What are Eigenkets and Eigenvectors and how do they relate to quantum mechanics?

In summary, eigenvectors refer to vectors in a specific vector space called Hilbert space, representing physical states. In the context of linear algebra, eigenvectors are vectors that are either preserved or reversed in direction and can be stretched, compressed, or unchanged in magnitude. This concept is closely related to the interpretation of a matrix as a representation of a linear map between vector spaces. In order to fully understand quantum mechanics, it is important to have a strong understanding of linear algebra and the concept of eigenvectors.
  • #1
sol66
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Ok I understand how to find eigenvectors, but I don't understand what they are. I am also uneasy with eigenkets and I don't understand what they are also. I need to understand both these topics to get a grasp on quantum mechanics. thank you
 
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  • #2
The last question is just a matter of language. Basically, a ket is a vector in a particular vector space called Hilbert space, in which (classes of) vectors represent physical states. Thus, if you read "ket" for "vector" your question reduces again to "what are eigenvectors?".

Let me first ask you some preliminary questions.

How firm is your knowledge of linear algebra? Do you know what eigenvectors are in that context?
Have you ever heard about the interpretation of a matrix as representation of a linear map between vector spaces?
 
  • #3
Eigenvectors in terms of geometrical signficance are vectors in which the direction is either preserved or reversed and the magnitude of the vector is either stretched, compressed or unchanged.
 
  • #4
Well I have not taken a formal course in linear algebra and the math methods course that was suppose to teach me this only showed me how to solve for these things and not how to interpret it. I don't know what you mean by interpreting a matrix as a linear map between vector spaces, however I have heard that that vectors are mapped onto operating matrices.

This is my understanding so far, given that you use eigenvalues to diagnolize a matrice, once you find your values you can then find a vector which is preserved in direction whenever your operator acts on that vector. That vector that has a preserved direction given your matrice operator is called your eigenvector.

Please let me know if my understanding is correct
 
  • #5
which is preserved in direction

To be more specific it conserves its direction along a particular line.

Since you can reverse its direction, it's not necessairly pointing in the same direction, but pointing in a direction along the same line.
 

FAQ: What are Eigenkets and Eigenvectors and how do they relate to quantum mechanics?

1. What is an eigenket?

An eigenket is a vector in a vector space that represents a state of a quantum system. It is also known as an eigenvector or characteristic vector.

2. What is an eigenvector?

An eigenvector is a vector that, when multiplied by a square matrix, results in a scalar multiple of itself. It is a special type of vector that represents a unique direction in a vector space.

3. What is the significance of eigenkets and eigenvectors in quantum mechanics?

Eigenkets and eigenvectors play a crucial role in quantum mechanics as they represent the states of a quantum system and the physical quantities that can be measured on that system. They also help in solving quantum mechanical equations and understanding the behavior of quantum systems.

4. How are eigenkets and eigenvectors related?

An eigenket is a specific type of eigenvector that is normalized to have a magnitude of 1. In other words, an eigenket is a unit eigenvector. Both eigenkets and eigenvectors have similar properties and are used in the same way in quantum mechanics.

5. How are eigenkets and eigenvectors calculated?

To calculate an eigenket or eigenvector, you need to find the solution to the eigenvalue equation (A-λI)x=0, where A is a square matrix, λ is an eigenvalue, and x is the eigenket or eigenvector. This can be done using various methods, such as the characteristic polynomial method or the diagonalization method.

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