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arXiv:quant-ph/9911095

The Schrödinger System H=−12(tot)a∂xx+12ω2(tto)bX2

Michael Martin Nieto, D. Rodney Truax

Published Aug 25, 2001• 2 citations
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Abstract

Abstract We attack the specific time-dependent Hamiltonian problem H=− 1 2 ( t o t ) a ∂ xx + 1 2 ω 2 ( t t o ) b x 2 . This corresponds to a time-dependent mass ( TM ) Schrodinger equation. We give the specific transformations to a different time-dependent quadratic Schrodinger equation ( TQ ) and to a different time-dependent oscillator ( TO ) equation. For each Schrodinger system, we give the Lie algebra of space-time symmetries, the number states, the squeezed-state 〈 x 〉 and 〈 p 〉 (with their classical motion), (Δ x ) 2 , (Δ p ) 2 , and the uncertainty product.

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We attack the specific time-dependent Hamiltonian problem H=−lfrac12(toovert)apartialxx+time−dependentmass({H}=-\\lfrac{1}{2}({{t_o}\\over{t}})^a\\partial_{xx}+ time-dependent mass (H=−lfrac12(to​overt)apartialxx​+time−dependentmass(TM)Schr"odingerequation.Wegivethespecifictransformationstoadifferenttime−dependentquadraticSchr"odingerequations() Schr\\"odinger equation. We give the specific transformations to a different time-dependent quadratic Schr\\"odinger equations ()Schr"odingerequation.Wegivethespecifictransformationstoadifferenttime−dependentquadraticSchr"odingerequations(TQ)andtoadifferenttime−dependentoscillator() and to a different time-dependent oscillator ()andtoadifferenttime−dependentoscillator(TO$) equation. For each Schr\\"odinger system, we give the Lie algebra of space-time symmetries, the number states, the squeezed-state <x><x><x> and $ $ (with their classical motion), (Deltax)2(\\Delta x)^2(Deltax)2, (Deltap)2(\\Delta p)^2(Deltap)2, and the uncertainty product.
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