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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}{

Two-dimensional hydrogen atom as a harmonic oscillator

Bound state energy levels \(E<0\) of a two-dimensional hydrogen atom are determined by the Schrödinger equation \[\left( -\frac{\hbar^2}{2 m} (\partial_x^2 + \partial_y^2) - \frac{q^2}{\sqrt{x^2 + y^2}} \right) \psi = E \psi \,,\] where \(m\) and \(q\) are the mass and charge of the electron, respectively, and \(\psi(x,y)\) is the electronic wave function. This equation can be solved in terms of Kummer's confluent hypergeometric function (see this post for details). Here we show how the two-dimensional hydrogen atom can be mapped onto a two-dimensional harmonic oscillator. This mapping has been discussed, e.g., in Quantum Mechanics of H-Atom from Path Integrals . We begin by making a coordinate transformation from \((x,y)\) to \((u,v)\) defined by \[\begin{align} x &= u^2 - v^2 , \\ y &= 2 u v \,. \end{align}\] From \[\begin{pmatrix} \partial_u \\ \partial_v \end{pmatrix} = J \begin{pmatrix} \partial_x \\ \partial_y \end{pmatrix}\] with \[J = \begin{pmatrix

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

Schrödinger equation in momentum space

Consider a quantum particle of mass \(m\) moving in an \(n\)-dimensional space in the presence of an external (scalar) potential \(V(\boldsymbol{x})\). The Hamiltonian governing the motion is \[H = \frac{\boldsymbol{P} \cdot \boldsymbol{P}}{2 m} + V(\boldsymbol{X}) \,,\] where \(\boldsymbol{X}\) and \(\boldsymbol{P}\) are the position and momentum operators, respectively. The time-dependent state of the particle \(| \Psi_t \rangle\) satisfies the Schrödinger equation \[i \hbar \frac{\partial | \Psi_t \rangle}{\partial t} = H | \Psi_t \rangle \,.\] In position representation, we have \[\boldsymbol{X} = \boldsymbol{x} \,, \qquad \boldsymbol{P} = -i \hbar \frac{\partial}{\partial \boldsymbol{x}} \,,\] and so the Schrödinger equation has the following familiar form: \[i \hbar \frac{\partial \psi(\boldsymbol{x},t)}{\partial t} = -\frac{\hbar^2}{2 m} \frac{\partial}{\partial \boldsymbol{x}} \cdot \frac{\partial \psi(\boldsymbol{x},t)}{\partial \boldsymbol{x}} + V(\boldsymbol{x}) \

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