Journal of Mathematical Physics

The quantum Lane-Emden-type Kanai-Caldirola oscillators

Vagner Henrique Loiola Bessa, I. Guedes

Published Dec 1, 2012• 6 citations

Abstract

We obtain the exact wave functions of two Lane-Emden-type Kanai-Caldirola (LE-KC) oscillators by using the dynamical invariant method. To do so, we analytically solve the respective Milne-Pinney equation for each oscillator. We also calculate the uncertainty product, the transition probability and the quantum-mechanical energy expectation value for each oscillator. We compare the results with those of the well-known KC oscillator. The quantum-mechanical energy expectation value of the KC oscillator goes to zero faster than that of the LE-KC oscillators, indicating that the KC system is more damped than the LE-KC ones.

DOI: 10.1063/1.4768702

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Journal of Mathematical Physics

The quantum Lane-Emden-type Kanai-Caldirola oscillators

Vagner Henrique Loiola Bessa, I. Guedes

Published Dec 1, 2012• 6 citations

Publisher

Abstract

DOI: 10.1063/1.4768702

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m(t)=m_0frac{t}{t_0}

, (II)

m(t)=m_0

and (III)

m(t)=m_0ajust{frac{t}{t_0}}^2

. In all three cases

omega(t)=omega_0frac{t_0}{t}

. For class (b) we study the Caldirola-Kanai oscillator (IV)where

omega(t)=omega_0

and

m(t)=m_0 ext{exp}ajust{gamma t}

and the oscillator with

omega(t)=omega_0

and

m(t)=m_0ajust{1+frac{t}{t_0}}^alpha

, for

alpha=2

(V) and

alpha=4

(VI). To obtain the classical solution for each oscillator we solve the respective equation of motion and analyze the behavior of

q(t)

p(t)

as well as the phase diagram

q(t)

p(t)

. To obtain the quantum solutions we use a unitary transformation and the Lewis and Riesenfeld quantum invariant method. The wave functions obtained are written in terms of a function ($ ho$) which is solution of the Milne-Pinney equation. Futhermore, for each system we solve the respective Milne-Pinney equation and discuss how the uncertainty product evolves with time.