Study of a Relativistic Quantum Harmonic Oscillator by the Method of State-Dependent Diagonalization

Lee, H. C.; Liu, K. L.; Lo, Chi‐Fai

doi:10.1142/s0129183196000545

Cited by 1 publication

(5 citation statements)

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“…The second one is the relativistic motion of the electron in stable Bohr orbits which acts like a harmonic oscillator. The kinetic energy of electron thus obeys the special relativistic relation in the usual form [18, 19]

T_{italicrel} = m_{e} c^{2} {((), 1 goodbreak+ \frac{p_{italicrel}^{2}}{m_{e}^{2} c^{4}})}^{1 / 2} - m_{e} c^{2} = \frac{p_{rel}^{2}}{2 m_{e}} + \frac{p_{rel}^{4}}{8 m_{e}^{3} c^{2}} + \dots,

where

m_{e}

is the rest mass of the electron and

p_{italicrel} = γ m_{e} v_{n}

is the relativistic linear momentum of the electron in Bohr circular orbits with constant speed

v_{n} = {italicze}^{2} / n ℏ = zαc / n

(

1 / 4 {italicπε}_{0} = 1

) and the Lorentz factor

γ = {((), 1 - z^{2} α^{2} / n^{2})}^{- 1 / 2}

[4, 7, 12]. As a result, the relativistic kinetic energy () can be approximated up to the fourth power

{((), zα)}^{4} / n^{4}

after substituting the expansions of

γ^{2}

and

γ^{4}

T_{italicrel} = \frac{1}{2} m_{e} c^{2}

…”

Section: Relativistic Kinetic Potential and Vibrational Energiesmentioning

confidence: 99%

“…The relativistic kinetic energy

E_{italickinetic}^{italicrel} (t)

is determined by substituting the relativistic linear momentum

< \overset{truê}{p} (t) >_{italicrel}

from () into the general relation of relativistic kinetic energy (). The relativistic potential energy

E_{italicpotential}^{italicrel} (t)

is defined by the usual relativistic relation

0.5 m_{e} {((), ω_{italicvib}^{italicrel})}^{2} < x (t) >_{italicrel}^{2}

[18] in which the physical role of two relativistic parameters

r_{n}^{italicrel}

and

ω_{italicvib}^{italicrel}

is reserved in (). It is worth noting to know that the research about finding an exact relation for the potential energy of a relativistic harmonic oscillator in the general case

γ > 1

is now under progress.…”

Section: Second‐order Rvh Trueĥitalicrel()2 and Energy Conservationmentioning

confidence: 99%

Section: Relativistic Kinetic Potential and Vibrational Energiesmentioning

confidence: 99%

“…The variations of ω rel 0 =ω 0 versus z is demonstrated in Figure 1. According to relation (18), the slope (α 2 z) of ω rel 0 =ω 0 slowly increases with z due to the infinitesimal value of the coefficient α 2 (α 2 < < 1). The plotted curve in Figure 1 shows an increase in ω rel 0 =ω 0 from 1 to 1.01 for lighter elements with 1 ≤ z ≤ 20 and a sharper increase from 1.01 to 1.27 for heavier elements with 20 < z ≤ 100.…”

Section: Relativistic Vibrational Hamiltonianmentioning

confidence: 99%

“…Finally, it is necessary to verify the accuracy of rotational and vibrational relations (1)-( 26) by demonstrating the energy conservation in the form [18] in which the physical role of two relativistic parameters r rel n and ω rel vib is reserved in (25). It is worth noting to know that the research about finding an exact relation for the potential energy of a relativistic oscillator in the general case γ > 1 is now under progress.…”

Section: Energy Conservationmentioning

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

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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.

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T_{italicrel} = m_{e} c^{2} {((), 1 goodbreak+ \frac{p_{italicrel}^{2}}{m_{e}^{2} c^{4}})}^{1 / 2} - m_{e} c^{2} = \frac{p_{rel}^{2}}{2 m_{e}} + \frac{p_{rel}^{4}}{8 m_{e}^{3} c^{2}} + \dots,

where

m_{e}

is the rest mass of the electron and

p_{italicrel} = γ m_{e} v_{n}

is the relativistic linear momentum of the electron in Bohr circular orbits with constant speed

v_{n} = {italicze}^{2} / n ℏ = zαc / n

(

1 / 4 {italicπε}_{0} = 1

) and the Lorentz factor

γ = {((), 1 - z^{2} α^{2} / n^{2})}^{- 1 / 2}

[4, 7, 12]. As a result, the relativistic kinetic energy () can be approximated up to the fourth power

{((), zα)}^{4} / n^{4}

after substituting the expansions of

γ^{2}

and

γ^{4}

T_{italicrel} = \frac{1}{2} m_{e} c^{2}

…”

Section: Relativistic Kinetic Potential and Vibrational Energiesmentioning

confidence: 99%

“…The relativistic kinetic energy

E_{italickinetic}^{italicrel} (t)

is determined by substituting the relativistic linear momentum

< \overset{truê}{p} (t) >_{italicrel}

from () into the general relation of relativistic kinetic energy (). The relativistic potential energy

E_{italicpotential}^{italicrel} (t)

is defined by the usual relativistic relation

0.5 m_{e} {((), ω_{italicvib}^{italicrel})}^{2} < x (t) >_{italicrel}^{2}

[18] in which the physical role of two relativistic parameters

r_{n}^{italicrel}

and

ω_{italicvib}^{italicrel}

is reserved in (). It is worth noting to know that the research about finding an exact relation for the potential energy of a relativistic harmonic oscillator in the general case