Hermitian kinetic-energy

Matrix elements of one-body hermitian operators (such as kinetic energy, nuclear attraction, the Fock operator, etc.) have the form... 

The kinetic energy operator corresponding to G z) is hermitian above. Loss of hermiticity may lead to spurious eigenvalues appearing in the calculations[42]. To avoid this, a proper basis should be chosen. For instance, to treat the term involving G z) in the RCL Hamiltonian, a Fourier D R basis was uscd[38]. 

The j(qA ) form a set of independent coordinates if the u lk) are such a set since they are related by the unitary transformation defined by (2.11). We note, however, as a new feature, that the <(qA ) are complex. Both the kinetic energy T [from (2.14)] and the potential energy O [from (2.18)] are hermitian and therefore the Lagrangian L derived from these forms is hermitian as required. In tensor notation we obtain... 

Here, a few comments are in order. The matrix of derivative couplings F is antihermitian. The matrix of scalar couplings G is composed of an hermitian as well as an antihermitian part. Of course, the dressed kinetic energy operator —(1/2M)(V - - F) in our basic Eq. (10) is hermitian, as is also the case for the nonadiabatic couplings A in Eq. (9a). The latter follows immediately from the relation (lie). The notation (V F) is self evident from Eq. (lid). Since F is a vector matrix, it can be written as F = (Fi, F2,..., Fjv ), where the matrices Fq, are simply defined by their... 

We haven t proven this. Briefly, the potential energy term must be Hermitian because it is a multiplicative operator, and the order of multiplication in the integrand does not affect the integral. For the kinetic energy, we show that the derivative operator is Hermitian as follows. Integration by parts yields... 

The kinetic energy matrices obey the relation TI = (ll y, so, despite appearances, the Dirac matrix in the scalar basis given above is Hermitian. 

The UESC kinetic energy matrix is not Hermitian fa- the X derived for all eigenvalues (Kutzelnigg and Liu 2005). 

For a Particle in a 1-dimensional Box, determine whether the following operators are Hermitian a) position b) momentum and c) kinetic energy. 

While the first is derived from classical momentum (p) considerations via the transformation p — V, the classical expression is retained in the case of the potential energy that only depends on the local coordinate. % is, thus, given in the end by the space functions and the second space derivatives (kinetic energy oc (momentum) ). It can be shown that is a Hermitian operator, i.e. (a H b) = (b t a). The star denotes the complex conjugate. Such Hermitian operators have, as they must, real eigenvalues Because (a t a) = e(a a) and (alWla) = e (a a = (a a), it follows that e = . ... 

Hermitian operators are very important in quantum mechanics because their eigenvalues are real. As a result, hermitian operators are used to represent observables since an observation must result in a real number. Examples of hermitian operators include position, momentum, and kinetic and potential energy. An operator is hermitian if it satisfies the following relation ... 

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