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CRC Press eBooks

Quantum Mechanics on a Hilbert Space

Martin Bojowald A

Published May 8, 2026• 0 citations
Publisher

Abstract

The Schrödinger equation is a linear partial differential equation for the wave function. Therefore, any linear combination of two solutions is also a solution. The possibility of forming linear combinations is a well-known property of vectors. Moreover, we normalize solutions for the statistical interpretation, and we have seen several examples of orthogonality relations between bound and other states. These concepts are reminiscent of the length of a vector and the angle between two vectors or the dot product. Scattering is described by a matrix, another concept from linear algebra.

DOI
10.1201/9781003588702-4

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•Mar 5, 2009•1 citation
The Schrödinger equation is a linear partial differential equation for the wave function. Therefore, any linear combination of two solutions is also a solution. The possibility of forming linear combinations is a well-known property of vectors. Moreover, we normalize solutions for the statistical interpretation, and we have seen several examples of orthogonality relations between bound and other states. These concepts are reminiscent of the length of a vector and the angle between two vectors or the dot product. Scattering is described by a matrix, another concept from linear algebra.
Quantum physics is the most successful description of the physical world we live in. Quantum theoretical predictions have been experimentally tested and confirmed up to the astonishing 14 digits of precision in terms of the anomalous magnetic moment of the electron. The bra-ket notation, composed of angle brackets and vertical bars, enhances the clarity of mathematical operations and improves the comprehension of quantum mechanical expressions. Quantum wave functions participate in interference phenomenona in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. When a quantum system is measured, it undergoes indeterministic transition that actualizes only one of the states from the set of all physically possible future states of the system. The Hilbert vector space structure of the space of quantum states follows from the linearity of the Schrodinger equation. The quantum state vector and the quantum wave function in a Hilbert space are abstract mathematical concepts that are basis independent.
Abstract Chapter 9 begins laying a foundation for quantum statistical mechanics with a discussion of relevant topics in quantum mechanics. The fundamental postulates of quantum mechanics are reviewed, beginning with the definition of a quantum state and the introduction of the concept of a state vector in a Hilbert space. The representation of observables as Hermitian operators on Hilbert space vectors is described; it is shown that the eigenvalues of Hermitian operators are real and, therefore, constitute possible outcomes of measurements made on these observables. The notions compatible and non-compatible observables and implication for the measurement problem are presented. The time-dependent Schrödinger equation is introduced as generating the time-evolution of the state vector, and it is shown how to cast the Schrödinger equation into a coordinate basis to yield the wave equation of quantum mechanics. Coordinate and momentum bases and their properties are desribed. Several analytically solvable quantum systems are presented as illustrative examples. The chapter concludes with a discussion of spin and the treatment identical particles
Classical mechanics is usually valid at the macro scale. The changes in the position and the momentum of the object over time are described by the Hamiltonian equation of motion. The wave function in quantum mechanics, if unobserved, evolves in a smooth and continuous way according to the Schrodinger equation, which is related to the Hamiltonian equation of motion. This equation describes a linear superposition of different states at time. A quantum mechanical description of a physical system is related to a probabilistic representation; it is described by a vector in Hilbert space. This description extends the two- or three-dimensional Euclidean space into spaces that have any finite or infinite number of dimensions. The state of a compound system is projected to the subspace that corresponds to the observed state and the vector representing the state is renormalized to the unit length. An observable describes a subspace for some dimensions with a special case of one dimension.
For quantum bound states, classical style probabilities P(p)=a(p)a(p) and P(x)=W(x)W(x) [where W(x)= the wavefunction = Sum over p a(p) exp(ipx)] exist. These are real and normalized to one, but unusually, they separate p and x completely. For example, P(p) is the probability to find a momentum p anywhere in the system - no information about x is known. This seems to suggest that the P(p/x) is not defined in a classical sense. (Similar arguments hold for P(x).) We have argued in a previous note (1), that due to a lack of time reversal balance, there is no clear classical notion of P(p/x) in quantum mechanics while in classical statistical mechanics, there is a time reversal balance of two-body collisions and so P(p/x) is well-defined. In particular, a momentum p collides with a pj to form another momentum pk. There is ambiguity if p is changing, but if at the same time a pk is changing (colliding) back into a p at x, then the ambiguity is removed. In quantum mechanics, the ambiguity is not removed at x, but one may discern one p state from another throughout the entire x space. This suggests orthonormal probabilities e.g. say exp(ip1 x) and exp(i p2 x). These exp(i p x) probabilities comprise the wavefunction which is linked to P(p/x) = a(p) exp(ipx)/W(x), but although Sum over p P(p/x)=1, P(p/x) is complex with the real and complex portions taking on positive and negative values. Nevertheless, it is W(x) which is the solution of the Schrodinger (and other) equations. It is known that solutions Wn(x) of the Schrodinger equation have discrete En (energy) values. It is possible to have [H, Operator]=0 and possibly to have H written in terms of this operator. In such a case, Wn(x) might be both an eigenfunction of H and the Operator. Alternatively, a linear combination of Operator eigenfunctions may be used. If En is discrete, one may expect the operator eigenfunctions to also be discrete, leading to quantization of the Operator eigenvalues. In some cases, one may have two operators which commute with H, e.g. LL and Lz (related to angular momentum). Lz eigenfunctions might not be linked to H so one might ask why Lz should be quantized. Another H, however, may distinguish between Lz eigenfunctions and so they also have discrete eigenvalues. In this note, we try to consider why one should have orthonormal eigenfunctions of these commuting operators which make up the wavefunction and try to interpret their physical meaning.
Research has been carried out to determine the physical meaning and derivation of the Schrodinger equation with the literature study research method. Based on the research results obtained through data processing in the form of information obtained from various literature sources, it can be concluded that the general solution of the Schrodinger equation is a wave function. In general, based on mathematical calculations by applying the technique of solving differential equations to the Schrodinger equation, it can be seen that this equation has many solutions, but not all of these solutions can be said to be functions that meet the wave function criteria. This wave function must be able to interpret the real physical criteria of a particle.