Harmonic oscillator Hamiltonian operator

Evaluate fm H f ) if (a) H is the harmonic-oscillator Hamiltonian operator and f and f are harmonic-oscillator stationary-state wave functions with vibrational quantum numbers m and n (b) H is the particle-in-a-box H and / and f are particle-in-a-box energy eigenfunctions with quantum numbers m and n. 

Consider a system with a time-independent Hamiltonian H that involves parameters. An obvious example is the molecular electronic Hamiltonian (13.5), which depends parametrically on the nuclear coordinates. However, the Hamiltonian of any system contains parameters. For example, in the one-dimensional harmonic-oscillator Hamiltonian operator - f /2m) (f/dx ) + kx, the force constant is a parameter, as is the mass m. Although is a constant, we can consider it as a parameter also. The stationary-state energies E are functions of the same parameters as H. For example, for the harmonic oscillator... 

When the quantum mechanical Hamiltonian for vibration is constructed from Eq. (22.4-16) there are 3 — 5 or 3n — 6 terms, each one of which is a harmonic oscillator Hamiltonian operator. The variables can be separated, and the vibrational Schrodinger equation is solved by a vibrational wave function that is a product of 3 - 5 or 3n — 6 factors ... 

The first two terms on the right-hand side [17] are precisely those obtained from the standard harmonic oscillator Hamiltonian ( em) for the electromagnetic field. The evolution operator can then be written as... 

In this expression, according to the theory of the quantum harmonic oscillator, the operator q appearing on the right-hand side, may couple two successive eigenstates /c ) of the Hamiltonian of the harmonic oscillator. Consequently, by ignoring the scalar term p(0,0), which does not couple these states, we may write the dipole moment operator according to... 

Owing to Eq. (N.6), this last result shows that the action of the translation operator on the ground state of the quantum harmonic oscillator Hamiltonian, generates a coherent state ... 

The operator between the brackets is the harmonic oscillator Hamiltonian, which has the basis functions (112) as its eigenfunctions the remaining term is taken into account via Eq. (114). The rotational kinetic energy operator L(cuP) [Eq. (26)] can be written in terms of the shift operators J = J, + iJi, and the operator J, which act on the basis as... 

For future reference we cite here without proof a useful identity that involves the harmonic oscillator Hamiltonian H = p /2m + (1 /2)ma> q and an operator of the general formH = explaip + a2q with constant parameters ai and that is, the exponential of a linear combination of the momentum and coordinate operators. The identity, known as the Bloch theorem, states that the thermal average A )t (under the hannonic oscillator Hamiltonian) is related to the thennal average ((aip + Q 2 )2)t according to... 

In accord with Eqs. (108)-(109), the full vibrational Hamiltonian H will be decomposed into the sum of a zero-order Hamiltonian and a coupling operator W. In the expression for the zero-order Hamiltonian (Eq. (109)), / refers to one nondegenerate normal or local mode Qt or to a pair of normal modes (( , QIV) for a doubly degenerate mode. In the case of a nondegenerate mode, h" represents a Morse oscillator. For the doubly degenerate mode, H] is a 2D harmonic oscillator Hamiltonian (i.e., h" = +... 

We now need a systematic way to evaluate matrix elements like — Rgf v y. This is provided by the second quantization formulation [5] of the one-dimensional harmonic oscillator problem, which parallels in some ways the ladder operator treatment of angular momentum. The harmonic oscillator Hamiltonian is... 

The Hamiltonian operator of the system is a sum of four harmonic oscillator Hamiltonians ... 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

The Hamiltonian operator H for the harmonic oscillator is given in equation (4.12). The quantity c) is then determined as follows... 

The Hamiltonian operator for the unperturbed harmonic oscillator is given by equation (4.12) and its eigenvalues and eigenfunctions are shown in equations (4.30) and (4.41). The perturbation H is... 

Here a and are the usual oscillator creation and annihilation operators with bosonic commutation relations (73), and 0i,..., 1 ,..., 0Af) denotes a harmonic-oscillator eigenstate with a single quantum excitation in the mode n. According to Eq. (80a), the bosonic representation of the Hamiltonian (79) is given by... 

Or is the frequency of the harmonic oscilator and b) are boson (phonon) creation (annihilation) operators. In order to use the perturbation theory we have to split the Hamiltonian (16) onto the unperturbed part Hq and the perturbation H ... 

Hamiltonian operators, 88, 151, 153, 197 core, 204 commutation with O, 200, 218 invarianoe of, 151. harmonic foroe constants, 165. harmonic oscillator, approximation, 165 equation, 170. 

Now consider an ensemble of harmonic oscillators in three dimensions. Each of these harmonic oscillators has a different frequency oo = k c, their own Hamiltonian and raising and lowering operators... 

Hereafter we put /ig = 1. Below we express our results in terms of the statistical properties (correlators) of the environment s noise, X(t). Depending on the physical situation at hand, one can choose to model the environment via a bath of harmonic oscillators [6, 3]. In this case the generalized coordinate of the reservoir is defined as X = ]T)Awhere xi are the coordinate operators of the oscillators and Aj are the respective couplings. Eq. 2 is then referred to as the spin-boson Hamiltonian [8]. Another example of a reservoir could be a spin bath [11] 5. However, in our analysis below we do not specify the type of the environment. We will only assume that the reservoir gives rise to markovian evolution on the time scales of interest. More specifically, the evolution is markovian at time scales longer than a certain characteristic time rc, determined by the environment 6. We assume that rc is shorter than the dissipative time scales introduced by the environment, such as the dephasing or relaxation times and the inverse Lamb shift (the scale of the shortest of which we denote as Tdiss, tc [Pg.14]

A complete treatment of this derivation can be found in Ref. [19]. The first three terms in the kinetic energy operator indicates the presence of a 3D harmonic oscillator, and the final two terms indicate the presence of a 2D rotator (as for the hydrogen atom). A similar conclusion was made by Auberbach et al. [22] where they use a semi-classical quantisation method and the molecule is said to undergo a unimodal distortion and then the semi-classical Hamiltonian is found to be separated into two parts - a harmonic oscillator part with three vibrational coordinates and a rotational part with two rotational coordinates. However, more progress in terms of specifying the wavefunctions of the system can be made by following a different approach. 

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