Yes/no question about non-commuting Hermitian operators

In summary, non-commuting Hermitian operators are mathematical operators used in quantum mechanics to represent physical observables. They are important because they describe the uncertainty and probabilistic nature of quantum systems. They are related to Heisenberg's uncertainty principle and cannot be simultaneously diagonalized. In quantum computing, they are used to represent quantum gates and implement quantum algorithms for efficient problem solving.
  • #1
nomadreid
Gold Member
1,729
229
Is the following a theorem? yes or no
If A and B are non-commuting Hermitian operators (or matrices), there does not exist Hermitian operators C and D such that AB-BA = CD.
(Or, as special case, ...there does not exist a Hermitian operator C s.t. C= AB-BA)
Thanks
 
Physics news on Phys.org
  • #2
If A and B are Hermitian, then AB-BA is anti-Hermitian, (##M^\dagger = - M##). ##i(AB-BA)## is therefore Hermitian.
 
  • #3
fzero: Thank you very much.
 

FAQ: Yes/no question about non-commuting Hermitian operators

What are non-commuting Hermitian operators?

Non-commuting Hermitian operators are mathematical operators that do not commute, meaning that the order in which they are applied affects the outcome. They are commonly used in quantum mechanics to represent physical observables, such as position and momentum.

Why are non-commuting Hermitian operators important?

Non-commuting Hermitian operators play a crucial role in quantum mechanics, as they represent physical quantities that cannot be simultaneously measured with complete accuracy. They also help to describe the fundamental uncertainty and probabilistic nature of quantum systems.

How are non-commuting Hermitian operators related to Heisenberg's uncertainty principle?

Heisenberg's uncertainty principle states that certain pairs of physical quantities, such as position and momentum, cannot be known with absolute certainty at the same time. This is mathematically represented by the non-commutativity of the corresponding Hermitian operators.

Can non-commuting Hermitian operators be simultaneously diagonalized?

No, non-commuting Hermitian operators cannot be simultaneously diagonalized, as their eigenvectors do not form a complete basis. This means that it is not possible to find a set of eigenstates that simultaneously diagonalize both operators.

How are non-commuting Hermitian operators used in quantum computing?

In quantum computing, non-commuting Hermitian operators are used to represent quantum gates, which are operations that manipulate the state of the qubits. By applying these gates in a specific order, quantum algorithms can be implemented to solve certain problems more efficiently than classical computers.

Similar threads

I Spectral theorem for Hermitian matrices-- special cases
Replies
2
Views
2K
B How does matrix non-commutivity relate to eigenvectors?
Replies
26
Views
4K
If A, B Hermitian, then <v|AB|v>=<v|BA|v>*. Why?
Replies
2
Views
2K
Hermitian Matrix and Commutation relations
Replies
2
Views
2K
I Is the product of two hermitian matrices always hermitian?
3
Replies
93
Views
11K
I A strange definition for Hermitian operator
Replies
3
Views
978
Hermiticity of AB where A and B are Hermitian operator?
Replies
11
Views
3K
I Composite quantum systems: Kronecker and Hadamard/Schur products
Replies
0
Views
539
Understanding Hermitian Operators: Exploring Their Properties and Applications
Replies
1
Views
1K
Understanding Commutativity and Eigenvalues in the Product of Hermitian Matrices
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top