## Posts

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 $$\{ | b \rangle \}$$
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