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Time-dependent mass Schrödinger equations; 3, Example 2

Michael Martin Nieto, D. Rodney Truax

Published Nov 23, 1999• 0 citations
OpenAlex

Abstract

We attack the specific time-dependent Hamiltonian problem ${H}=-\\lfrac{1}{2}({{t_o}\\over{t}})^a\\partial_{xx}+ time-dependent mass ($TM$) Schr\\"odinger equation. We give the specific transformations to a different time-dependent quadratic Schr\\"odinger equations ($TQ$) and to a different time-dependent oscillator ($TO$) equation. For each Schr\\"odinger system, we give the Lie algebra of space-time symmetries, the number states, the squeezed-state $<x>$ and $ $ (with their classical motion), $(\\Delta x)^2$, $(\\Delta p)^2$, 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.ForeachSchr"odingersystem,wegivetheLiealgebraofspace−timesymmetries,thenumberstates,thesqueezed−state) equation. For each Schr\\"odinger system, we give the Lie algebra of space-time symmetries, the number states, the squeezed-state )equation.ForeachSchr"odingersystem,wegivetheLiealgebraofspace−timesymmetries,thenumberstates,thesqueezed−state<x>andandand (withtheirclassicalmotion),(with their classical motion),(withtheirclassicalmotion),(\Delta x)^2,, ,(\Delta p)^2$, and the uncertainty product.
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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In this paper, we attack the specific time-dependent Hamiltonian problem H=−12eϒ(t−to)∂xx+12ω2e−ϒ(t−to)x2. This corresponds to a time-dependent mass (TM) Schrodinger equation. We give the specific transformations to (i) the more general quadratic (TQ) Schrodinger equation and to (ii) a different time-dependent oscillator (TO) equation. For each Schrodinger system, we give the Lie algebra of space-time symmetries, the number states, the coherent states, the squeezed-states, and the time-dependent 〈x〉, 〈p〉, (Δx)2, (Δp)2, and uncertainty product.
We present some general results for the time-dependent mass Hamiltonian problem with H=-{1/2}e^{-2ν}\partial_{xx} +h^{(2)}(t)e^{2ν}x^2. This Hamiltonian corresponds to a time-dependent mass (TM) Schrödinger equation with the restriction that there are only P^2 and X^2 terms. We give the specific transformations to a different quantum Schrödinger(TQ) equation and to a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space-time symmetries and (x,t) representations for number states, coherent states, and squeezed states. These general results include earlier work as special cases.
We present some general results for the time-dependent mass Hamiltonian problem with H=-½e-2ν(t)∂xx+h(2)(t)e2ν(t)x2, where ν(t) is a continuous function of t. This Hamiltonian corresponds to a time-dependent mass (TM) Schrödinger equation with the restriction that there are only P2 and X2 terms. We give the specific transformations to a different quadratic Schrödinger (TQ) equation and to a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space–time symmetries and (x,t) representations for number states, coherent states and squeezed states. These general results include earlier works as special cases.