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

Jahanpanah, J.

doi:10.1063/5.0036017

Cited by 2 publications

(13 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The relativistic vibrational energy was then used to construct the relativistic vibrational Hamiltonian (RVH) in a way similar to deriving NRVH [13]. By applying the first‐order RVH to the Heisenberg equation, the normalized relativistic natural frequency

ω_{0}^{italicrel} / ω_{0}

was calculated for the electron oscillation in the lowest vibrational state of HLAs.…”

Section: Discussionmentioning

confidence: 99%

“…We have recently constructed the nonrelativistic vibrational Hamiltonian (NRVH) of HLAS in terms of the simple harmonic oscillator Hamiltonian

{\overset{H}{true}}_{0}

by expanding the corresponding energy eigenvalue

E_{italicvib} = - 0.5 m_{e} c^{2} {((), zα)}^{2} / n^{2}

around

n = 1

so that it is transformed to a power series in the form

E_{italicvib} = 0.5 μ c^{2} {((), zα)}^{2} {false\sum}_{k = 0}^{\infty} {((), - 1)}^{k + 1} (k + 1) {((), n - 1)}^{k}

(

μ \approx m_{e}

) [13]. Moreover, by ignoring the vacuum fluctuations, the infinite ladder energy spectrum of a harmonic oscillator can be written as

E_{n}^{0} = (n - 1) ℏ ω_{0}

.…”

Section: Relativistic Vibrational Hamiltonianmentioning

confidence: 99%

“…Moreover, by ignoring the vacuum fluctuations, the infinite ladder energy spectrum of a harmonic oscillator can be written as

E_{n}^{0} = (n - 1) ℏ ω_{0}

. Now if one substitutes the energy level

n

by the number operator

\overset{truê}{N}

in both energy eigenvalues

E_{italicvib}

and

E_{n}^{0}

, a power series is obtained for the corresponding Hamiltonians

{\overset{H}{true}}_{italicvib}

and

{\overset{H}{true}}_{0}

as [13]

{\overset{H}{true}}_{italicvib} = 0.5 ℏ ω_{0} {false\sum}_{k = 0}^{\infty} {((), goodbreak- 1)}^{k + 1} (k + 1) {((), \frac{{\overset{truê}{H}}_{0}}{ℏ ω_{0}})}^{k},

where

ω_{0} = {normalℏ}^{- 1} μ c^{2} {((), Zα)}^{2}

is the nonrelativistic natural frequency associated with the lowest energy level

n = 1

as given in relation () with

γ = 1

. The RVH

{\overset{H}{true}}_{italicvib}^{italicrel}

can now be derived by substituting the NRVH

{\overset{H}{true}}_{italicvib}

for the corresponding energy eigenvalue

E_{italicvib}

in relation () as…”

Section: Relativistic Vibrational Hamiltonianmentioning

confidence: 99%

“…The primary aim of the present paper is to introduce a new method for calculating the relativistic vibrational Hamiltonian (RHV)

H_{italicvib}^{italicrel}

of HLAs. This method has recently been exploited to derive the nonrelativistic vibrational Hamiltonian (NRVH)

{\overset{H}{true}}_{italicvib}

as a power series in

H_{0}

(the Hamiltonian of harmonic oscillator) [13]. The well‐known Langevin equation [14, 15] is derived by applying

{\overset{H}{true}}_{italicvib}

to the Heisenberg equation.…”

Section: Introductionmentioning

confidence: 99%

“…Accordingly, the relativistic energy eigenvalue E rel vib is derived by applying the principles of special relativity to the kinetic and potential energies of the electron. The RVH, H rel vib , is then constructed by substituting the number operator b N for the energy level n in the relativistic energy eigenvalue E rel vib with a similar approach to derive energy eigenvalue of a simple harmonic oscillator [13,17]. In these calculations, ω 0 is the natural frequency of harmonic oscillator that has been introduced by Equation (10) of Reference [13] as ω…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Relativistic ro‐vibrational feature of electron in Bohr's orbits of hydrogen‐like atoms in Heisenberg picture

Jahanpanah

Vahedi

Khosrojerdi

2022

Int J of Quantum Chemistry

Self Cite

View full text Add to dashboard Cite

The relativistic effects have been widely reported to affect the chemical and physical properties of the heavy elements such as lanthanides (57≤z≤71), actinides (89≤z≤103), and transactinides (z≥104). This effect is definitely weakened by reducing the atomic number z in lighter elements such as hydrogen‐like atoms (HLAs). The aim of present paper is to investigate the relativistic effects of electron motion in Bohr orbits on the chemical and physical properties of HLAs. The theoretical model is based on the Heisenberg (rather than Schrodinger) picture where the relativistic vibrational Hamiltonian (RVH) Hitalicvibitalicrel is expanded as a power series of harmonic oscillator Hamiltonian H0 for the first time. By applying the first‐order RVH (correct to H0) to the Heisenberg equation, a pair of coupled equations is obtained for the relativistic position and linear momentum of electron. A simple comparison of the first‐order relativistic and nonrelativistic equations reveals that the relativistic natural frequency of an HLA (like entropy) is slowly raised by increasing z beyond z≈20. In general, RVH plays a fundamental role because it specifies the temporal relativistic variations of position, velocity, and linear momentum of the oscillating electron. The results are finally verified by demonstrating energy conservation.

show abstract

ω_{0}^{italicrel} / ω_{0}

was calculated for the electron oscillation in the lowest vibrational state of HLAs.…”

Section: Discussionmentioning

confidence: 99%

“…We have recently constructed the nonrelativistic vibrational Hamiltonian (NRVH) of HLAS in terms of the simple harmonic oscillator Hamiltonian

{\overset{H}{true}}_{0}

by expanding the corresponding energy eigenvalue

E_{italicvib} = - 0.5 m_{e} c^{2} {((), zα)}^{2} / n^{2}

around

n = 1

so that it is transformed to a power series in the form

E_{italicvib} = 0.5 μ c^{2} {((), zα)}^{2} {false\sum}_{k = 0}^{\infty} {((), - 1)}^{k + 1} (k + 1) {((), n - 1)}^{k}

(

μ \approx m_{e}

) [13]. Moreover, by ignoring the vacuum fluctuations, the infinite ladder energy spectrum of a harmonic oscillator can be written as

E_{n}^{0} = (n - 1) ℏ ω_{0}

.…”

Section: Relativistic Vibrational Hamiltonianmentioning

confidence: 99%

“…Moreover, by ignoring the vacuum fluctuations, the infinite ladder energy spectrum of a harmonic oscillator can be written as

E_{n}^{0} = (n - 1) ℏ ω_{0}

. Now if one substitutes the energy level

n

by the number operator

\overset{truê}{N}

in both energy eigenvalues

E_{italicvib}

and

E_{n}^{0}

, a power series is obtained for the corresponding Hamiltonians

{\overset{H}{true}}_{italicvib}

and

{\overset{H}{true}}_{0}

as [13]

{\overset{H}{true}}_{italicvib} = 0.5 ℏ ω_{0} {false\sum}_{k = 0}^{\infty} {((), goodbreak- 1)}^{k + 1} (k + 1) {((), \frac{{\overset{truê}{H}}_{0}}{ℏ ω_{0}})}^{k},

where

ω_{0} = {normalℏ}^{- 1} μ c^{2} {((), Zα)}^{2}

is the nonrelativistic natural frequency associated with the lowest energy level

n = 1

as given in relation () with

γ = 1

. The RVH

{\overset{H}{true}}_{italicvib}^{italicrel}

can now be derived by substituting the NRVH

{\overset{H}{true}}_{italicvib}

for the corresponding energy eigenvalue

E_{italicvib}

in relation () as…”

Section: Relativistic Vibrational Hamiltonianmentioning

confidence: 99%

“…The primary aim of the present paper is to introduce a new method for calculating the relativistic vibrational Hamiltonian (RHV)

H_{italicvib}^{italicrel}

of HLAs. This method has recently been exploited to derive the nonrelativistic vibrational Hamiltonian (NRVH)

{\overset{H}{true}}_{italicvib}

as a power series in

H_{0}

(the Hamiltonian of harmonic oscillator) [13]. The well‐known Langevin equation [14, 15] is derived by applying

{\overset{H}{true}}_{italicvib}

to the Heisenberg equation.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Relativistic ro‐vibrational feature of electron in Bohr's orbits of hydrogen‐like atoms in Heisenberg picture

Jahanpanah

Vahedi

Khosrojerdi

2022

Int J of Quantum Chemistry

Self Cite

View full text Add to dashboard Cite

show abstract

Relativistic potential energy of a non-dissipative classical harmonic oscillator

Jahanpanah

2024

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

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

Cited by 2 publications

References 26 publications

Relativistic ro‐vibrational feature of electron in Bohr's orbits of hydrogen‐like atoms in Heisenberg picture

Relativistic ro‐vibrational feature of electron in Bohr's orbits of hydrogen‐like atoms in Heisenberg picture

Relativistic potential energy of a non-dissipative classical harmonic oscillator

Contact Info

Product

Resources

About