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Showing posts with the label One-dimensional systems

Harmonic oscillator with a time-dependent frequency

Consider a quantum particle in a harmonic trap with a time-dependent frequency \[\omega = \omega(t) \,.\] In atomic units (\(\hbar = m = 1\)), the Schrödinger equation describing this system reads \[i \dot{\psi} = -\frac{1}{2} \psi'' + \frac{1}{2} \omega^2 x^2 \psi \,,\] where \(\psi = \psi(x,t)\) is the particle's wave function, and \(\; '\) and \(\dot{}\) denote \(\frac{\partial}{\partial x}\) and \(\frac{\partial}{\partial t}\), respectively. A general treatment of this problem can be found, for instance, in Propagator for the general time-dependent harmonic oscillator with application to an ion trap . Here we only construct an example solution to the Schrödinger equation above. Let us look for solutions \(\psi\) in the form of a Gaussian wave packet centered at the origin: \[\psi = \left( \frac{2 \alpha}{\pi} \right)^{1/4} e^{-(\alpha + i \beta) x^2 + i \gamma} \,,\] where \(\alpha = \alpha(t)\), \(\beta = \beta(t)\), and \(\gamma = \gamma(t)\) are ye

Motion under a spatially uniform time-dependent force

Consider a quantum particle of mass \(m\) moving along a line, taken to be the \(x\) axis, under the action of a spatially uniform time-dependent force \(F(t)\). The time-evolution of the particle's wave function \(\Psi(x,t)\) is governed by the Schrödinger equation, \[i \hbar \frac{\partial \Psi(x,t)}{\partial t} = -\frac{\hbar^2}{2 m} \frac{\partial^2 \Psi(x,t)}{\partial x^2} - F(t) x \Psi(x,t) \,.\] The solution to this equation can be written as \[\Psi(x,t) = \exp \left[ i \frac{p(t) x - s(t)}{\hbar} \right] \Psi_0 \big( x-q(t), t \big) \,,\] where \(\Psi_0\) is the free-particle wave function that initially coincides with \(\Psi\), \[\Psi_0(x,0) = \Psi(x,0) \,,\] and the functions \(p(t)\), \(q(t)\), and \(s(t)\) are defined as \[p(t) = \int_0^t d\tau \, F(\tau) \,,\] \[q(t) = \frac{1}{m} \int_0^{t} d\tau \, p(\tau) = \frac{1}{m} \int_0^{t} d\tau \int_0^{\tau} d\tau' \, F(\tau') \,,\] \[s(t) = \frac{1}{2 m} \int_0^t d\tau \, p^2(\tau) = \frac{1}{2 m} \int_