91影视

This integral can be thought of as the averagevalue that would be obtained from a large number of measurements. It could also be thought of as the average position value for a large number of particles described by the same wavefunction.

The expectation value of the position is the integral of the wave function squared ${\mathit{蠄}}^{\mathbf{*}}\mathit{蠄}$multiplied by the position, X.

The wave function of the particle is$蠄\left(x\right)$.

$\mathbf{蠄}\left(x\right)\mathbf{=}\{\begin{array}{l}2\sqrt{a^3{\mathrm{xe}}^{-\mathrm{ax}}}X>0\\ 0X<0\end{array}$

To find the expectation value of the position of the particle.

The expectation value of the position is the integral of the wave function squared, multiplied by the position, X.

$\begin{array}{rcl}\mathbf{鉄�}\mathbf{x}\mathbf{鉄�}& \mathbf{=}& {\mathbf{鈭�}}_{\mathbf{-}\mathbf{鈭�}}^{\mathbf{鈭�}}\mathbf{唯}\mathbf{路}\mathbf{x}\mathbf{路}\mathbf{唯诲虫}\\ & \mathbf{=}& {\mathbf{鈭�}}_{\mathbf{-}\mathbf{鈭�}}^{\mathbf{0}}\mathbf{唯}\mathbf{路}\mathbf{x}\mathbf{路}\mathbf{唯诲虫}\mathbf{+}{\mathbf{鈭�}}_{\mathbf{0}}^{\mathbf{鈭�}}\mathbf{唯}\mathbf{路}\mathbf{x}\mathbf{路}\mathbf{唯诲虫}\\ & \mathbf{=}& \mathbf{0}\mathbf{+}{\mathbf{鈭�}}_{\mathbf{0}}^{\mathbf{鈭�}}\mathbf{唯}\mathbf{路}\mathbf{x}\mathbf{路}\mathbf{唯诲虫}\\ & \mathbf{=}& {\mathbf{鈭�}}_{\mathbf{0}}^{\mathbf{鈭�}}{\left(2\sqrt{a^3}{\mathrm{xe}}^{-\mathrm{ax}}\right)}^{\mathbf{2}}\mathbf{xdx}\\ & \mathbf{=}& {\mathbf{鈭�}}_{\mathbf{0}}^{\mathbf{鈭�}}\mathbf{4}{\mathbf{a}}^{\mathbf{3}}{\mathbf{x}}^{\mathbf{3}}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{ax}}\mathbf{dx}\\ & \mathbf{=}& \mathbf{4}{\mathbf{a}}^{\mathbf{3}}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{ax}}{\left[-\frac{x^3}{2a}-\frac{3x^2}{4a^2}-\frac{6x}{8a^3}-\frac{6}{16a^4}\right]}_{\mathbf{0}}^{\mathbf{鈭�}}\\ & \mathbf{=}& \mathbf{0}\mathbf{-}\left[4a^3\left(-\frac{3}{8a^4}\right)\right]\\ & \mathbf{=}& \frac{\mathbf{3}}{\mathbf{2}\mathbf{a}}\end{array}$

Hence, the expectation value of the position of the particle is .