What are the important lessons in linear algebra for Quantum?

  • Thread starter Ayham
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  • #1
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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?
 
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  • #2
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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.
 
  • #3
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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.
 
  • #4
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Ok, thanks guys
 

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