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Free particle: Long time evolution

Consider a free quantum particle of mass \(m\) moving in an \(n\)-dimensional space. Let \(\Psi(\boldsymbol{r}, t)\) be the particle's wave function, with \(\boldsymbol{r} \in \mathbb{R}^n\) and \(t \ge 0\) denoting position and time, respectively. Suppose that initially, at \(t = 0\), the wave function \(\Psi(\boldsymbol{r},0)\) is localized around \(\boldsymbol{r} = 0\). Then, at long times (\(t \to \infty\)), the wave function is approximately given by \[\Psi(\boldsymbol{r},t) \simeq \left( \frac{m}{2 \pi i \hbar t} \right)^{n/2} \exp \left( \frac{i m }{2 \hbar t} |\boldsymbol{r}|^2 \right) \Phi \left( \frac{m \boldsymbol{r}}{t} \right) \,,\] where \[\Phi(\boldsymbol{p}) = \int_{\mathbb{R}^n} d^n\boldsymbol{r} \, \Psi(\boldsymbol{r},0) e^{-i \boldsymbol{p} \cdot \boldsymbol{r} / \hbar}\] is the initial wave function in momentum space. Proof: Using the free-particle propagator \[K(\boldsymbol{x},t) = \left( \frac{m}{2 \pi i \hbar t} \right)^{n/2} \exp \left( \frac{

Wigner function: Observables

Let a (pure or mixed) quantum state of a one-dimensional particle be represented by a Wigner function \(W(x,p)\). Here, \(x\) and \(p\) are the particle's position and momentum, respectively. Density operator The density operator \(\rho\) representing the state can be expressed in terms of the Wigner function: \[\rho = \int_{-\infty}^{+\infty} dx \int_{-\infty}^{+\infty} dx' \int_{-\infty}^{+\infty} dp \, | x' \rangle \, e^{i p (x' - x) / \hbar} W \left( \frac{x + x'}{2}, p \right) \langle x | \,.\] Proof: According to the definition of the Wigner function, \[W(x,p) = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dx' \, e^{-i p x' / \hbar} \langle x + \tfrac{1}{2} x' | \rho | x - \tfrac{1}{2} x' \rangle \,.\] Hence, \[\begin{align} \int_{-\infty}^{+\infty} dp \, e^{i p \xi' / \hbar} W(\xi,p) &= \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} d\xi'' \, \langle \xi + \tfrac{1}{2} \xi'' |\rho | \xi - \tfrac{1}{2}

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_