The interpretation of quantum mechanics due to Lande' is applied to the connection between wave mechanics and matrix mechanics. The connection between the differential eigenvalue equation and the matrix eigenvalue equation for an operator is elucidated. In particular, we show that the elements of a matrix vector state are probability amplitudes with a structure rather than being mere constants. We obtain the most general expressions for the probability amplitudes for the description of spin-1/2 measurements. As a result, we derive spin-1/2 operators and vectors from first principles. The procedure used is analogous to that by which orbital angular momentum wavefunctions and operators are transformed to matrix mechanics vectors and matrices. The most generalized forms of the spin operators and their eigenvectors for spin-1/2 are derived and shown to reduce to the Pauli spin matrices and vectors in an appropriate limit.
The interpretation of quantum mechanics due to Lande' is applied to the connection between wave mechanics and matrix mechanics. The connection between the differential eigenvalue equation and the matrix eigenvalue equation for an operator is elucidated. In particular, we show that the elements of a matrix vector state are probability amplitudes with a structure rather than being mere constants. We obtain the most general expressions for the probability amplitudes for the description of spin-1/2 measurements. As a result, we derive spin-1/2 operators and vectors from first principles. The procedure used is analogous to that by which orbital angular momentum wavefunctions and operators are transformed to matrix mechanics vectors and matrices. The most generalized forms of the spin operators and their eigenvectors for spin-1/2 are derived and shown to reduce to the Pauli spin matrices and vectors in an appropriate limit.
Recently, we have shown how the interpretation of quantum mechanics due to Lande' can be used to derive from first principles generalized formulas for the operators and some eigenvectors for spin 1/2 Though we gave the operators for all the components of the spin, we did not give the eigenvectors of the operators for the x and y components of the spin. We now give these vectors. In addition, we present a new and simple method of deriving the operators for the x and y components of the spin as well as their vectors from those for the z component. We give a general proof that the operator for the square of the spin is the unit matrix multiplied by the value of the square of the spin.
In a recent paper, we introduced a new way of treating systems of compounded angular momentum. We obtained the probability amplitudes for measurements on the systems and used these to derive the matrix treatment of compounded spin. However, the matrix forms are in 3- and 4- dimensional space and are therefore entirely different from the standard forms. This raises the question of the connection between these forms and the standard forms. In this paper, we answer this question. We not only derive the standard matrix treatment of spin addition - we discover a more generalized form of the theory. We apply the new generalized theory to the singlet and triplet states arising from the addition of the spins of two systems of spin 1/2 each. We obtain new generalized forms of the vectors and operators for these cases, and show that they reduce to the standard forms in the appropriate limit.
This chapter deals with the matrix and vector formulation of quantum mechanics. Topics covered under the chapter are: matrices and their basic algebra, bra and ket notations, vector space and the basis vectors, inner and outer products of vectors, the orthogonalization procedure for vectors, the Schwarz inequality, linear transformation of a vector, eigenvalues and eigenvectors of a matrix, numerical methods to find eigenvalue and eigenvector, different types of matrices used in quantum mechanics, diagonalization of a matrix, the Cayley-Hamilton theorem, change of basis in a vector space and the unitary transformations, infinite-dimensional and Hilbert spaces, representation of quantum states and operators in Hilbert space, the Schrödinger equation in matrix form, assumptions of quantum mechanics in matrix formulations, the general uncertainty principle, the one dimensional harmonic oscillator, solved examples and unsolved exercises. The objectives of this chapter are to introduce the important aspects of vector and matrix algebra that have been vital in the development and the description of modern quantum mechanics and in exhibiting that the Heisenberg and Schrödinger approaches to quantum mechanics are equivalent.
The approach to quantum mechanics which we have used to derive the matrix treatment of spin from first principles is now employed to treat systems of compounded angular momentum. A general treatment is first given, which is then applied to the concrete cases of a spin-0 and a spin-1 system obtained by adding the spins of two spin-1/2 systems. Thus the probability amplitudes for measurements on the systems are derived, as well as the matrix vectors and operators corresponding to the systems. The matrix operators and states obtained are different from the standard forms and are much more generalized. The new results are applied to the case of joint measurements on the subsystems of such a system; this is a problem that has been made very topical by the high level of interest in the foundations of quantum mechanics. As a consequence of the insights arising from this treatment, we show that the Clebsch-Gordan coefficients are amenable to generalization, and we give the generalized forms for these cases.
The spin operator s = (ħ/2) σ in the Pauli equation fulfills the commutation relation of the angular momentum and leads to half-integer eigenvalues of the eigenfunctions for s. If one tries to express s by canonically conjugated operators Φ and π = (ħ/i) ∂/∂Φ the formal angular momentum term s = Φ X π fails because it leads only to whole-integer eigenvalues. However, the modification of this term in the form s = 1/2 {π + Φ(Φ π) + Φ X π} leads to the required result. The eigenfunction system belonging to this differential operator s(Φ π) consists of (2s + 1) spin eigenfunctions ξm (Φ) which are given explicitly. They form a basis for the wave functions of a particle of spin s. Applying this formalism to particles with s = 1/2, agreement is reached with Pauli’s spin theory. The function s(Φ π) follows from the theory of rotating rigid bodies. The continuous spinvariable Φ = ((Φx , Φy, Φz) can be interpreted classically as a “turning vector” which defines the orientation in space of a rigid body. Φ is the positioning coordinate of the rigid body or the spin coordinate of the particle in analogy to the cartesian coordinate x. The spin s is a vector fixed to the body.
In this article, we discard the bra-ket notation and its correlative definitions, given by Paul Dirac. The quantum states are only described by the wave functions. The fundamental concepts and definitions in quantum mechanics is simplified. The operator, wave functions and square matrix are represented in the same expression which directly corresponds to the system of equations without additional introduction of the matrix representation of operator. It can make us to convert the operator relations into the matrix relations. According to the relations between the matrices, the matrix elements will be determined. Furthermore, the first order differential equations will be given to find the solution of equations. As a result, we unified the descriptions of the matrix mechanics and the wave mechanics.