<i>S</i>-Matrix Solution for the Forced Harmonic Oscillator

Fuller, R. W.; Harris, S.; Slaggie, E.L.

doi:10.1119/1.1969575

Cited by 22 publications

(17 citation statements)

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“…The projectile is assumed to couple linearly with the surface atom coordinate q ,

V (q) = - F (t) q

, along some pre‐defined trajectory which determines the force F ( t ) (

F (t) \to 0

when

t \to \pm \infty

for a scattering trajectory). The resulting model is analytically solvable and is known as the Forced Oscillator Model . Its Hamiltonian takes the simple form

H = ℏ ω_{S} true(a^{†} a + \frac{1}{2} true) - ℏ f (t) (a^{†} + a) = H_{0} + V (t) V (t) = - ℏ f (t) (a^{†} + a)

where

a^{†} = \frac{q}{2 Δ q} - i \frac{p}{2 Δ p}

(

a = \frac{q}{2 Δ q} + i \frac{p}{2 Δ p}

) is the usual raising (lowering) operator,

ℏ f (t) = F (t) Δ q

is the scaled force and the fundamental widths are

Δ q = \sqrt{ℏ / 2 m_{S} ω_{S}}

and

Δ p = ℏ / 2 Δ q

.…”

Section: Energy Transfermentioning

confidence: 99%

Classical and quantum dynamics at surfaces: Basic concepts from simple models

Bonfanti

Martinazzo

2016

Int J of Quantum Chemistry

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Elementary processes involving atomic and molecular species at surfaces are reviewed. The emphasis is on simple classical and quantum models that help to single out unifying dynamical themes and to identify the basic physical mechanisms that underlie the rich variety of phenomena of surface chemistry. Starting from an elementary description of the energy transfer between a gas‐phase species and a surface—for both classical and quantum lattices—the key processes establishing the formation of an adsorbed phase (sticking, diffusion and vibrational relaxation) are discussed. This is instrumental for introducing the simplest chemical transformations involving adsorbed species and/or scattering of gas‐phase molecules: Langmuir–Hinshelwood, Hot‐Atom, and Eley–Rideal reactions forming complex molecules from elementary constituents, and dissociative chemisorption of molecules into smaller fragments. Applications are also provided illustrating the ideas developed along the way at work in real‐world gas‐surface problems.

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“…The projectile is assumed to couple linearly with the surface atom coordinate q ,

V (q) = - F (t) q

, along some pre‐defined trajectory which determines the force F ( t ) (

F (t) \to 0

when

t \to \pm \infty

for a scattering trajectory). The resulting model is analytically solvable and is known as the Forced Oscillator Model . Its Hamiltonian takes the simple form

H = ℏ ω_{S} true(a^{†} a + \frac{1}{2} true) - ℏ f (t) (a^{†} + a) = H_{0} + V (t) V (t) = - ℏ f (t) (a^{†} + a)

where

a^{†} = \frac{q}{2 Δ q} - i \frac{p}{2 Δ p}

(

a = \frac{q}{2 Δ q} + i \frac{p}{2 Δ p}

) is the usual raising (lowering) operator,

ℏ f (t) = F (t) Δ q

is the scaled force and the fundamental widths are

Δ q = \sqrt{ℏ / 2 m_{S} ω_{S}}

and

Δ p = ℏ / 2 Δ q

.…”

Section: Energy Transfermentioning

confidence: 99%

Classical and quantum dynamics at surfaces: Basic concepts from simple models

Bonfanti

Martinazzo

2016

Int J of Quantum Chemistry

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“…It has been shown by many authors using very different approaches, such as the Green's function method, 1,2 Wick's theorem, 3,4 the canonical transformation method, 6 and nested commutator expansion, 7 that for the particular interaction ͑5͒ and the scattering boundary conditions ͑10͒, the operator S I takes the form:…”

Section: Problem and Notationmentioning

confidence: 99%

“…which is a standard expression that has been used as a point of departure in calculations of the transition probabilities. [2][3][4] Although these calculations are instructive in their own right, the simplest and most direct way to proceed is to expand the operators e ϪiG ϩ and e ϪiG Ϫ in Eq. ͑18͒ in a power series.…”

Section: Probability Amplitudes For State-to-state Transitions Omentioning

confidence: 99%

“…Many authors have used this model as a testing ground for nearly all the many-body techniques developed during that time. [1][2][3][4][5] In addition, a number of realistic problems can be fruitfully treated within this simple model [5][6][7][8][9][10][11][12][13][14][15][16] or its generalizations. 7,[17][18][19][20][21] The present work revisits the same problem once more with a different agenda.…”

Section: Introduction: the Forced Oscillator Modelmentioning

confidence: 99%

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Perturbations of the excited quantum oscillator: From number states to statistical distributions

Sunko

Gumhalter

2004

American Journal of Physics

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We discuss the transitions that an external time-dependent perturbation can induce upon a quantum harmonic oscillator in an excited initial state. In particular, we show how to describe transitions of the oscillator from initial states characterized by statistical distributions. These results should be useful for interpretations of the properties of weakly dispersive bosonic excitations in quantum systems whose dynamics is investigated by time or energy resolved spectroscopies.

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“…In this case the energy transfer can be calculated in the first approximation by considering that the unperturbed (recoilless) motion of the particle along the classical trajectory acts as an external time dependent perturbation on the dynamical degrees of freedom of the target. This approach, often termed the semiclassical trajectory approximation (SCTA), has been followed in a number of works on inelastic scattering [15,[25][26][27][28] due to its attractiveness which stems from the fact that there exists a nonperturbative closed form solution to the problem of linear coupling of an external perturbation to harmonic vibrations [15,17,22,[29][30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%