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Showing posts with the label One-dimensional systems

Every purely attractive potential in 1D has at least one bound state

Every purely attractive one-dimensional potential well, no matter how shallow, has at least one bound state. More precisely, this statement can be formulated as follows. Consider a one-dimensional non-relativistic quantum particle of mass \(m\) inside a purely attractive well potential \(V(x)\): the function \(V(x)\) is such that \(V(x) < 0\) for all \(x\), and \(V(x) \to 0\) as \(x \to \pm \infty\). This system has at least one bound state. That is, the ground state energy \(E_{\text{ground}}\) of the Hamiltonian \[H = -\frac{\hbar^2}{2 m} \frac{d^2}{d x^2} + V(x)\] is strictly negative: \[E_{\text{ground}} < 0 \,.\] Proof: [D. ter Haar, Selected Problems in Quantum Mechanics (Academic, New York, 1964).] According to the variational principle, \[E_{\text{ground}} \le \langle \psi | H | \psi \rangle = \int_{-\infty}^{\infty} dx \, \psi^*(x) H \psi(x)\] for any normalized state \(\psi(x)\). Hence, in order to prove the negativity of \(E_{\text{ground}}\),

Free-particle Gaussian wave packets

Below are basic properties of quantum Gaussian packets describing the motion of a free particle on a line (taken to be the \(x\) axis). Position representation A time-dependent Gaussian wave function, in its most general form, is given by \[\psi(x,t) = \left( \frac{2 \operatorname{Re} \alpha_t}{\pi} \right)^{1/4} \exp \left( -\alpha_t (x - x_t)^2 + \frac{i}{\hbar} p_0 (x - x_t) + i \frac{p_0^2 t}{2 \hbar m} + \frac{i}{2} \arg \frac{\alpha_t}{\alpha_0} \right) \,,\] where \[x_t = x_0 + \frac{p_0 t}{m} \,, \qquad \alpha_t = \frac{\alpha_0}{1 + i \frac{2 \hbar t}{m} \alpha_0} \,.\] Here, \(m\) is the particle's mass, \(x_0\) and \(p_0\) are real-valued parameters, and \(\alpha_0\) is a complex-valued parameter, such that \(\operatorname{Re} \alpha_0 > 0\). The wave function \(\psi\) is normalized, \[\int_{-\infty}^{+\infty} dx \, |\psi(x,t)|^2 = 1 \,,\] and satisfies the free-particle Schrödinger equation, \[i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{

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_