Self Adjoint and Anti-Self Adjoint questoin

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In summary, Self-adjoint operators represent physical observables in quantum mechanics, while anti-self-adjoint operators represent the rate of change of those observables. Self-adjoint operators are equivalent to Hermitian matrices, while anti-self-adjoint operators are equivalent to anti-Hermitian matrices. A self-adjoint operator cannot be anti-self-adjoint, and examples of these operators include the position, momentum, angular momentum, and Hamiltonian operators in quantum mechanics. These operators are important in quantum mechanics as they provide information about the possible outcomes and states of a quantum system, and satisfy important mathematical properties for the development and application of quantum mechanics.
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Alupsaiu
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I'm having trouble showing that any normal linear transformation T is the sum of a self-adjoint transformation T1 and anti-self adjoint linear transformation T2, (so T=T1+T2) so that T1 and T2 commute. Anti-self adjoint being <Ta,b>=-<a,Tb>.
Specifically I'm not sure how to use the information of T being normal.
Any help is appreciated, thank you.
 
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  • #2
Hi Alupsaiu! :smile:

What about this standard trick:

[tex]T=\frac{T+T^*}{2}+\frac{T-T^*}{2}[/tex]
 
  • #3
...Moments like these I feel so small...haha thanks a bunch
 

FAQ: Self Adjoint and Anti-Self Adjoint questoin

1. What is the difference between self-adjoint and anti-self-adjoint operators?

Self-adjoint operators represent physical observables in quantum mechanics, while anti-self-adjoint operators represent the rate of change of those observables. Self-adjoint operators have real eigenvalues, while anti-self-adjoint operators have purely imaginary eigenvalues.

2. How do self-adjoint and anti-self-adjoint operators relate to Hermitian and anti-Hermitian matrices?

Self-adjoint operators are equivalent to Hermitian matrices, while anti-self-adjoint operators are equivalent to anti-Hermitian matrices. This means that the complex conjugate of a self-adjoint operator is equal to the operator itself, while the complex conjugate of an anti-self-adjoint operator is equal to the negative of the operator.

3. Can a self-adjoint operator be anti-self-adjoint?

No, a self-adjoint operator cannot be anti-self-adjoint, as this would mean that its eigenvalues are both real and purely imaginary, which is not possible. A self-adjoint operator can only be self-adjoint.

4. What are some examples of self-adjoint and anti-self-adjoint operators?

Examples of self-adjoint operators include the position operator and the momentum operator in quantum mechanics. Examples of anti-self-adjoint operators include the angular momentum operator and the Hamiltonian operator in quantum mechanics.

5. Why are self-adjoint and anti-self-adjoint operators important in quantum mechanics?

Self-adjoint and anti-self-adjoint operators are important in quantum mechanics because they represent physical observables and their rates of change, respectively. The eigenvalues and eigenvectors of these operators provide information about the possible outcomes and states of a quantum system. These operators also satisfy important mathematical properties that allow for the development of quantum mechanics and its applications.

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