Dynamical Aspects of Macroscopic and Quantum Transitions due to Coherence Function and Time Series Events

Abstract

This study presents the application of dynamical equations able to generate alternating deformations with increasing amplitude and delayed pulses in a certain material medium. It is considered that an external force acts at certain time interval (similar to a time series) upon the material medium in the same area. Using a specific differential equation (considering nonzero initial values and using a function similar to the coherence function between the external force and the deformations inside the material), it results that modulated amplitude oscillations appear inside the material. If the order of the differential dynamical equation is higher, supplementary aspects as different delayed pulses and multiscale behaviour can be noticed. These features are similar to non-Markov aspects of quantum transitions, and for this reason the mathematical model is suitable for describing both quantum phenomena and macroscopic aspects generated by sequence of pulses. An example of a quantum system, namely, the Hydrogen atom, is discussed.

1. Introduction

For simulating the generation of specific deformations inside a material medium under the action of external forces it can be considered that some short wavelength vibrations appear in the area where the force acts. The corresponding deformation is generated inside the material medium, using linear differential equations or equations with partial derivatives (similar to the wave equation or to the equation of diffusion). Yet such linear equations cannot explain the distance between the space area where the external force acts and the space area where fracture phenomena appear. Using differential equations of higher order, some slow variations of deformation along a certain direction could be obtained. Due to the fact that the mathematical model should explain the sharp deformations at a certain distance of the point of space where the force acts (leading to fracture phenomena), some different types of differential equations had been investigated. An analysis of oscillations generated by step functions acting upon second-order systems working at limt of stability was presented in [1].

Better qualitative results were obtained using dynamical equations able to generate practical test functions (similar to wavelets) and delayed pulses (when a free term which corresponds to an external force is added) [2] for justifying fracture phenomena appearing in a certain material medium [3]. It has been considered that an external force (described by a short wavelength sine function multiplied by a gaussian function) acts upon the material medium in a certain area. Using a specific differential equation (able to generate symmetrical functions for a null free term) for describing the generation of the corresponding deformation along an axis inside the material medium, it has been shown that a significant deformation could appear at a certain distance. This significant deformation justifies the fracture phenomena, while the inner structure of the material cannot allow significant sharp deformations without breaking.

However, this mathematical model cannot explain the breaking effect of a sequence of external pulses (e.g., applied as transverse force upon a beam fixed at both ends), when the time interval between these pulses is large enough so as the final effect not to be considered a superposition of individual effects of each pulse. It is well known from practice that workers using traditional tools have to apply some medium-power shocks at certain intervals upon a beam fixed at both ends, before a final great-power shock to be applied for breaking the material. Each medium-power shock generates specific damping vibrations inside the material medium, and the subsequent shock has to be applied right before the annihilation of these damping vibrations by the fluctuations of the external medium (the noise). Thus a certain degree of coherence for the effects of external pulses can be achieved, for the maximum possible value of the amplitude of fundamental harmonic corresponding to envelope of generated vibrations (if the time interval between external pulses is shortened, the final value of envelope function at the end of this interval is no more equal to zero and the difference between extreme values of envelope function decreases—thus the amplitude of fundamental harmonic component decreases also). This implies the use of some non-Markov aspects, while the memory of the previous similar events should be involved (similar to [4]).

For this reason, some specific differential equations based on the coherence function between the generated deformation and the alternating input should be taken into consideration. Since this coherence function vanishes if the output equals zero, the initial condition should be set at a small nonzero value.

2. Mathematical Model for Generating Amplitude-Modulated Deformations

As was shown in the previous paragraph, a differential equation modelling aspects similar to quantum phenomena should be based on the use of coherence function as free term. For the beginning, the external command (corresponding to the sequence of external pulses) should be considered as a superposition of cosine functions and we will analyze just the output generated by a certain cosine function (with the period set to the value 𝑇=1). The free term of the differential equation is represented by the coherence function

𝐶ℎ(𝑥)=^𝑥_𝑥in𝑦(𝑡)cos𝜔𝑡𝑑𝑡=^𝑥_𝑥in𝑦(𝑡)cos2𝜋𝑡𝑑𝑡, (2.1) where variable 𝑥 corresponds to time. The initial value 𝑦₀ of the output function 𝑦(𝑥) is set to a small nonzero value 𝑦₀=0.3 at the initial time moment 𝑥₀=0. By choosing an undamped second-order system with time constant 𝑇₀=1 (the period being 2𝜋≈6𝑇 so as very weak resonance aspects to appear) it results the equation

𝑦^′′1=−𝑇²₀𝑦+^𝑥₀𝑦(𝑡)cos𝜔𝑡𝑑𝑡=−𝑦+^𝑥₀𝑦(𝑡)cos2𝜋𝑡𝑑𝑡. (2.2) Using simulation in MATLAB (based on Runge-Kutta functions) it results for the derivative of the output 𝑦^′(𝑥) (denoted as 𝑧(𝑥)) the function represented in Figure 1. The continuous line corresponds to function 𝑧(𝑥)=𝑦^′(𝑥); the discontinuous line corresponds to the function 𝑓(𝑥)=−0.1cos2𝜋𝑥 necessary for studying the correlation between 𝑧(𝑥) and 𝑓(𝑥).

Figure 1 

Output generated for second-order system by cosine function.

It can be easily noticed that 𝑧(𝑥) and 𝑓(𝑥) exhibit in-phase oscillations. However, the oscillations corresponding to 𝑧(𝑥) present a supplementary aspect: the local maximum values and the local minimum values of each oscillation (the envelopes) are both amplitude-modulated by a periodical signal with a time period six times greater than the period 𝑇 of the external command. This means that for three successive oscillations the local maximum values increase, and for next three successive oscillations the local maximum values decrease (the peak-to-peak value for each oscillations being the same). This aspect can be put in correspondence with an increasing velocity of particles vibrating in a bar under the influence of an external alternating force, generating fracture phenomena after a few oscillations (when the velocity becomes higher than a certain threshold value). It can be also put in correspondence with quantum aspects, where a transition can appear just after a certain number of oscillations for the wave-trains of particles involved in interaction. Supplementary simulations performed for greater time periods of the proper oscillations of the second order system show that the local maximum values and the local minimum values of each oscillation (the envelopes) are in fact amplitude-modulated by a periodical signal corresponding to the proper oscillations of the second order system, so the number of successive oscillations for which the local maximum increases can vary from 2-3 to some higher values.

At next step, the external command should be considered as a sequence of rectangular pulses similar to 𝛿^′(𝑥−𝑘). Let us consider that the same second order system has the same initial conditions at the initial moment of time 𝑥_in=−1.2 and that the command function is represented by a sequence of step pulses 𝑠(𝑥) with amplitude 𝐴=5 for short time before 𝑥_𝑘=−1,0,1,2,3… and with amplitude 𝐴=−5 short time after these time moments 𝑥_𝑘=−1,0,1,2,3…. This means

𝑠=5for𝑥∈[−1.2,−1]∪[−0.2,0]∪[0.8,1]∪[1.8,2]∪[2.8,3]…,𝑠=−5for𝑥∈[−1,−0.8]∪[0,0.2]∪[1,1.2]∪[2,2.2]∪[3,3.2]…. (2.3) Using simulation in MATLAB (based on Runge-Kutta functions) it results for the derivative of the output 𝑦^′(𝑥) (denoted as 𝑧(𝑥)) the function represented in Figure 2. The continuous line corresponds to function 𝑧(𝑥)=𝑦^′(𝑥); the discontinuous line corresponds to the function 𝑓(𝑥)=0.02𝑠(𝑥) necessary for studying the correlation between 𝑧(𝑥) and 𝑓(𝑥). It can be noticed that 𝑧(𝑡) is a saw-tooth function. The external command 𝑠(𝑥) and 𝑧(𝑥) are no more in-phase functions, but the main feature of Figure 1 is still present: an alternance of three increasing local maximum values and of three decreasing local maximum values of 𝑧(𝑡) can be noticed, with the period of these alternances being six time greater than the period of the external command 𝑠(𝑥) (represented by a sequence of pulses similar to 𝛿^′(𝑥−𝑘),𝑘∈−1,0,1…. It should be emphasized the fact that these aspects cannot be noticed if the maximum and minimum values of external command 𝑠(𝑥) are close to unity.

Figure 2 

Output generated for second-order system by sequence of alternating pulses.

If the time period of the proper oscillations of the second order system decreases so as 𝑇₀≈𝑇 the well-known phenomena of beat (interference between two oscillations of slightly different frequencies generating amplitude-modulated oscillations with a frequency corresponding to the difference between the two frequencies) or resonance (oscillations with amplitude 𝐴→∞) appear. For 𝑇₀<𝑇, some proper oscillations of the second order system modulated by the external cosine function can be noticed. While none of these mathematical models can generate oscillations with increasing local maximum values for a few oscillations of the external command function, they are not useful for modeling transitions.

3. Mathematical Model for Generating Multiscale Delayed Deformations

The analysis can be extended by investigating differential equations able to generate delayed phenomena, similar to non-Markov aspects of transitions in quantum physics (where the intutive model requires a certain interaction time between the wave-trains corresponding to interacting particles before a suddenly emerging transition phenomenon to occur). For this purpose, the order of the differential equation should be increased, while a higher order implies usually a greater transient time. So as new aspects to be revealed accurately, the order of the differential equation is increased three times and the time constant 𝑇₀ is increased √10 times from unity value for avoiding any resonance effect. It results the differential equation

𝑦⁽⁶⁾1=−𝑇²₀𝑦⁽⁴⁾+^𝑥₀𝑦(𝑡)cos𝜔𝑡𝑑𝑡=−0.1𝑦⁽⁴⁾+^𝑥₀𝑦(𝑡)cos2𝜋𝑡𝑑𝑡 (3.1) for an external command represented by the same oscillating function cos2𝜋𝑡 (the period 𝑇 equals 1). The initial moment of time is set to 𝑥_in=0 and the initial value of 𝑦(𝑥) is set to 𝑦₀=0.3 (as in te case of second order differential equation presented at the beginning of previous paragraph).

The simulation performed in Matlab for 𝑧(𝑥)=𝑦^′(𝑥) reveals new aspects. On the time interval 𝑥∈(0,40) it can be noticed a positive value for 𝑥∈(0,37) with a maximum value 𝑀𝑣₁≈1.6 units for 𝑥₁≈30 (the delay time being 𝑡𝑑₁≈5) as shown in Figure 3.

Figure 3 

First positive alternance of $z(x)$ generated for six-order system by cosine function.

On the time interval 𝑥∈(0,100) it can be noticed at first sight (due to the scale of vertical axis) a negative value for 𝑥∈(0,85) (the positive values for 𝑥∈(0,37) being hard to be noticed) with a minimum value 𝑀𝑣₂≈−40 units for 𝑥₂≈75 (the delay time can be approximated as 𝑡𝑑₂≈50) as shown in Figure 4.

Figure 4 

First negative alternance of $z(x)$ generated for six-order system by cosine function.

On the time interval 𝑥∈(0,150) it can be noticed at first sight (due to the scale of vertical axis) a positive value for 𝑥∈(0,130) (the negative values for 𝑥∈(37,85) being hard to be noticed) with a maximum value 𝑀𝑣₃≈1000 units for 𝑥₃≈120 (the delay time can be approximated as 𝑡𝑑₃≈100) as shown in Figure 5.

Figure 5 

Second positive alternance of $z(x)$ generated for six-order system by cosine function.

On the time interval 𝑥∈(0,200) it can be noticed at first sight (due to the scale of vertical axis) a negative value for 𝑥∈(0,185) (the positive values for 𝑥∈(0,37) and 𝑥∈(85,130) being hard to be noticed) with a minimum value 𝑀𝑣₄≈30000 units for 𝑥₄≈170 (the delay time can be approximated as 𝑡𝑑₄≈140) as shown in Figure 6.

Figure 6 

Second negative alternance of $z(x)$ generated for six-order system by cosine function.

On the time interval 𝑥∈(0,250) it can be noticed at first sight (due to the scale of vertical axis) a positive value for 𝑥∈(0,230) (the negative values for 𝑥∈(37,85) and 𝑥∈(130,185) being hard to be noticed) with a maximum value 𝑀𝑣₅≈800000 units for 𝑥₅≈220 (the delay time can be approximated as 𝑡𝑑₅≈180) as shown in Figure 7.

Figure 7 

Third positive alternance of $z(x)$ generated for six-order system by cosine function.

And so on, it can be noticed that different alternances of 𝑧(𝑡) can be noticed, depending on the time interval selected for analysis. The ratio between peak values corresponding to consecutive alternances is about |𝑀𝑣_𝑘+1/𝑀𝑣_𝑘|≈25-30 and the time interval between moments corresponding to peak values for consecutive alternances is about 𝑡_𝑘+1−𝑡_𝑘≈45-50 (approximately two times greater than the period 2𝜋𝑇₀=2√10𝜋 of the proper oscillations). A similar value can be noticed for the difference between delay times for two consecutive alternances (𝑡𝑑_𝑘+1−𝑡𝑑_𝑘≈40-50). As a consequence, a higher-order differential equation with a free term corresponding to the coherence function between an external cosine command and the output 𝑦(𝑥) generates for 𝑧(𝑥)=𝑦^′(𝑥) alternances with the same temporal pattern and with an increasing amplitude according to a geometrical progression. For this reason, the higher-order differential equation previously presented is suitable for modeling multiscale phenomena and for explaining multiscale threshold transitions—see also [5, 6].

4. Quantum-Mechanical Case Study: The Hydrogen Atom

Aspects mentioned in the previous paragraphs have shown that both macroscopic and quantum transitions require a certain set of events (similar to a time series events) or a certain repetitive phenomenon for generating the coherence function which drives the dynamics of internal oscillations. When the peak values of these internal oscillations are above a certain threshold, a possible transition to another state could appear.

However, these aspects should be valid also for standing waves corresponding to stable quantum or macroscopic states. This means that a certain wavefunction corresponding to a noninteracting particle should be described in an intuitive manner by a zero energy state and should be noticed by an external observer as a term of a quantum series (each term of such a series representing a quantum noninteracting state). For this purpose, the well-known case of the hydrogen atom should be analyzed at first.

The principle of mass-energy equivalence normally never comes to mind when the quantum-mechanical analysis of the hydrogen atom is undertaken. It is well known that by applying Schrödinger's equation to the problem of the electron in the hydrogen atom, the Balmer energy levels are obtained by means of a purely classical (i.e., nonrelativistic) analysis [7, 8]. There is, however, a very interesting connection between the problem of the hydrogen atom and the principle of mass-energy equivalence that was previously unexplored. If we write the time-dependent Schrödinger equation

𝑖ℏ𝜕𝜓𝜕𝑡=𝐻𝜓 (4.1) and its solution

𝜓=𝜓₀𝑖exp(−ℏ𝐻𝑑𝑡), (4.2) where 𝐻 is the total energy of the moving particle, namely, the electron, we must ask what conclusion can we make if we assume that the electron is in a stable orbit around the nucleus? Obviously, we must assume that the wave function 𝜓=𝜓₀ (i.e., a constant, or stable wave function that does not evolve over time; even though it is dependent on the radius and the spherical harmonics). This, of course, is well known, since the electron's wave function in the hydrogen atom represents a standing wave and has no time dependence. Hence, the conclusion that inevitably emerges in this case is that the total energy of the electron 𝐻 must be equal to zero everywhere along the path of the electron. In view of some fundamental research on the principle of mass-energy equivalence that was previously published by the author [9, 10], this conclusion, as a matter of fact, is not surprising.

In the earlier publications by the author, it was demonstrated that a number of fundamental problems in quantum mechanics cannot be understood on the basis of the relativistic law of mass-energy equivalence, 𝐻=𝑚𝑐². The problem of the hydrogen atom is one such problem. It was further demonstrated that 𝐻=𝑚𝑐² can be regarded as a special case of a more general law of mass-energy equivalence that does in fact explain that category of problems that the relativistic law fails to explain. That general law is 𝐻=𝑚𝑣², where the relativistic constant 𝑐² has been replaced by 𝑣², with 𝑣 being the velocity of the moving particle (see [9–11] for a complete historical accounting of the origin and the applications of that law). We will now proceed to solve the problem of the total energy of the electron in the hydrogen atom and demonstrate that the general mass-energy equivalence law 𝐻=𝑚𝑣² correlates with and explains the result predicted by Schrödinger's equation. We will further demonstrate that a new “zero-energy wavefunction” that will be obtained under that law is actually a quantized version of the classical wavefunction that has been known for decades.

4.1. The Law of Mass-Energy Equivalence and the “Zero-Energy” Wave Equation

It is not difficult to see how the general mass-energy equivalence law 𝐻=𝑚𝑣² (which, admittedly, may seem strange to the readers who are not familiar with it) correlates with the result predicted by Schrödinger's equation. In the hydrogen atom, the electron is in equilibrium due to the equality of the two forces

𝑒²𝑟²=𝑚𝑣²𝑟, (4.3) where 𝑒²/𝑟² is the Coulomb electrostatic force (here, 𝑒²=𝑞²/4𝜋𝜖₀, where 𝑞 is the electron's charge), and where 𝑚𝑣²/𝑟 is the centrifugal force. But the electrostatic potential 𝑉 acting on the electron is equal to −𝑒²/𝑟, by definition. From the above equation, it is therefore clear that 𝑉=−𝑚𝑣². If we now assume that the total energy of the free electron is given by the quantity +𝑚𝑣², then it must be further clear that the total energy of the bound electron must be equal to zero (due to the addition of the electrostatic potential 𝑉). (It is to be pointed out that this conclusion concerns the TOTAL ENERGY of the electron. In practice, the atom is observed to emit and absorb energy during bound-state transitions because such transitions involve only kinetic energy and potential energy changes. Mass-energy equivalence obviously does not play a role in electronic bound-state transitions. That is why the present conclusions are not in disagreement with the classical theory or with experimental results). This is the classical view according to Bohr's theory. Let us now examine the view according to the Schrödinger Hamiltonian theory.

The classical Schrödinger Hamiltonian is given by

ℏ𝐻=−²∇2𝑚²+𝑉. (4.4) This Hamiltonian represents the sum Kinetic Energy + Potential Energy, and it is the Hamiltonian used to derive the Balmer energy levels and the classical wave function of the electron. If we want to write the Hamiltonian in a manner that takes mass-energy equivalence into account, the Hamiltonian will be written as follows:

ℏ𝐻=−²𝑚∇²+𝑉, (4.5) where we have replaced the kinetic energy 1/2𝑚𝑣² by the total energy 𝑚𝑣². But since the total energy must be equal to zero, then we have the following wave equation:

−ℏ²𝑚∇²𝜓₀+𝑉𝜓₀=0. (4.6) We will now demonstrate that the wave function 𝜓₀ that satisfies this zero-energy wave equation is the same as the wave function derived through the classical analysis, with the surprising restriction that the wave function itself must be radially quantized!

4.2. The Connection between the Zero-Energy Wave Equation and the Classical Wave Equation

For the purpose of comparison, we write the classical equation that is based on the Schrödinger Hamiltonian together with the new wave equation that incorporates mass-energy equivalence:

ℏ−(²2𝑚)∇²𝜓₀+𝑉𝜓₀=𝑊𝜓₀ℏ(classical),−(²𝑚)∇²𝜓₀+𝑉𝜓₀=0(totalenergy), (4.7) where 𝑊 represents the Balmer energy levels and where 𝑉=−𝑒²/𝑟 is the potential of the nucleus. While the two equations obviously seem to be two very different equations, we will now demonstrate that the second equation does indeed revert to the first equation if 𝜓₀ is restricted to be a radially quantized function, rather than a continuous function! We first write the zero-energy equation as follows:

(ℏ²𝑚)∇²𝜓₀=𝑉𝜓₀𝑒=−²𝑟𝜓₀. (4.8) Since the radial distance 𝑟 takes only quantized values as multiples of the Bohr radius, 𝑎=ℏ²/𝑚𝑒², we substitute for 𝑟 in the equation by using this quantity, getting

(ℏ²𝑚)∇²𝜓₀=−𝑒²𝑚𝑒²ℏ²𝜓₀=−𝑚𝑒⁴ℏ²𝜓₀. (4.9) Dividing both sides of the equation by 2 gives

(ℏ²2𝑚)∇²𝜓₀=−𝑚𝑒⁴2ℏ²𝜓₀. (4.10) It is not difficult to verify that the coefficient of 𝜓₀ on the r.h.s. of the equation is the Balmer energy 𝑊. That is, we have the result that

(ℏ²2𝑚)∇²𝜓₀=𝑊𝜓₀. (4.11) Now, by virtue of (4.11), the zero-energy wave equation in (4.7) can be finally written as

ℏ−(²2𝑚)∇²𝜓₀+𝑉𝜓₀ℏ=+(²2𝑚)∇²𝜓₀=𝑊𝜓₀. (4.12) This last equation is of course the classical wave equation.

If we decompose the zero-energy wave equation into its radial and spherical-harmonic components, it becomes a simple matter to verify that the classical unnormalized wave function

𝜓₀(𝑟)=exp(−𝑚𝑒²ℏ²𝑟) (4.13) will indeed satisfy the radial wave equation at 𝑟=𝑛𝑎, or integer multiples of the Bohr radius (see the proof in the appendix). The fact that the classical wave function satisfies the zero-energy wave equation at multiples of the Bohr radius can be understood physically as follows: the classical wave function is a continuous, differentiable function that defines the boundary of a space that theoretically extends from 𝑟=0 to 𝑟=∞ (see the plot in Figure 8). The solution of the zero-energy wave equation, on the other hand, is a discrete, sparse set in 𝑟 that is defined only at integer multiples of the Bohr radius (see Figure 8). This discrete function therefore inhabits the space defined by the classical wave function (a simple analogy might be a wave in a plastic sheet on top of which tiny droplets of mercury always flow to the minimum of that wave, as if the wave was a “potential well”). This is not a surprise, since, as was concluded earlier, the total energy of the electron is equal to zero at multiples of the Bohr radius. The discrete solution, therefore, is indeed a solution in which the minimum energy principle is manifested; as opposed to the classical solution in which only the kinetic and potential energies are accounted for.

Figure 8 

The classical wave function ${\psi }_0$ and the quantized solution of the zero-energy wave equation. The latter exists only at integer multiples of the Bohr radius and inhabits the space defined by the former.

5. Conclusions

This study has presented the application of dynamical equations able to generate alternating deformations with increasing amplitude and delayed pulses in a certain material medium. It has been considered that an external force acts at certain time interval (similar to a time series) upon the material medium in the same area. Using a specific differential equation (considering nonzero initial values and using a function similar to the coherence function between the external force and the deformations inside the material), certain modulated amplitude oscillations were generated. For a higher order of the differential dynamical, some delayed pulses and a specific multiscale behaviour could be noticed. These features are similar to non-Markov aspects of quantum transitions, and for this reason the mathematical model is suitable for describing both quantum phenomena and macroscopic aspects generated by sequence of pulses. The amplitude modulation of local maximum/minimum values can be put in correspondence with PDE equations (see [12]) and with propagating wavelets through dispersive media [13]. The multiscale analysis of delayed pulses is similar to multiscale analysis of wave propagation [14] and breaking phenomena can be put in correspondence with studies upon localized fractals (as in [15]) and could be extended for solving natural integral equations (as in [16]). A time-series application considering the Hydrogen atom as a case study was presented.

Appendix

Solution of the Zero-Energy Wave Equation and the Quantization Condition

To solve (4.6) for 𝜓₀, we must replace the operator ∇² by its equivalent expression in spherical coordinates and substitute the potential 𝑉 by the traditional quantity −𝑒²/𝑟. The process of replacing ∇² in (4.6) by its equivalent expression in spherical coordinates is well known in the literature [7, 8], and we simply write the result

ℏ(−²𝑚𝑑²𝑑𝑟²+𝑙(𝑙+1)ℏ²𝑚𝑟²−𝑒²𝑟)(𝑟𝜓₀(𝑟))=0. (A.1) Here, 𝜓₀(𝑟) is the radial component of 𝜓₀ and 𝑙 is the orbital quantum number. Typically, a second equation is needed to solve for the spherical-harmonic component of 𝜓₀, but since this solution is well known in the literature it will not be discussed here. The usual approach for solving (A.1) is to let the product 𝑟𝜓₀(𝑟) be equal to another function, say Γ(𝑟). Equation (A.1) is then rewritten as

−Γ^′′(𝑟)+(𝑙(𝑙+1)𝑟²−𝑚𝑒²ℏ²⋅1𝑟)Γ(𝑟)=0. (A.2) In the classical solution, the Balmer series for hydrogen is obtained by simply setting 𝑙=0. While the above equation cannot be solved for the Balmer energy, setting 𝑙=0 results in

Γ^′′(𝑟)+𝑚𝑒²ℏ²⋅1𝑟Γ(𝑟)=0. (A.3) We now note that the quantity ℏ²/𝑚𝑒² represents the Bohr radius, 𝑎. We will follow however the standard procedure of replacing 𝑎 by 𝑛𝑎, where 𝑛 is the principal quantum number. We therefore rewrite the above equation as follows:

Γ^′′1(𝑟)+⋅1𝑛𝑎𝑟Γ(𝑟)=0. (A.4) Solving this simple differential equation is a simple but rather lengthy and uninformative mathematical exercise. It can be quickly verified, however, that the classical wave function

Γ(𝑟)=𝑟𝜓₀𝑟(𝑟)=𝑟exp(−)𝑛𝑎 (A.5) does in fact satisfy (A.4), provided that the radial distance 𝑟 in the final expression is replaced by an integer multiple of the Bohr radius, or 𝑛𝑎.