Abstract The Eigen problem is a problem that exists when there is an operator that is affected by the function, then that function does not change except being multipled by the constant in the form of an eigenvalue. In general, the eigenvalue problem uses the Hermitian operators by using observable variable in the form of mathematical operations. Hermitian operator is an operator that happens when the Hermit conjugation is performed, then the operator will return to its original condition. This research aims to determine the base matrix of Hermitian operators with order n ≤ 4 (size 2 × 2, 3 × 3 and 4 × 4). Entry element of the hermitian matrix operator uses complex numbers of which each order consists of few Hermitian operators. The method used is an analytical method by inserting problems into the eigen equation then normalized to obtain a diagonal matrix, so as to obtain a complete solution of the eigen problem on form of a matrix (including eigenvalues, normalized eigenvectors, and base matrix). Base matrix that has been obtained will be a diagonal matrix with diagonal elements of the eigenvalue. The results of this research indicate that the base matrix obtained in each order forms a square matrix that has entry elements in the form of complex numbers.
Abstract The Eigen problem is a problem that exists when there is an operator that is affected by the function, then that function does not change except being multipled by the constant in the form of an eigenvalue. In general, the eigenvalue problem uses the Hermitian operators by using observable variable in the form of mathematical operations. Hermitian operator is an operator that happens when the Hermit conjugation is performed, then the operator will return to its original condition. This research aims to determine the base matrix of Hermitian operators with order n ≤ 4 (size 2 × 2, 3 × 3 and 4 × 4). Entry element of the hermitian matrix operator uses complex numbers of which each order consists of few Hermitian operators. The method used is an analytical method by inserting problems into the eigen equation then normalized to obtain a diagonal matrix, so as to obtain a complete solution of the eigen problem on form of a matrix (including eigenvalues, normalized eigenvectors, and base matrix). Base matrix that has been obtained will be a diagonal matrix with diagonal elements of the eigenvalue. The results of this research indicate that the base matrix obtained in each order forms a square matrix that has entry elements in the form of complex numbers.
Classical physics began to be superseded by quantum physics because it could not explain microscopic phenomena. Quantum physics employs mathematical approaches such as wave mechanics and matrix mechanics, one of which is the eigenvalue problem involving matrix operators. There are many types of matrix operators; however, not all are suitable for measuring physical quantities in quantum physics. Operators in quantum physics must be Hermitian to ensure that their eigenvalues are real. Eigenvalue problems for matrix operators can be solved either analytically or numerically. Therefore, the objective of this research is to analyze symmetric and antisymmetric matrix operators in the measurement of physical quantities through eigenvalue problems. The research method utilized is a theoretical computational approach. The results demonstrate that symmetric matrix operators are Hermitian and yield real eigenvalues. In contrast, antisymmetric matrix operators are non Hermitian, yielding imaginary or complex eigenvalues. The results of solving eigen problems using Python obtained relative errors between 0 and orders., while the absolute error remained below 10%, thus confirming the validity of the results. Based on the findings, symmetric matrix operators satisfy the requirements for use in physical problems through eigenvalue analysis. The eigenvalues of antisymmetric matrices cannot represent the measurement of physical quantities directly and serve only as rotation operators. However, antisymmetric matrices can be utilized in quantum physics for physical measurements if they are transformed into Hermitian matrices. Furthermore, solving eigenvalue problems with Python is effective due to its high accuracy and small error margins.
Ψ=Ψ Iˆ In view of the expansion (29.28), which holds for an arbitrary state function in the form Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ Ψ∑ ∑ we may define the unit or identity operator as i iI ΨΨ=∑ =1 ˆ (33.4) Conversely, for every linear operator we can form a square matrix, in terms of a complete set of functions by multiplying the left-hand side of Eq.(33.3) by and integrating, which gives fˆ ,iΨ ∗Ψi ΨΨ=ΨΨ=ΦΨ=ΦΨ∫ ∗ Iffd iiii ˆˆˆV ΨΨ=ΨΨΨΨ= n ˆˆ Since ii b=ΦΨ this result is identical with that given by Eq.(33.1), provided we set Vˆˆ dfff kikiik ΨΨ=ΨΨ= ∫ ∗ (33.5) which are called the matrix elements of the linear operator in terms of the particular complete set of functions It is a straightforward matter to show that the rules of matrix algebra for addition, multiplication, associativity and distributivity (see Appendix II) apply to the matrix elements (33.5), following from their definition. Two properties of fˆ .iΨ matrix elements that are needed in quantum mechanics can be readily derived from Eq.(33.5). Using Eq.(29.19) we obtain ∗∗ =ΨΨ=ΨΨ= kiikkiik ffff ˆˆ (33.6) which means that if is a Hermitian operator, then its matrix representation in a complete set of functions is Hermitian. Hence, the elements in the principal diagonal of a Hermitian matrix must be real. When all the elements are real, the matrix must be symmetric. From Eq.(29.20) we have fˆ ikiikiik f δλλ =ΨΨ=ΨΨ ˆ (33.7) that is: if the functions are the eigenfunctions of the Hermitian operator the matrix representation is diagonal, and the matrix elements are the eigenvalues of the operator. A quantum mechanical problem can be stated in a differential equation form or in a matrix form. In both cases the solution consists of finding the eigenvalues and the eigenfunctions of an operator associated with a physical observable. In view of a matrix representation in any complete set of functions, the eigenvalue equation (29.20) must be formulated as iΨ ,fˆ (33.8) Ψ=Ψ λfˆ where the arbitrary eigenfunction Ψ can be expanded, as given by Eq.(29.4), in terms of a complete set of functions If we multiply Eq.(33.8) by and integrate, we obtain .iΨ ∗Ψi ˆ ˆ ˆV V ori i if d d f Iλ λ∗ ∗Ψ Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ∫ ∫ i or also, in view of Eq.(33.4) k kki f λ ˆ This can be rewritten as = n 1 λ which reads (33.9) ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ a a a a a a fff fff fff …… … ……… … … λ Equation (33.6) is called the matrix eigenvalue equation and consists of n linear homogenous equations for the given by ,ka ( ) 0... 1212111 =+++− nnafafaf λ ( ) 0... 2222121 =++−+ nnafafaf λ (33.10) ……… ( ) 0...2211 =−+++ nnnnn afafaf λ This system of simultaneous equations has a nontrivial solution for the if and only if the characteristic determinant vanishes ( ) ( ) ( ) ( ) 01ˆˆdet = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ − − − =− λ λ λ λ nnnn fff fff fff f … ……… … … (33.11) If expanded, the determinant gives a polynominal of nth degree called the characteristic polynominal of fˆ with n roots which are the desired eigenvalues. For each eigenvalue Eqs.(33.10) can be solved for the giving the corresponding eigenfunction that is a column matrix.
Eigenproblem arises in a large number of disciplines of sciences and engineering. Combining quantum algorithms to solve linear differential equations, quantum singular value estimation and quantum phase estimation, we propose quantum algorithms to compute the eigenvalues of diagonalizable matrices of two types: matrices that only have real eigenvalues and normal matrices. The quantum algorithms return a quantum state such that the first register stores eigenvalues and the second register stores the corresponding eigenvectors. For normal matrices, the complexity to obtain this quantum state is dominated by the complexity to perform quantum singular value estimation. For the other case, the complexity is determined by the solving of a linear system of differential equations. Under certain conditions (e.g. sparse), the complexities are polylog on the size n of the given matrix. If we perform measurements, then we can obtain all the eigenvalues classically. If M is s sparse, then the complexity is O(sn∥M∥maxκ2/ϵ2), where κ is the condition number of the matrix of eigenvectors.
The bases used in quantum mechanics, in general, are orthonormal. The relations in orthonormal basis vectors are simpler than those in nonorthogonal basis vectors. Also, the operators that represent different physical quantities are Hermitian and the eigenvectors of these operators belonging to different eigenvalues are orthonormal to each other. But, in many cases, such as sets of screened hydrogenic wavefunctions, nonorthogonal sets of basis vectors occur. It is, therefore, necessary to consider the representation of vectors and operators in nonorthogonal bases. Making use of the set of vectors reciprocal to the given set of nonorthogonal base vectors, the expression for the norms of vectors and for the matrix elements has been found. Also, adjoint and Hermitian matrices in nonorthogonal bases have been worked out using a set of reciprocal basis vectors.
This chapter illustrates how one can obtain a matrix representation of a state vector, an operator, and an eigenvalue equation in a basis of orthonormal states. It helps to learn that matrix representations are very useful in calculations of the properties of operators, because algebra of matrices is very simple and completely developed. Readers familiar with the algebra of matrices agree that mathematical operations on matrices such as multiplication by a scalar, addition, subtraction, multiplication and diagonalization are simple and easy to perform. In fact, the application of the algebra of matrices to quantum physics results from the direct relations between operators and matrices that any mathematical relation that holds between operators also holds between the corresponding matrices. The eigenvalues and eigenvectors of the matrix representing a given operator  in the basis of orthonormal states are the same as the eigenvalues and eigenvectors of Â.
We present a prescription for forming matrices with specified eigenvalues and known eigenvectors. With this method, we can form Hermitian, anti-Hermitian, symmetric and general matrices with arbitrary eigenvalues. In addition we propose an algorithm for diagonalizing such matrices. The functions required for the realization of this are probability amplitudes connecting observables with discrete eigenvalue spectra and can be obtained from spin theory. For the example case of 5×5 matrices, these functions are given.