The quantum damped harmonic oscillator

Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

“…In many applications as radio-frequency ion traps [1][2][3][4][5][6][7][8], quantum optics [9][10][11][12], cosmology [13,14], quantum field theory [15], quantum dissipation [16][17][18][19][20][21][22], magneto transport in lateral heterostructures [23][24][25][26] and even gravitational waves [27] the time evolution of particles in quadratic potentials is frequently examined. The one-dimensional, generalized time-dependent quadratic Hamiltonian is given bŷ H = a 1 (t) + a 2 (t)x + a 3 (t)p + a 4 (t)x 2 + a 5 (t)p 2 + a 6 (t) (xp +px) ,…”

“…Aside from the simple harmonic oscillator, a large number of interesting systems arise from this Hamiltonian as the linear potential [28,29], the driven harmonic oscillator [30,31], Kanai-Caldirola Hamiltonians [16][17][18][19][20][21], and time dependent harmonic oscillators i.e. an oscillator with time-varying frequency [21,32,33].…”

On the Canonical Transformation of Time-Dependent Harmonic Oscillator

“…The result of (10) shows that the cancellation of the differential in (4) generates the Lagrangian describing the harmonic oscillator of the system. Another condition of (4) is the cancellation of quadratic term in co-ordinate and these yields…”

“…The QCT is also used in solving time-dependent frequency [3]. However, the solutions of time-dependent harmonic oscillator have been obtained through various methods including invariant operator [8], Path Integral [9,10], and the space-time transformation [11,12]. We write the Schrödinger equation to be solved as…”

A concise quantum mechanical treatment of the forced damped harmonic oscillator

“…Several techniques, such as the invariant operator method, the propagator method, the unitary transformation method and so on, are used for the dissipative systems [1][2][3][4][5][6][7].…”