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American Journal of Physics

Matrix Methods in Quantum Mechanics

Eugen Merzbacher

Published Sep 1, 1968• 30 citations
Publisher

Abstract

Some standard results of matrix theory are derived in a manner designed to appeal to physicists and are illustrated by examples from quantum mechanics. Sylvester's formulas, expressing a function of any n-dimensional square matrix as a linear combination of the first n−1 powers of A, are applied to the evaluation of the rotation matrices D(i)(R) for low values of j. The neutral K meson is used as an example of the usefulness of matrix techniques for systems with two basis states.

DOI
10.1119/1.1975154

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Some standard results of matrix theory are derived in a manner designed to appeal to physicists and are illustrated by examples from quantum mechanics. Sylvester's formulas, expressing a function of any n-dimensional square matrix as a linear combination of the first n−1 powers of A, are applied to the evaluation of the rotation matrices D(i)(R) for low values of j. The neutral K meson is used as an example of the usefulness of matrix techniques for systems with two basis states.
In this work we present a pedagogically motivated analysis of spin-correlation calculations in a quantum system composed of two spin-1/21/21/2 particles. Rather than aiming at new physical results, our purpose is to clarify and bring attention to different strategies for evaluating expectation values of the form ⟨ψ∣Su^(1)Sv^(2)∣ψ⟩\langle ψ| S^{(1)}_{\hat{\boldsymbol{u}}} S^{(2)}_{\hat{\boldsymbol{v}}} | ψ\rangle⟨ψ∣Su^(1)​Sv^(2)​∣ψ⟩, which play an important role in discussions of entanglement and Bell-type correlations. We compare three complementary approaches. The first follows a direct algebraic evaluation in the product basis, closely related to standard textbook methods. The second uses a matrix representation of bipartite states, in which the tensor-product structure is expressed in terms of 2×22\times22×2 complex matrices. This representation keeps the calculation close to the familiar Pauli-matrix algebra and makes the independent action of operators on each subsystem more transparent. The third explores a symmetry-based argument, highlighting both its usefulness and its limitations when applied beyond the singlet state. We show explicitly that the singlet state is rotationally invariant, which explains why the symmetry argument successfully reproduces its correlation function, while a naive extension fails for triplet states. The discussion illustrates how entanglement, tensor-product structure, and rotational symmetry interplay in spin correlations.
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A general expression for spherical harmonics as an explicit function of the angle coordinates is derived using standard methods of group-representation theory. This expression is more useful to physicists than a corresponding expression available in the mathematical literature for two reasons: (1) Its normalization is the one most commonly used by physicists, rather than one of several other normalization conventions used mainly by mathematicians. (2) Its form is especially convenient for use in the operator equations which occur in quantum mechanics. Furthermore, the derivation serves the pedagogical purpose of providing a simple illustration of the methods of group-representation theory. A set of relations useful in solving the Dirac equation are listed, both because their verification provides an example of the utility of the expression derived, and because the form in which these relations are stated in the literature is apt to result in confusion stemming from the various different normalization conventions in use.
The quantum theoretical description of microscopic systems is based on differential operators and the solution of the underlying eigenvalue equations. This approach is called Schrödinger approach or wave mechanics. Heisenberg developed another equivalent approach called matrix mechanics. In matrix mechanics all observables are represented by matrices. All possible eigenvalues of a matrix representing an observable are the possible outcomes of experimental measurement on that observable. This chapter gives the preliminary concepts relative to the use of the matrix method. It presents the various elements of matrix mechanics and discusses the matrix representation of operators of certain systems. The chapter introduces Dirac's notation of state vectors, Hilbert space, projection operator, displacement operators and the generators of translations. The theory of quaternions has found applications in classical mechanics, quantum mechanics and relativity. It has been used in aerospace applications, flight simulators, computer graphics, animation, visualization, fractals and virtual reality.
We apply the framework of block-encodings, introduced by Low and Chuang (under the name standard-form), to the study of quantum machine learning algorithms and derive general results that are applicable to a variety of input models, including sparse matrix oracles and matrices stored in a data structure. We develop several tools within the block-encoding framework, such as singular value estimation of a block-encoded matrix, and quantum linear system solvers using block-encodings. The presented results give new techniques for Hamiltonian simulation of non-sparse matrices, which could be relevant for certain quantum chemistry applications, and which in turn imply an exponential improvement in the dependence on precision in quantum linear systems solvers for non-sparse matrices. In addition, we develop a technique of variable-time amplitude estimation, based on Ambainis' variable-time amplitude amplification technique, which we are also able to apply within the framework. As applications, we design the following algorithms: (1) a quantum algorithm for the quantum weighted least squares problem, exhibiting a 6-th power improvement in the dependence on the condition number and an exponential improvement in the dependence on the precision over the previous best algorithm of Kerenidis and Prakash; (2) the first quantum algorithm for the quantum generalized least squares problem; and (3) quantum algorithms for estimating electrical-network quantities, including effective resistance and dissipated power, improving upon previous work.