## Posts

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 $$E_{\text{ground}}$$,

### 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_