On the Foundations of Quantum Mechanics

1. Introduction

Although matrix mechanics and wave mechanics are believed to be equivalent formulations of quantum mechanics, they are not equal participants. They are based on widely separated initial assumptions, distinct experiments, and different mathematical models. Contributions to the foundations by wave mechanical models predominate, while matrix mechanics has not advanced significantly in its contributions to the foundations beyond initial assessments 100 years ago. The reasons given for the imbalance are its lack of visualizability and ease of calculation, properties not of matter but of the formulation. The relative importance of a theory should not depend upon how it was formulated. Rather than ignoring matrix mechanics during discussions of the foundations we will reformulate it to obtain insights that complement the contributions of wave mechanics.

2. Matrix Mechanics Reformulated

Classical mechanics is the study of the motion of objects and their causes; in other words, the equations of motion. Ideally the objective of quantum mechanics should be to find out the subatomic equations of motion. Wave mechanics is clearly an attempt at this, but with limited success. However, matrix mechanics makes no attempt at all for it describes the properties not of an atom but of hydrogen gas. The spectrum of hydrogen is obtained by analyzing the black body radiation that is emitted by a hot gas, the simultaneous emission of radiation by many atoms. The possibility of using a single hydrogen atom to interpret matrix mechanics was not conceived of by the originators.

There are difficulties involved in deriving the laws of mechanics for a hydrogen atom as noted by Nobel laureate Richard Feynman [1].

But suppose we look at the whole hydrogen atom as a “particle.” If we didn’t know that the hydrogen atom was made out of a proton and an electron, we might have started out and said: “Oh, I know what the base states are—they correspond to a particular momentum of the hydrogen atom.” No, because the hydrogen atom has internal parts. It may, therefore, have various states of different internal energy, and describing the real nature requires more detail.

The internal energy of a hydrogen atom may be described quantum mechanically by the Hamiltonian matrix H_ij because it has an infinite number of matrix elements and there are an infinite number of possible energy levels that an electron can transition between. However, it is impossible to determine the transition energies of all states. As energy states go higher, the energy differences between successive states become smaller, and the spectral lines converge and become continuous. Heisenberg chose to disregard spectral lines that cannot be observed directly stating that quantum mechanics should be “founded exclusively upon relationships between quantities which in principle are observable” [2]. We will include all electron transitions in a theory of the hydrogen atom, irrespective of whether the corresponding spectral lines are observable.

A matrix is a snapshot of an atom’s energy state at a particular point in time. The diagonalized matrix describing the hydrogen spectrum due to black body radiation is not the result of a single hydrogen atom, rather it is due to many atoms acting in unison. It is the result of many events, where each one occurs at an independent point in space and time. It is nevertheless possible for a solitary, electrically neutral hydrogen atom, isolated in space and time, to emit the entire hydrogen spectrum if we apply heat incrementally and describe the energy states sequentially in a series. Initially the states are of infinitesimal energy and undetectable, but they eventually register lines in the infrared, the visible spectrum, and beyond to the ultraviolet spectrum. The complete spectrum as viewed by a single atom appears in the manner of a motion picture with each frame consisting of a Hamiltonian matrix. We observe the spectrum as a continuously increasing background thermal energy that surrounds the hydrogen atom and is periodically interrupted by spectral lines of increasing energy in the Lyman, Balmer. Paschen, and Brackett series. Thus the absorption of heat causes the discrete energy eigenvalues of the atom to appear sequentially as diagonalized elements of the Hamiltonian matrix. This is in contrast to the complete hydrogen spectrum which is generated by simultaneous emissions from the many atoms of a gas.

3. Energy Conservation

It is well known that energy is not always conserved in quantum mechanics [3]. “Wave functions collapse when they are observed. This collapse process is unpredictable, and doesn’t obey the Schrödinger equation. As a result, the average energy is not conserved in the process of quantum measurement.” In fact the original proof of energy conservation given by Heisenberg in his study of the anharmonic oscillator is not rigorous. He demonstrated energy conservation for a closed quantum system; however, the atom absorbs thermal energy from the environment as a continuous series of infinitesimal transitions during black body radiation, so it must be treated as an open system. He did not prove that energy is conserved for open quantum systems.

In order to prove that energy is conserved for an open quantum system the energy absorbed must be identical to the energy emitted. In fact due to energy conservation energy input equals energy output for all isolated material systems, whether the system is a hydrogen atom, a machine, a hydroelectric power plant, or a galaxy. The quantum system conserves energy for changes in electromagnetic potential in the same way that a classical system conserves energy for changes in the gravitational potential. Energy conservation causes matrix mechanics to derive physical meaning from energy absorption and it necessarily equates with energy emission and the physical meaning attributed to wave mechanical formulations.

We shall illustrate energy conservation in a quantum system formally by using the conjugate variables energy and time to show that matrix and wave mechanics work together to describe the emission and absorption of radiation. We have described energy absorption in real space and real time with matrices as incremental increases. It can also be described in real time by one rotation of the wave function. Emission follows by means of a second rotation of the wave function demonstrating that absorption and emission are part of the same quantum process. We interpret the first rotation of the wave function not as a rotation in abstract space, but as the change in phase of electromagnetic fields from 0 to 2π in real space with phase information given by the off-diagonal matrix elements. Thus one “rotation” of the wave function is interpreted as one full cycle of an electromagnetic wave and an increase in the electron’s energy from the ground state to an excited state. The second rotation occurs as the electron returns to the ground state and is interpreted as a localization of electromagnetic field energy and emission of a photon. The dual wave-particle nature of the photon is thereby realized in a continuous transformation of fields. When a continuous function is integrated between two fixed endpoints, the orbitals, and two specified times, the period, the result is a relativistic invariant, the time integral of a Lagrangian ∫Ldt, or action function.

4. The Lagrangian in Quantum Mechanics

The photons of a spectral line are emitted when electrons “jump” from the ground state to an excited state and return to the ground state. The paths of excitation and decay occur between two specific endpoints, the atomic orbitals, and two specific times, the period τ, which are the physical conditions that define Hamilton’s principle of least action [4]. Thus the physical characteristics of a quantum system naturally define the correct equations of motion needed to describe it. The principle of least action simply means that of all the paths that an electron can take between two atomic orbitals, the ones actually taken are found by computing the action for each of the trajectories, and selecting the ones that have the least action. The paths selected will be the ones that have an action equal to the reduced Planck’s constant ћ.

The action is the time integral of the Lagrangian ∫Ldt, where L=T-V, T is the kinetic energy, and V is the potential energy. We compute the action of an electron transitioning from the ground state to an excited state by using generalized coordinates, three to describe its position on the electron shells R₁ = (x₁,y₁,z₁) and R₂ = (x₂,y₂,z₂), and three to describe its trajectory.

$$S\left[r\left(t\right)\right]={\displaystyle \underset{R_1}{\overset{R_2}{\int }}{\displaystyle \underset{t_1}{\overset{t_2}{\int }}\left(\mathrm{T}-\mathrm{V}\right)dt}}$$

(1)

The action, S[r(t)], is a functional that describes the absorption process in four dimensions. It has as its argument an infinite number of functions, the possible electron paths r(t). We know that it is relativistically correct because gravitational fields influence the action identically to the way that they influence clocks; that is, by causing time dilation [5].

Emission initiates from the excited state R₂ = (x₂,y₂,z₂) at time t₂ and it finalizes at the ground state R₁ = (x₁,y₁,z₁) at time t₁. Each of the electron shells R₂ and R₁ determines a locus of points where the fields vanish [6]. The Lagrangian density over the region of space-time between the excited and ground states is “stationary for all small variations of the coordinates inside the region.” Changes in action during emission are evaluated by integrating the Lagrangian density four-dimensionally thereby yielding a relativistic formulation.

$$S\left[{\varphi }_i\left(t\right)\right]={\displaystyle \underset{R_2}{\overset{R_1}{\int }}{\displaystyle \underset{t_2}{\overset{t_1}{\int }}L}}\left({\varphi }_i{\varphi }_{i,\mu }\right)d^{3}xdt=E\tau$$

(2)

Solving 2) we obtain an expression for the localized fields of a photon Eτ=ћ, which is equivalent to the more familiar E=hν. The emission of a photon is described by the action integral of a Lagrangian density ∫£(ϕ_i ,ϕ_{i ,μ})dt in the region of space-time between the excited and ground states given by the fields φ_i and its first derivatives ϕ_i,μ. This allows for a complete accounting of the energy interactions that occur during an electron transition within the volume ∫d³x between the electron orbitals where ϕ_i is the current density described radially and ϕ_i,μ is the electromagnetic field strength described transversely that results in the creation of a photon. The photon is described therefore as a four-dimensional localization of fields contained within the volume ∫d³x and the time interval t₂-t₁ = τ. The discrete and continuous properties of the emission process are described by the Lagrangian density over the time interval of a period which results in a photon being created. A more complete derivation is given elsewhere [6].

5. Conclusion

There are two ways to formulate a theory on quantum mechanics, non-relativistically using Hamiltonian mechanics and position-momentum conjugate variables or relativistically using Lagrangian mechanics and energy-time conjugate variables. Dirac knew already in 1930 that both were necessary [7]. “The two formulations are, of course closely related, but there are reasons for believing that the Lagrangian one is the more fundamental. In the first place the Lagrangian method allows one to collect together all the equations and express them as the stationary property of a certain action function. (This action function is just the time-integral of the Lagrangian.) There is no corresponding action principle in terms of the coordinates and momenta of the Hamiltonian theory. Secondly the Lagrangian method can easily be expressed relativistically, on account of the action function being a relativistic invariant; while the Hamiltonian method is essentially non-relativistic in form, since it marks out a particular time variable as the canonical conjugate of the Hamiltonian function. For these reasons it would seem desirable to take up the question of what corresponds in the quantum theory to the Lagrangian method of the classical theory.”

Experimental evidence obtained from black body radiation and interpreted by the matrix mechanical model provides proof that Dirac is correct when he states that the Lagrangian formulation is the more fundamental than the Hamiltonian. This is because it originates earlier in the quantization process; that is, during absorption; is relativistically correct, and unlike Hamiltonian formulations rigorously satisfies energy conservation.