From equations (6-23) and (6-29) obtain the dispersion coefficient for matter waves (in vacuum), then show that probability density (6-35) follows from (6-28)

The relation between angular frequency and the wave number of matter wave is termed as the Dispersion relation

Write the expression of matter wave dispersion relation from (6–23).

$\mathrm{\omega }\left(\mathrm{k}\right)=\frac{{\mathrm{hk}}^2}{2\mathrm{m}}$

Here, $\omega$is angular frequency of matter wave, k is wave number and m is mass of corresponding particle.

Write the expression of coefficient of dispersion from (6–29).

$\mathrm{D}=\frac{{\mathrm{d}}^2\mathrm{\omega }\left(\mathrm{k}\right)}{{\mathrm{dk}}^2}$

Substitute $\frac{{\mathrm{hk}}^2}{2\mathrm{m}}$for $\omega \left(k\right)$in the above expression.

$\begin{array}{rcl}D& =& \frac{d^2\left(\frac{h{k}^2}{2m}\right)}{d{k}^2}\\ & =& \frac{h}{m}\end{array}$

Write the expression of probability density from (6–28).

$\left|\mathrm{\Psi }\right(\mathrm{x},\mathrm{t}){|}^2=\frac{{\mathrm{c}}^2}{{\left(1+\frac{{\mathrm{D}}^2{\mathrm{t}}^2}{4{\mathrm{\epsilon }}^4}\right)}^{\frac{1}{2}}}\exp \left(\frac{-{(\mathrm{x}-\mathrm{st})}^2}{2{\mathrm{\epsilon }}^2\left(1+\frac{{\mathrm{D}}^2{\mathrm{t}}^2}{4{\mathrm{\epsilon }}^4}\right)}\right)$

Here, C is amplitude of wave function and t is any arbitrary time.

Substitute$\frac{h}{m}$for D in above expression.

$\begin{array}{rcl}\left|\mathrm{\Psi }\right(\mathrm{x},\mathrm{t}){|}^2& =& \frac{{\mathrm{c}}^2}{{\left(1+\frac{{\left(\frac{\mathrm{h}}{\mathrm{m}}\right)}^2{\mathrm{t}}^2}{4{\mathrm{\epsilon }}^4}\right)}^{\frac{1}{2}}}\exp \left(\frac{-{(\mathrm{x}-\mathrm{st})}^2}{2{\mathrm{\epsilon }}^2\left(1+\frac{{\left(\frac{\mathrm{h}}{\mathrm{m}}\right)}^2{\mathrm{t}}^2}{4{\mathrm{\epsilon }}^4}\right)}\right)\\ & =& \frac{{\mathrm{C}}^2}{{\left(1+\frac{{\mathrm{h}}^2{\mathrm{t}}^2}{4{\mathrm{m}}^2{\mathrm{\epsilon }}^4}\right)}^{1/2}}\exp \left(\frac{-{(\mathrm{x}-\mathrm{st})}^2}{2{\mathrm{\epsilon }}^2\left(1+\frac{{\mathrm{h}}^2{\mathrm{t}}^2}{4{\mathrm{m}}^2{\mathrm{\epsilon }}^4}\right)}\right)\end{array}$

It can be observed that the above expression resembles equation (6-35) therefore it is verified that it follows from equation (6-28).

Therefore, the dispersion coefficient of matter waves in vacuum is $\frac{h}{m}$and it is verified that probability density is followed from equation (6–28).