Sigma Mij |i > < j|. = I , |i> is the basis! and I is the identity matrix

But isn't that in linear algebra
We call the set of basis is complete when
Any vector can be expressed into their linear combination

I wonder why here we seems have 2 definition of completeness
 
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  • #2
I recommend reading of Sakurai's Book on QM - read the introductory chapters.
 
  • #3
I wonder why here we seems have 2 definition of completeness
There is only one definition if the two renderings are equivalent, right ? And they are equivalent!
 

Related to Completeness of basis in quantum mechanics

What is the completeness of basis in quantum mechanics?

The completeness of basis in quantum mechanics is a fundamental concept in the theory of quantum mechanics. It refers to the idea that any state of a quantum system can be described as a linear combination of a set of basis states. This means that the basis states form a complete set, in that they can be used to express any state of the system.

Why is completeness of basis important in quantum mechanics?

The completeness of basis is important in quantum mechanics because it allows us to fully describe the state of a quantum system. This is essential for understanding the behavior and properties of quantum systems, and for making predictions about their behavior. Without a complete basis, we would not be able to fully understand or accurately describe quantum phenomena.

What is the relationship between completeness of basis and the uncertainty principle?

The completeness of basis and the uncertainty principle are both fundamental principles in quantum mechanics. The uncertainty principle states that it is impossible to know both the position and momentum of a particle with absolute certainty. Completeness of basis plays a role in this because it allows us to express the state of a particle in terms of a basis set, and therefore to make predictions about its behavior within the constraints of the uncertainty principle.

How is completeness of basis related to the concept of superposition in quantum mechanics?

Completeness of basis is closely related to the concept of superposition in quantum mechanics. Superposition refers to the idea that a quantum system can exist in multiple states simultaneously. Completeness of basis allows us to express the state of a system in terms of a basis set, which includes all possible states. This means that a system in superposition can be described as a linear combination of basis states.

Can the completeness of basis be violated in quantum mechanics?

No, the completeness of basis cannot be violated in quantum mechanics. It is a fundamental principle that is essential for understanding and describing quantum systems. Any attempt to violate the completeness of basis would result in a failure to accurately describe the system and make predictions about its behavior.

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