What are the important lessons in linear algebra for Quantum?

[Ayham]

This may sound like a dumb question, I heard that to understand Quantum Maths, I have to know Linear Algebra, Calculus, Differential Equations...
I don't have any problems with Calculus and Differentials but Linear Algebra was a bit foggy sometimes... What are the topics in Linear Algebra that i should fully understand for Quantum Mechanics? Are Images and pre-images important?

 

[dydxforsn]

You need to understand vector spaces and transformations between different vector spaces (matrices).

You're going to be told in quantum that quantum systems are defined by "abstract vectors" and to find out something about the quantum system (such as the momentum of a particle), you're going to have to project said abstract vector into the "momentum space" representation of the quantum system. You do this by operating on a "ket" with the momentum operator (much like a matrix operates on a vector in one space to transform it to the same vector in a different space representation.)

Having a natural understanding of linear algebra (not just some silly computational linear algebra course either - one that starts with abstract vectors spaces, then goes to linear transformations between vectors spaces, and then finally discusses matrices later in this context), the terminology and common ideas, REALLY helps in the understanding and discussion of the ideas in quantum. Much of the flow is the same.

 

[Arsenic&Lace]

On the computational side, know how to compute eigenvalues and eigen vectors. Understand why hermitian operators are diagonalized in their eigenbasis. Understanding the relationship between differential operators and linear algebra as well as dual vector spaces gave me more theoretical comfort.

 

[Ayham]

Ok, thanks guys

 

[Matrix0]

I can tell you that understanding linear algebra is indeed crucial for understanding quantum mechanics. Linear algebra provides the mathematical framework for describing the quantum states of particles, which are represented as vectors in a complex vector space.

Some important concepts in linear algebra that are relevant to quantum mechanics include vector spaces, basis vectors, linear transformations, and eigenvalues and eigenvectors. These concepts are used to describe the properties of quantum systems, such as their position, momentum, and energy.

In addition, understanding matrix operations and matrix representations is important for understanding how quantum operators act on quantum states. This is essential for solving quantum mechanical equations and predicting the behavior of quantum systems.

Images and pre-images, also known as range and null space, are important concepts in linear algebra that are relevant to quantum mechanics. These concepts help us understand the relationships between different quantum states and how they are affected by quantum operations.

Overall, a solid understanding of linear algebra is necessary for grasping the fundamental principles of quantum mechanics. It is important to review and fully understand the topics in linear algebra that are relevant to quantum mechanics, such as vector spaces, basis vectors, linear transformations, eigenvalues and eigenvectors, and matrix operations. This will greatly aid in your understanding of quantum mechanics and its applications.