Hermitian operators without considering them as Matrices

In summary, a Hermitian matrix is a special type of square matrix that is equal to its conjugate transpose. This concept can also be applied to linear transformations, where a self-adjoint linear transformation satisfies a specific property involving inner products. The more general term for this concept is "self-adjoint".
  • #1
Master J
226
0
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?
 
Physics news on Phys.org
  • #2
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.
 

1. What is a Hermitian operator?

A Hermitian operator is a mathematical object that represents a linear transformation on a complex vector space. It is defined as an operator that is equal to its own conjugate transpose. In other words, the operator is equal to its adjoint, which is the complex conjugate of its transpose.

2. How are Hermitian operators different from other operators?

Unlike other operators, Hermitian operators have special properties that make them useful in quantum mechanics and other areas of physics. These properties include real eigenvalues and orthogonal eigenvectors, which allow for easier mathematical calculations and interpretations of physical systems.

3. Can a Hermitian operator be represented as a matrix?

Yes, a Hermitian operator can be represented as a matrix. However, it is important to note that the matrix representation of a Hermitian operator is not unique. This means that different matrices can represent the same Hermitian operator, depending on the chosen basis for the vector space.

4. What is the significance of Hermitian operators in quantum mechanics?

In quantum mechanics, Hermitian operators represent physical observables, such as position, momentum, and energy. The eigenvalues of these operators correspond to the possible outcomes of measurements on a quantum system, and the eigenvectors represent the states of the system.

5. How are Hermitian operators used in quantum computing?

In quantum computing, Hermitian operators are used to represent quantum gates, which are operations that manipulate the state of a quantum system. These gates are essential for performing calculations and algorithms in quantum computers, and the Hermitian properties of these operators ensure the preservation of unitarity.

Similar threads

  • Linear and Abstract Algebra
Replies
2
Views
494
  • Linear and Abstract Algebra
Replies
2
Views
1K
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
Replies
9
Views
1K
  • Linear and Abstract Algebra
Replies
7
Views
7K
  • Classical Physics
Replies
2
Views
877
  • Linear and Abstract Algebra
Replies
2
Views
1K
Replies
15
Views
1K
  • Advanced Physics Homework Help
Replies
4
Views
1K
Back
Top