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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

Two-dimensional hydrogen atom

Consider a particle of mass \(M\) moving in the \(xy\) plane in the presence of the Coulomb potential \[V = -\frac{\alpha}{\sqrt{x^2 + y^2}} \qquad (\alpha > 0) \,.\] Bound states \(\psi\) and energy lelvels \(E < 0\) are determined by the time-independent Schrödinger equation \[-\frac{\hbar^2}{2 M} \nabla^2 \psi + V \psi = E \psi \,.\] In polar coordinates (\(x = r \cos \theta\), \(y = r \sin \theta\)), the equation reads \[-\frac{\hbar^2}{2 M} \left( \frac{\partial^2}{\partial r^2} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} \right) \psi - \frac{\alpha}{r} \psi = E \psi \,.\] Using separation of variables, along with the condition that \(\psi\) is single-valued, we get \[\psi(r,\theta) = R(r) e^{i m \theta} \qquad (m \in \mathbb{Z}) \,,\] where the radial wave function \(R(r)\) satisfies \[\frac{d^2 R}{d r^2} + \frac{1}{r} \frac{d R}{d r} + \left( \frac{2 M E}{\hbar^2} + \frac{2 M \alpha}{\hbar^2 r} - \frac{m^2}{r^2}

Madelung equations

Consider a quantum particle of mass \(m\) moving along a line, the \(x\) axis, in the presence of a potential \(V(x,t)\). The time-evolution of the particle's wave function \(\Psi(x,t)\) is governed by the Schrödinger equation, \[i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2 m} \frac{\partial^2 \Psi}{\partial x^2} + V \Psi \,.\] The Schrödinger equation can be written in a hydrodynamic form, known as the Madelung equations: \[\frac{\partial \rho}{\partial t} + \frac{\partial (\rho v)}{\partial x} = 0 \,,\] \[\frac{\partial v}{\partial t} + v \frac{\partial v}{\partial x} + \frac{1}{m} \frac{\partial (V + Q)}{\partial x} = 0 \,.\] Here, \(\rho(x,t)\) is the probability density associated with \(\Psi\), \[\rho = |\Psi|^2 \,,\] \(v(x,t)\) is the velocity field associated with the flow of \(\rho\), \[\rho v = \frac{\hbar}{m} \operatorname{Im} \left\{ \Psi^* \frac{\partial \Psi}{\partial x} \right\} \,,\] and \(Q(x,t)\) is the Bohm quantum potential , \[Q = -\f

Density operator at thermal equilibrium

The density operator of a quantum system in thermal equilibrium at temperature \(T\) is given by \[\rho_T = \frac{e^{-H / k_{\text{B}} T}}{Z} \,,\] where \(H\) is the system Hamiltonian, \(k_{\text{B}}\) is the Boltzmann constant, and \[Z = \operatorname{Tr} e^{-H / k_{\text{B}} T}\] is the partition function. This expression for the density matrix can be obtained by extremizing the von Neumann entropy \[S = -k_{\text{B}} \operatorname{Tr} \rho \ln \rho\] over all states \(\rho\) of the same mean energy \[E = \operatorname{Tr} \rho H \,.\] Proof: We want to extremize \(S\) subject to (i) the normalization constraint, \(\operatorname{Tr} \rho = 1\), and (ii) the mean energy constraint, \(\operatorname{Tr} \rho H = E\). To this end, we introduce two Lagrange multipliers, \(\alpha k_{\text{B}}\) and \(\beta k_{\text{B}}\), and perform unconstrained extremization of \[\mathcal{F} = -k_{\text{B}} \operatorname{Tr} \rho \ln \rho - \alpha k_{\text{B}} \left( \operatorname{Tr}

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_

Density operator of a subsystem

Let us consider a closed quantum system \(S\) consisting of two subsystems \(S_1\) and \(S_2\). The Hilbert space \(\mathcal{H}\) of system \(S\) is a tensor product of Hilbert spaces \(\mathcal{H}_1\) and \(\mathcal{H}_2\) of subsystems \(S_1\) and \(S_2\), respectively, i.e., \[\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2 \,.\] Let \(\{ | a \rangle \}\) and \(\{ | b \rangle \}\) be complete orthonormal bases in \(\mathcal{H}_1\) and \(\mathcal{H}_2\), respectively, so that \[\langle a | a' \rangle = \delta_{aa'} \,, \qquad \langle b | b' \rangle = \delta_{bb'} \,,\] and any state \(| \Psi \rangle\) of \(S\) can be written as \[| \Psi \rangle = \sum_{a,b} \Psi_{ab} | a b \rangle \,, \qquad | a b \rangle \equiv | a \rangle \otimes | b \rangle \,.\] We take \(| \Psi \rangle\) to be normalized to one: \[\langle \Psi | \Psi \rangle = \sum_{a,b} |\Psi_{ab}|^2 = 1 \,.\] For notational simplicity, we treat sets \(\{ | a \rangle \}\) and \(\{ | b \rangle \}\)

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