Question

Show that${\left|\psi \right|}^{\mathbf{2}}\mathbf{=}{\left|\psi \right|}^{\mathbf{2}}$ , with related as in Eq. 38-14. That is, show that the probability density does not depend on the time variable.

Short answer

The value 1 is does not depend on the time variable that shows that the probability density does not depend on the time variable.

Step-by-step solution

Determine the concept and the formula

In the year 1900, derived a formula that shows the relationship between all the wavelength and all the temperatures and that is given as Equation 38-14 in book and also known as the Planck’s radiation law

$S\left(\lambda \right)=\frac{2\pi c^{2}h}{{\lambda }^5}\frac{1}{e^{{\displaystyle \raisebox{1ex}{$hc$}\!\left/ \!\raisebox{-1ex}{$\lambda Kt$}\right.}}-1}$

Here, h is the Planck constant, k is the Boltzmann constant, and T is the temperature of the radiating surface (in kelvins), $\lambda$is wavelength S is thermal radiation.

Determine the derivation as:

By referring the equation number 38-14 solve for the derivation as:

$\begin{array}{rcl}{\left|\psi \right|}^2& =& {\left|e^{ikx}\right|}^2\\ {\left|\psi \right|}^2& =& \left(e^{ikx}\right)*\left(e^{ikx}\right)=e^{-ikx}{e}^{ikx}\\ {\left|\psi \right|}^2& =& e^0=1\\ {\left|\psi \right|}^2& =& {\left|\psi \right|}^{\text{2}}\end{array}$

The value 1 is does not depend on the time variable that shows that the probability density does not depend on the time variable.