Hermitian operators without considering them as Matrices

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SUMMARY

A Hermitian operator is defined as a linear transformation that is self-adjoint, meaning it satisfies the condition = for vectors u and v in an inner product space. In the context of Hermitian matrices, they are equal to their conjugate transpose, which is crucial for understanding their properties. The discussion emphasizes the relationship between Hermitian operators and functions, specifically how the expression [H.f] relates to [f].H when not considering matrix representations. This highlights the broader concept of self-adjointness in both real and complex valued square matrices.

PREREQUISITES
  • Understanding of Hermitian operators and their properties
  • Familiarity with inner product spaces
  • Knowledge of complex conjugates and their role in linear algebra
  • Basic concepts of linear transformations
NEXT STEPS
  • Study the properties of self-adjoint operators in functional analysis
  • Explore the implications of Hermitian matrices in quantum mechanics
  • Learn about the spectral theorem for Hermitian operators
  • Investigate the relationship between Hermitian operators and eigenvalues
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Mathematicians, physicists, and students studying linear algebra or quantum mechanics who seek a deeper understanding of Hermitian operators and their applications.

Master J
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A Hermitian matrix is a square matrix that is equal to it's conjugate transpose.
Now let's say I have a Hermitian operator and a function f:

[ H.f ]

The stuff in the square is the complex conjugate as the functions are in general complex. If I do not consider the matrix representation of the function and Hamiltonian, and just consider them functions, then how do I motivate the result that the above equals:

[ f ].H

I have seen this before but I am a bit confused.
Any enlightenment?
 
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The more general concept is "self- adjoint". A linear transformation from an inner product space to itself is "self-adjoint" if and only if <Au, v>= <u, Av> where u and v are vectors in the vector space and < , > is the inner product. A real valued square matrix is "self-adjoint" if and only if it is symmetric (it is equal to its transpose) and a complex valued square matrix is "self adjoint" if and only if it is equal to its conjugate transpose.
 

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