## Posts

### Wigner function: States

Consider a quantum particle moving along the $$x$$ axis. Let the particle's state be represented by a density operator $$\rho$$. If the particle is in a pure state $$| \psi \rangle$$, then $$\rho = | \psi \rangle \langle \psi |$$. The particle's state can also be described in phase space, with position and momentum variables $$x$$ and $$p$$, by means of an appropriate quasi-probability distribution . The Wigner function $$W(x,p)$$ is one of the most prominent choices. Definition The Wigner function is defined as $W(x,p) = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dx' \, e^{-i p x' / \hbar} \langle x + \tfrac{1}{2} x' |\rho | x - \tfrac{1}{2} x' \rangle \,.$ Equivalently, it can be written as an integral over momentum: $W(x,p) = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dp' \, e^{i p' x / \hbar} \langle p + \tfrac{1}{2} p' |\rho | p - \tfrac{1}{2} p' \rangle \,.$ Proof: Making use of the completeness relation

### Chebyshev polynomial expansion of the time-evolution operator

Here we outline an efficient numerical method of solving the time-dependent Schrödinger equation, $i \hbar \frac{\partial | \psi(t) \rangle}{\partial t} = H | \psi(t) \rangle \,.$ The method is based on the Chebyshev polynomial expansion of the time-evolution operator. Only the basic idea is presented in these notes. Further details on the method can be found, for instance, in the following papers: An accurate and efficient scheme for propagating the time dependent Schrödinger equation Unified framework for numerical methods to solve the time-dependent Maxwell equations (in particular, section 3.3) The Hamiltonian $$H$$ is assumed to have no explicit dependence on time. For instance, if the system under consideration is a particle of mass $$m$$ moving in the presence of a static potential $$V(x)$$, then $\psi(x,t) = \langle x | \psi(t) \rangle$ is the particle's wave function, and $H = -\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2} + V(x) \,.$ In what