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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, \[i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2 m} \frac{\partial^2 \psi}{\partial x^2} \,.\]

Momentum represnetation

In momentum representation, the Gaussian wave function reads \[\begin{align} \widetilde{\psi}(p,t) &\equiv \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{+\infty} dx \, e^{-i x p / \hbar} \psi(x,t) \\[0.1cm] &= \left( \frac{1}{2 \pi \hbar^2} \operatorname{Re} \frac{1}{\alpha_t} \right)^{1/4} \exp \left( -\frac{(p - p_0)^2}{4 \hbar^2 \alpha_t} - \frac{i}{\hbar} x_t (p - p_0) - i \frac{p_0^2 t}{2 \hbar m} - i \frac{p_0 x_0}{\hbar} - \frac{i}{2} \arg \alpha_0 \right) \,. \end{align}\]

Expectation value and uncertainty of position

The mean position of the particle is \[\overline{x} \equiv \int_{-\infty}^{+\infty} dx \, x |\psi(x,t)|^2 = x_t \,.\] The uncertainty in position is \[\Delta x \equiv \sqrt{ \int_{-\infty}^{+\infty}dx \, (x - \overline{x})^2 |\psi(x,t)|^2 } = \frac{1}{2 \sqrt{ \operatorname{Re} \alpha_t} } \,.\]

Expectation value and uncertainty of momentum

The mean momentum of the particle is \[\overline{p} \equiv \int_{-\infty}^{+\infty} dp \, p \left| \widetilde{\psi}(p,t) \right|^2 = p_0 \,.\] The uncertainty in momentum is \[\Delta p \equiv \sqrt{ \int_{-\infty}^{+\infty} dp \, (p - \overline{p})^2 \left| \widetilde{\psi}(p,t) \right|^2 } = \frac{\hbar |\alpha_0|}{\sqrt{\operatorname{Re} \alpha_0}} \,.\]

Expectation value and uncertainty of energy

The mean energy of the particle is \[\overline{E} \equiv \int_{-\infty}^{+\infty} dp \, \frac{p^2}{2 m} \left| \widetilde{\psi}(p,t) \right|^2 = \frac{p_0^2 + (\Delta p)^2}{2 m} \,.\] The uncertainty in energy is \[\Delta E \equiv \sqrt{ \int_{-\infty}^{+\infty} dp \, \left( \frac{p^2}{2 m} - \overline{E} \right)^2 \left| \widetilde{\psi}(p,t) \right|^2 } = \frac{(\Delta p)^2}{m} \sqrt{\frac{1}{2} + \left( \frac{p_0}{\Delta p} \right)^2} \,.\]