This chapter introduces Heisenberg’s trajectory from old quantum mechanics towards his quantum matrix mechanics. It shows his partnership with Born and for the study of the helium atom and molecules under the old quantum theory and his work on the core model to account for the anomalous Zeeman effect in collaboration with Landé. The collaboration between Heisenberg Kramers to present the treatment between matter and radiation under the scope of the old quantum mechanics is also displayed. Then, afterwards, it is shown the birth of the matrix mechanics by its three initial papers in 1925: the Heisenberg’s paper where he replaced the motion of the electron into its transition; the Born and Jordan’s paper where they placed Heisenberg equation into matrices and found the corresponding equation of motion and the commutator between linear momentum and position of the electron; the Heisenberg, Jordan and Born’s paper where they obtained the diagonal elements of the Hamiltonian matrix (the energy of a quantum state) and its application to one-electron harmonic oscillator. Dirac’s contribution to matrix mechanics is also introduced followed the application in the quantum angular momentum (184 words).
This chapter introduces Heisenberg’s trajectory from old quantum mechanics towards his quantum matrix mechanics. It shows his partnership with Born and for the study of the helium atom and molecules under the old quantum theory and his work on the core model to account for the anomalous Zeeman effect in collaboration with Landé. The collaboration between Heisenberg Kramers to present the treatment between matter and radiation under the scope of the old quantum mechanics is also displayed. Then, afterwards, it is shown the birth of the matrix mechanics by its three initial papers in 1925: the Heisenberg’s paper where he replaced the motion of the electron into its transition; the Born and Jordan’s paper where they placed Heisenberg equation into matrices and found the corresponding equation of motion and the commutator between linear momentum and position of the electron; the Heisenberg, Jordan and Born’s paper where they obtained the diagonal elements of the Hamiltonian matrix (the energy of a quantum state) and its application to one-electron harmonic oscillator. Dirac’s contribution to matrix mechanics is also introduced followed the application in the quantum angular momentum (184 words).
Quantum mechanics can be formulated either in terms of matrix mechanics or in terms of wave mechanics. This chapter centers on a discussion of the historical origins, mathematical structure, and extension to non-Hermitian quantum theory of matrix mechanics. The development of atomic discrete spectra and the old quantum theory showed that classical mechanics could not account for the discreteness of atomic energy levels and spectral lines, which motivated Heisenberg, Born, Jordan, and others to establish matrix mechanics centered on noncommuting physical quantities. Combining the Bohr atomic model, angular momentum quantization, and radiative transitions, one can see how matrix mechanics describes the energy level structure and quantum transitions of atomic systems, and how the Heisenberg equation of motion gives the fundamental form of quantum dynamics. The example of the harmonic oscillator further shows that the discreteness of the energy spectrum can be obtained directly from the matrix method, thereby establishing a clear connection among transition frequencies, matrix elements, and radiation-induced transitions. The matrix representation corresponds to the operator theory in Hilbert space, where state vectors, inner products, complete bases, and spectral decomposition constitute the fundamental mathematical framework of quantum mechanics. For closed quantum systems, physical observables are generally required to be represented by Hermitian or self-adjoint operators, so as to ensure that their eigenvalues possess the physically acceptable property of being real numbers. The Hermiticity of an operator is also closely related to its domain and boundary conditions, which provides a mathematical possibility for the emergence of non-Hermitian Hamiltonians. Based on these discussions, this chapter further explains how non-Hermitian dynamics is connected with open systems, dissipative processes, and special boundary conditions, thereby demonstrating the natural connection between traditional matrix mechanics and modern non-Hermitian quantum theory.
Chapter 3 introduces the student to the matrix formulation of quantum mechanics, pioneered by Heisenberg in collaboration with Born and Jordan. It provides the basic formal elements of the Heisenberg description in terms of operators and their matrix representation, derives the Heisenberg inequality, and applies the formalism to the harmonic oscillator as a central and particularly simple quantum problem. It introduces the state vectors and their Hilbert space, and concludes with a brief discussion hinting at the physical significance of the replacement of classical continuous dynamical variables with operators as a manifestation of the coupling of the particle to the radiation field.
The quantum theoretical description of microscopic systems discussed so far 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. Because the trajectory of an electron is unobservable and moreover line spectra implied discrete allowed states, Heisenberg discarded the idea of electron orbits and formulated a theory for line spectra of excited systems based on observables. When Heisenberg arranged the allowed set of transitions in a matrix form, to his surprise, each element of the matrix was found to be related to the probability of a particular transition. This gave the birth to matrix mechanics. Then Heisenberg, Born and Jordan worked out the rules of these matrices.
In this chapter shows some applications of matrix and wave mechnanics. The equivalence between matrix mechanics and wave mechanics is explored in this chapter. Another approach for this equivalence is indicated as well. The Heisenberg equation of motion is applied to the quantum virial theorem and quantum theory of atoms in molecules (QTAIM) in next section. It is introduced afterwards that the Heisenberg’s matrix mechanics is applied to the helium atom leads to the concept of quantum resonance and chemical bond. Next, the use of the matrix mechanics and wave mechanics to obtain the uncertainty principle is discussed and its derivation is also given. Subsequently, Dirac’s quantum mechanics – an amalgamation of matrix mechanics and wave mechanics – is introduced. Finally, the importance of the commutation of the quantum operators to the quantum mechanics is discussed followed by some basic quantum principles from the Copenhagen interpretation. A couple of examples and exercises is also available in this chapter.(156 words).
As mentioned in Chapter 1 , the Heisenberg matrix mechanics provides one of the three main avenues to quantum mechanics. This approach to quantum mechanics was disclosed in three papers authored by Heisenberg (1925) , Born and Jordan (1925) , and Born et al. (1926) . An iconic result from the Heisenberg–Born–Jordan contribution was the commutation rule ( 14.1 ) p q − q p = h 2 π i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315216720/70121b16-f2c6-4331-b1ba-eecb806b93da/content/eq595.tif"/>
The report by Born and Heisenberg on ‘quantum mechanics’ may seem surprisingly difficult to the modern reader. This is partly because Born and Heisenberg are describing various stages of development of the theory that are quite different from today's quantum mechanics. Among these, it should be noted in particular that the theory developed by Heisenberg, Born and Jordan in the years 1925–6 and known today as matrix mechanics (Heisenberg 1925b [1], Born and Jordan 1925 [2], Born, Heisenberg and Jordan 1926 [4]) a differs from standard quantum mechanics in several important respects. At the same time, the interpretation of the theory (the topic of Section II of the report) also appears to have undergone important modifications, in particular regarding the notion of the state of a system. Initially, Born and Heisenberg insist on the notion that a system is always in a stationary state (performing quantum jumps between different stationary states). Then the notion of the wave function is introduced and related to probabilities for the stationary states. At a later stage, probabilistic notions (in particular, what one now calls transition probabilities) are extended to arbitrary observables, but it remains somewhat unclear whether the wave function itself should be regarded as a fundamental entity or merely as an effective one. This may reflect the different routes followed by Born and by Heisenberg in the development of their ideas. The common position presented by Born and Heisenberg emphasises the probabilistic aspect of the theory as fundamental, and the conclusion of the report expresses strong confidence in the resulting picture.