Hamiltonian Operator

Wenhua Wang, ZhengLi Chen, W. Li

Abstract The Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermitian Hamiltonian which is similar to its adjoint. We first present some properties of the metric operators for pseudo-Hermitian Hamiltonians and obtain a sufficient and necessary condition for an invertible operator to be a metric operator for a given pseudo-Hermitian Hamiltonian. When the pseudo-Hermitian Hamiltonian has real spectra, we provide a new method such that any given metric operator can be transformed into the same positive-definite one and the new inner product with respect to the positive-definite metric operator is well defined. Finally, we illustrate the results obtained with an example.

Dmitri M Gitman, Igor Viktorovich Tyutin, Boris Leonidovich Voronov

In the article arXiv:0903.5277 [quant-ph], we have presented a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential V (x) = αx −2. In such a way, we have described all possible s.a. operators (s.a. Hamiltonians) associated with the formal differential expression ˇH = −d 2 x + αx−2 for the Calogero Hamiltonian. Here, we discuss a new aspect of the problem, the so-called oscillator representation for the Calogero Hamiltonians. As it is know, operators of the form ˆ N = â + â and  = ââ+ are called operators of oscillator type. Oscillator type operators obey several useful properties in case if the elementary operator â and â + are densely defined. It turns out that some s.a. Calogero Hamiltonians are oscillator type operators. We describe such Hamiltonians and find the corresponding mutually adjoint elementary operators. 1

A F Sugihartin, Bambang Supriadi, Subiki Subiki +3 more

Abstract In general some operators in quantum mechanics are not commutable, which means the measurement of two or more operators can’t be done simultaneously. Angular momentum operator and Hamiltonian are examples of operators that can only be measured using a mathematical approach in the form of a commutation relationship. Based on the fifth postulate of quantum mechanics about a constant of motion, if an operator commute with Hamiltonian operator then the operator is a constant of motion. This study was conducted to examine the commutation of the angular momentum operators to the Hamiltonian operator. The commutation of the angular momentum operators <?CDATA ${\widehat{L}}_{x}$?> L ^ x , <?CDATA ${\widehat{L}}_{y}$?> L ^ y , <?CDATA ${\widehat{L}}_{z}$?> L ^ z , <?CDATA ${\widehat{L}}_{+}$?> L ^ + , and <?CDATA ${\widehat{L}}_{-}$?> L ^ − to the Hamiltonian operator shows that the operators are commute because the values are zero. Otherwise, the results of the component quadratic angular momentum operators commutator <?CDATA ${\widehat{L}}_{x}{}^{2}$?> L ^ x 2 , <?CDATA ${\widehat{L}}_{y}{}^{2}$?> L ^ y 2 , <?CDATA ${\widehat{L}}_{z}{}^{2}$?> L ^ z 2 , <?CDATA ${\widehat{L}}^{2}$?> L ^ 2 , <?CDATA ${L}_{+}{}^{2}$?> L + 2 and <?CDATA ${L}_{-}{}^{2}$?> L