WikiMechanics

The Change of Momentum due to Emission

The interaction is caused when $\mathbf{A}$ emits $\sf{X}$ at an event $\mathbf{A}_{\it{i}}$ which is called the initial event of the interaction

$\mathbf{A}_{\it{i}\sf{ -1}} \to \mathbf{A}_{\it{i}} + \sf{X}_{\it{i}}$

Momentum is conserved so a total over all momenta $\bar{p}$ are the same before and after the interaction

$\bar{p}_{\it{i}\sf{-1}}^{\mathbf{A}} = \bar{p}_{\it{i}} ^{ \mathbf{A}} + \bar{p}_{\it{i}} ^{ \sf{X}}$

And the change in $\mathbf{A}$'s momentum due to the emission of $\sf{X}$ is given by

$\Delta \bar{p}^{ \mathbf{A}} = \bar{p}_{\it{i}} ^{ \mathbf{A}} - \bar{p}_{\it{i}\sf{-1}}^{ \mathbf{A}}$

so that

The Change of Momentum due to Absorption

Particle $\sf{X}$ has an effect on $\mathbf{B}$ by being absorbed at event $\mathbf{B}_{\it{f}}$ which is called the final event of the interaction.

$\mathbf{B}_{\it{f}} + \sf{X}_{\it{f}} \to \mathbf{B}_{\it{f}\sf{+1}}$

Momentum is conserved so a total over all momenta $\bar{p}$ are the same before and after the interaction

$\bar{p}_{\it{f}} ^{ \mathbf{B}} + \bar{p}^{ \sf{X}}_{\it{f}} = \bar{p}_{\it{f}\sf{+1}} ^{ \mathbf{B}}$

And the change in $\mathbf{B}$'s momentum due to the emission of $\sf{X}$ is given by

$\Delta \bar{p}^{ \mathbf{B}} = \bar{p}_{\it{f}\sf{+1}} ^{ \mathbf{B}} - \bar{p}_{\it{f}} ^{ \mathbf{B}}$

so that

The Elapsed Time

The difference in the time of occurrence $t$ between initial and final events is

$\Delta t = t_{\it{f}} ^{\mathbf{B}} - t_{\it{i}} ^{\mathbf{A}}$

If the frame of reference is ideal and $\sf{X}$ is isolated (apart from the interaction under consideration) then the period $\hat{\tau}$ of $\sf{X}$ does not vary between events. So the elapsed time is given by

$\Delta t = \left( \, f-i \right) \hat{\tau} ^{\sf{X}}$

or in terms of the frequency $\nu$ of $\sf{X}$

$\begin{align} \Delta t = \frac { f-i }{ \nu ^{\sf{X}} } \end{align}$

Then using Planck's postulate gives the elapsed time in terms of the mechanical energy $E$ of $\sf{X}$ as

The Distance Between Emission and Absorption Events

The distance between initial and final events is

$\ell \equiv \left\| \, \bar{r} ^{\, \mathbf{B}}_{\it{f}} - \bar{r} ^{\, \mathbf{A}}_{\it{i}} \vphantom{\sum^{2}} \right\|$

If the frame of reference is ideal and $\sf{X}$ is isolated (apart from the interaction under consideration) then the wavelength $\lambda$ of $\sf{X}$ does not vary so that

$\ell = \left( \, f-i \right) \lambda ^{\sf{X}}$

Then using de Broglie's expression for the wavelength in terms of the momentum $p$ of $\sf{X}$ gives