The forming mechanism of spontaneous emission noise flux radiated from hydrogen-like atoms by means of vibrational Hamiltonian

The spontaneous emission (SPE) process was initially described by Einstein in 1917 as the exchange of momentum between atoms and radiation so that the atomic system suffers a corresponding recoil in the opposite direction.¹ This recoil was experimentally confirmed by Frisch in 1933 by showing that a long and narrow beam of excited sodium atoms widens up after spontaneous emissions have taken place.² Subsequently, many articles have been published to describe the SPE by implementing the different sources, such as interacting with the zero-point energy (vacuum fluctuations) and radiation reaction field.^3–6 By contrast, we have here interpreted the SPE from a different physical point of view by considering the energy eigenvalue E_n = (n + 1/2) ℏω₀ of a simple harmonic oscillator (SHO), which has immediately been converted to its Hamiltonian operator $Ĥ0=(N^+1/2)ℏω0$ by substituting the number operator $N^$ for the energy level number n. This simple case persuaded us to construct the vibrational Hamiltonian of more complicated systems (such as atoms and molecules) in terms of the simple harmonic oscillator Hamiltonian $Ĥ0$ in the form of a power series. The novelty of the present research is to describe the formation mechanism of spontaneous emission radiation by using the vibrational Hamiltonian.

The first report was published in 2016⁷ where the vibrational Hamiltonian of a diatomic molecule had been formed up to the third power of $Ĥ0$ as $Ĥ(3)=Ĥ0+γ2Ĥ02+γ3Ĥ03$ in order that the second-order and third-order expansion coefficients γ₂ and γ₃ were determined by inspiring the stability theory of lasers.⁸ The vibrational motion equations of diatomic molecules have then been derived by applying $Ĥ(3)$ into the Heisenberg equation. The solutions led to the useful information about the molecular vibration frequencies in the different energy levels of the Morse potential, and the last vibrational stable level (dissociation level).⁷ 

The purpose of the present paper is to extend this method to cover the more complicated case of hydrogen-like atoms (HLAs) due to the Coulomb potential and its divergent behavior (singularity) on the nucleus. However, the vibrational Hamiltonian of the HLA has been formed by expanding the corresponding energy eigenvalue around the lowest energy level number n = 1 and excluding the zero-point energy (E₀ = ℏω₀/2) from the energy spectrum of the SHO. Similarly, by substituting the vibrational Hamiltonian of the HLA into the Heisenberg equation, an equation is derived for the vibrational motion of the electron around the rest nucleus that acts like a quantum harmonic oscillator (QHO) with the oscillatory frequencies consistent with Bohr’s classical model.

On the other side, the sudden transition of the oscillating electron from the upper to a lower atomic energy level is an unavoidable process that alters the oscillatory state of the atom. The energy conservation requires the electron transition energy to be emerged from the atom as the spontaneous emission radiation. However, the atom is converted to a damped oscillator, which is known as the Langevin oscillator.⁹ As a result, the Langevin equation—which describes the motion of a damped oscillator—is formed by adding the damping term (spontaneous emission rate) and the fluctuation (Langevin) force into the vibrational equation of the HLA according to the fluctuation–dissipation theorem (FDT).¹⁰ The solution of the Langevin equation gives the fluctuation imposition to both the potential and kinetic energies of the HLA during its rather fast vibrational oscillations. The correlation function of Langevin force is then used to calculate the noise flux of SPE, which was radiated from the HLA during the atomic transition. Finally, it will be demonstrated that the noise flux of SPE is provided not only by the potential and kinetic noise fluxes of the oscillating electron but also by its interchange noise fluxes according to the flux conservation law.

The first step is initiated by the vibrational energy eigenvalue of the HLA, which was derived from solving the time-independent equation of Schrodinger associated with a two-particle system as E_vib = E_n = −0.5μ c²(Zα)²/n², in which μ, c, Z, α, and n = 1, 2, 3, … are, respectively, the reduced mass, light speed, atomic number, fine constant, and energy level number.¹¹ The vibrational energy eigenvalue E_vib is now expanded around the lowest value n = 1 as

On the other hand, there is no connection between E_vib and the usual energy eigenvalue of the SHO E_n = (n + 1/2) ℏω₀, except the non-integer value 1/2 associated with the exclusion of the zero-point energy. Therefore, an infinite ladder energy spectrum with the equal steps ℏω₀ remains for the energy spectrum of the SHO as

which is consistent with the expanded eigenvalue E_vib given by (1). It is emphasized that an infinite ladder with the equal steps ℏω₀ can be represented by n ℏω₀ or equivalently by (n − 1) ℏω₀.

The next step is to substitute the number operator $N^$ for the energy level number n in both energy eigenvalues (1) and (2) to derive their corresponding Hamiltonian operators in the forms

and

The last step is to form the vibrational Hamiltonian of the HLA (⁠$Ĥvib$⁠) in terms of the Hamiltonian of the SHO (⁠$Ĥ0$⁠) by substituting $(N^−1)$ from (4) into (3) as

in which all the expansion coefficients will simultaneously be determined by calculating the value of fundamental frequency ω₀ in Sec. III.

The linear regime of HLA-Hamiltonian is corresponding to k ≤ 1 in the general relation (5) as

The first-order vibrational equations of motion are now achieved for the mean values of the relative position operator $x^$ and linear momentum operator $p^$ by using the well-known Heisenberg equation in quantum mechanics as¹² 

and

where the commutation relations $[Ĥ0,x^]=−iℏμ−1p^$ and $[Ĥ0,p^]=iℏμω02x^$ have been used.⁷ The final form of the vibrational equation of motion associated with the linear Hamiltonian (6) is found for the variable $x^(t)$ by substituting (8) into the derivative of equation (7) as

so that the first-order vibrational frequency $ωvib(1)$ is the same as the fundamental frequency ω₀ in the form

Consequently, the final form of the vibrational Hamiltonian HLA (5) is turned out as

in which the value of fundamental frequency ω₀ is given by (10).

On the other side, the vibrational oscillation frequency of the HLA had already been calculated by Bohr as $ωn=meμω0n3(n=1,2,3,…)$ in which the electron mass m_e must be substituted by the reduced mass μ (m_e = μ) as a correction to his classical model.¹¹ One can probe the complete agreement between our quantum model and Bohr’s classical model by expanding Bohr’s oscillation frequency $ωn(n)=ω0n3$ around the lowest value n (n = 1). The first term of expansion becomes equal to the fundamental frequency ω₀ given by (10).

The second-order non-linear vibrational Hamiltonian of the HLA is assigned by the values k ≤ 2 in the infinite power series (11) as

The motion equations of variables $x^$ and $p^$ are similarly rendered by substituting $Ĥvib(2)$ for $Ĥvib(1)$ into the Heisenberg equations (7) and (8) as

and

The motion equation for the variable $x^$ is finalized by the respective substitution of $p^(t)$ and $dp^(t)/dt$ from (13) and (14) into the derivative of (13) as

in which the second-order vibrational frequency $ωvib(2)$ is equal to (4 − 3n) ω₀. Clearly, the second-order term β⁽²⁾ = 3 ω₀ must be ignored because it is only a phase term associated with the infinite number of rotation axis directions in space. Therefore, the second-order equation (15) is simplified to the first-order equation (9) with the vibrational frequency equal to $ωvib(2)$ rather than $ωvib(1)$⁠. Meanwhile, the first-order and second-order expansions of Bohr’s vibrational frequency $ωn(n)=ω0n3$ around n = 1 are in complete accordance with the corresponding first-order and second-order vibrational frequencies $ωvib(1)=ω0$ and $ωvib(2)=(4−3n)ω0$ that appeared in the equations (9) and (15), respectively.

As a result, the common features of the first-order and second-order vibrational Hamiltonians $Ĥvib(1)$ and $Ĥvib(2)$ defined by (6) and (12) and their corresponding motion equations (19) and (15) [β⁽²⁾ = 0] imply the general Hamiltonian of the HLA (11), which imitates the motion equation of a quantum harmonic oscillator (QHO) in the general form

in which

and

It is noteworthy to emphasize that the vibrational level number n takes the values 1, 2, 3, …, and the lowest value n = 1 is only used for constructing the vibrational Hamiltonian without appearing in the final results (16)–(18). Meanwhile, the Hamiltonian (11) and the equation of motion (15) had similarly been derived for a diatomic molecule as a two-particle system in our previous work.⁷ However, we have achieved different categories of information, such as the stability conditions of diatomic molecules, the dissociation level of both homo-nuclei and hetero-nuclei diatomic molecules in an acceptable agreement with the experiment, and the cut-off frequency.

According to the fluctuation–dissipation theorem (FDT), the motion of all oscillators including the vibrational motion of the HLA suffers from fluctuations imposed by their damping conditions.^9,10,13 Therefore, the QHO equation of the HLA (16) must be modified to cover the damping role of SPE in the form

in which $A=τR−1=2γsp$ is Einstein’s A-coefficient equal to the inverse of radiative lifetime of the excited state.¹⁴ Equivalently, 2γ_sp is equal to the atomic decay rate of the HLA due to the spontaneous emission radiation.

The source of fluctuations is the Langevin force Γ_x(t) that should be added to the equation (19) to evaluate the random fluctuations $δx^(t)$ that superimposed on the position of the oscillating electron $x^(t)$ (⁠$x^(t)→x^(t)+δx^(t)$⁠).^9,15 Thus, the equation of motion (19) is transferred to an equation for the fluctuating variable $δx^(t)$ after adding the Langevin force Γ_x(t) and applying $x^(t)→x^(t)+δx^(t)$ in the form

in which μ is the reduced mass and ω_vib is the vibrational frequency of the electron in the HLA as defined in (18). It is noteworthy to remind that equation (20) is the well-known Langevin equation, which here is derived for the vibrational motion of the HLA for the first time by using our heuristic vibrational Hamiltonian of the HLA $Ĥvib$ given by (11).

The Langevin equation (20) can be easily solved in the frequency rather than the time domain by implementing the following Fourier integral:^16,17

so that the first-order and second-order temporal differentials d/dt and d²/dt² are, respectively, transformed to iω and −ω² in the frequency domain. As a result, the solutions for the fluctuations that are imposed to the position $δx^(ω)$ and linear momentum $δp^(ω)$ of the oscillating electron are rendered as

and

where the relation $δp^(ω)=iμωδx^(ω)$ is used.

Now, the energy conservation relation $Ê(ω)=Û(ω)+K^(ω)$ associated with the potential, kinetic, and total energies of the oscillating electron can be converted into a conservation relation for their corresponding fluctuation variables $δÛ(ω)$⁠, $δK^(ω)$⁠, and $δÊ(ω)$ by applying the superposition principle $Û(ω)→Û(ω)+δÛ(ω)$⁠, $K^(ω)→K^(ω)+δK^(ω)$⁠, and $Ê(ω)→Ê(ω)+δÊ(ω)$ in the form

Our first priority is to calculate the respective potential and kinetic fluctuations $δÛ(ω)$ and $δK^(ω)$⁠. In this way, we take the derivative of the potential energy $Û(t)=0.5μω02x^(t)2$ and keep the fluctuating terms in accordance with the first-order approximation [$δx^(t)2=0$] as

The Fourier transform of (25) leads to a relation for its counterpart in the frequency domain as

where the solution (17) is substituted into (25) before taking the Fourier transform. Meanwhile, the relation (26) also includes a term proportional to $δx^(ω+ωvib)$ that has been eliminated because it will produce an identical profile with a peak located at the unacceptable negative frequency of ω = −ω_vib.

The same procedure should be followed to derive the similar result for the kinetic energy as

where the fluctuation relation $δp^(ω−ωvib)=iμ(ω−ωvib)δx^(ω−ωvib)$ is used.

The next important stage is to consider the correlation functions of the Langevin force Γ_x(ω) in the forms^6,7,18

and

in which

is the mean number of thermal photons at the frequency mode ω (⁠$n¯th≪1$⁠).¹⁴ 

In the other perspective, the correlation function of an arbitrary fluctuating variable a(ω) with a white noise origin (Dirac function) is defined in the following complex conjugate form:^19,20

in which $N(ω)=h(ω)2$ is the dimensionless mean flux per unit angular frequency bandwidth at angular frequency ω and given by

The noise flux spectrum of the oscillating electron associated with its positional fluctuations is first calculated by choosing $a(ω)=δx^(ω−ωvib)$ as

where the relations (22), (29), and (32) are used. It is to be reminded that the fluctuation in the electron position, in turn, gives rise to a fluctuation in its linear momentum according to $δp^(ω)=iμωδx^(ω)$⁠. Consequently, the fluctuations $δx^(ω)$ and $δp^(ω)$ lead to the corresponding fluctuations for the potential and kinetic energies of the HLA due to the relations (26) and (27).

The next important task is to calculate the noise fluxes of potential and kinetic energies by substituting $a(ω)=δÛ(ω)$ and $a(ω)=δK^(ω)$ from (26) and (27) into the noise flux relation (32) as

and

One can now calculate the total noise flux emitted from the HLA in the form of spontaneous emission radiation by taking the correlation function of the fluctuation conservation relation (24). The noise flux of spontaneous emission $NSP(ω)=δÊ(ω)δÊ(ω′)*$ is, thus, related to the potential $NU(ω)=δÛ(ω)δÛ(ω′)*$ and kinetic $NK(ω)=δK^(ω)δK^(ω′)*$ noise fluxes in the following noise flux conservation:

in which $NUK(ω)=δÛ(ω)δK^(ω′)*$ and $NKU(ω)=δK^(ω)δÛ(ω′)*$ are the interchange noise fluxes between the potential and kinetic energies. They are turned out to be equal to each other as we have

We now consider the atomic transition 2P → 1S of the hydrogen atom as a typical case with the upper energy level n = 2, the fundamental frequency ω₀ = 4.15 × 10¹⁶ Hz, the vibrational frequency ω_vib = 5.19 × 10¹⁵ Hz, and the spontaneous emission (radiative) decay rate γ_sp = 4.69 × 10⁸ Hz.²¹ The noise flux spectrum of the electron position N_x(ω) is indicated in Fig. 1 for the atomic transition 2P → 1S. N_x(ω) is interpreted as the number of fluctuations that occur for the position of the oscillating electron in the scale of time (second) at the oscillation frequency ω. The most physical feature of positional noise flux N_x(ω) is to play a fundamental role in producing other noise fluxes according to (34)–(37).

The noise flux spectra of potential N_U(ω) (red), kinetic N_K(ω) (green), and their equal interchange N_UK(ω) = N_KU(ω) (blue) together with their sum in the form of spontaneous emission radiation N_SP(ω) (black) are illustrated in Fig. 2. The kinetic and interchange noise fluxes are evidently much smaller than the potential noise flux. The physical interpretation is concerned with the high acceleration of the oscillating electron at its oscillation amplitude where the potential energy has the maximum value, in contrast with the kinetic energy. The potential energy, thus, undertakes the maximum uncertainty (noise) due to a sudden change in the oscillation direction of the electron. Meanwhile, the negative values for the noise fluxes N_UK(ω) = N_KU(ω) are meaningless except interpreted as the interchanged noise fluxes between the kinetic and potential energies.

Finally, the spectrum of the spontaneous emission noise flux, which is formed by the linear contribution of four different internal noise fluxes N_U(ω), N_K(ω), and N_UK(ω) = N_KU(ω) in Fig. 2, has a Lorentzian profile with the bandwidth equal to A = 2γ_sp = 9.38 × 10⁸ Hz in complete agreement with the experimental work of weak measurement, based on atomic spontaneous emission.²² The peak of the different internal and external noise fluxes has commonly concentrated at the frequency ω ≈ 2ω_vib in Fig. 2, which is due to the peak location of the positional noise flux N_x(ω) in Fig. 1.

We have elaborated how the spontaneous emission is formed inside the hydrogen-like atoms (HLAs) by constructing their vibrational Hamiltonian. Our model has already been implemented to derive the vibrational⁷ and rovibrational²³ Hamiltonians of diatomic molecules. Here, the vibrational Hamiltonian of the HLA (5) is defined as an infinite power series of Hamiltonian of a simple harmonic oscillator (SHO). The series coefficients have been determined by applying the first-order vibrational Hamiltonian (6) into the Heisenberg equations (7) and (8). The second-order vibrational Hamiltonian (12) clarified the final form of the vibrational motion equation of the HLA (16) that acts like a quantum harmonic oscillator in the absence of any damping process.

Many physicists have attempted to explain the SPE process by involving different external sources.^3–6 One of the premier models is the “Weisskopf–Wigner theory” in which the spontaneous emission is described by coupling the atom to the electromagnetic vacuum field.²⁴ By contrast, the origin of SPE radiation is here investigated by using the vibrational Hamiltonian of the atom (11) and the Langevin equation (20) in the absence of any interaction of an atom with an external source such as the electromagnetic vacuum field. The solutions of the Langevin equation (22) and (23) provide very useful information about the fluctuations that occur for the position $δx^(ω)$ and linear momentum $δp^(ω)$ of the oscillating electron in the HLA. These fluctuations are then implemented to calculate the internal noise fluxes of the potential N_U(ω) (34), kinetic N_K(ω) (35), and their equal interchange N_UK(ω) = N_KU(ω) (37). It is finally demonstrated that the internal noise fluxes are responsible for the external noise flux of spontaneous emission radiation according to the flux conservation relation (36). The sum of internal noise flux spectra gives a Lorentzian profile for the spontaneous emission spectrum with a bandwidth equal to Einstein’s A-coefficient, as illustrated in Fig. 2.

The data that support the findings of this study are available from the author upon reasonable request.