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### Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

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# (a) Find the wavelength of a proton whose kinetic energy is equal 10 its integral energy. (b) ' The proton is usually regarded as being roughly of radius ${\mathbf{10}}^{\mathbf{-}\mathbf{15}}\mathbf{m}$. Would this proton behave as a wave or as a particle?

(a) The wavelength of a proton that has kinetic energy equal to its internal energy is $7.63×{10}^{-16}\mathrm{m}$

(b) The moving proton would behave like as a particle in nature.

See the step by step solution

## Step 1: Given data

$\mathrm{Mass}\mathrm{of}\mathrm{proton},m=1.67×{10}^{-27}\mathrm{kg}\phantom{\rule{0ex}{0ex}}\mathrm{Speed}\mathrm{of}\mathrm{light},c=3.0×{10}^{8}\mathrm{m}/\mathrm{s}\phantom{\rule{0ex}{0ex}}\mathrm{Plank}\text{'}\mathrm{s}\mathrm{constant},h=6.63×{10}^{-34}\mathrm{J}.\mathrm{s}$

## Step 2: Relativistic effect in de Broglie's equation

The internal energy E of an object is

${\mathbit{E}}{\mathbf{=}}{\mathbit{m}}{{\mathbit{c}}}^{{\mathbf{2}}}$

The equation for the kinetic energy ( KE) of an object traveling at relativistic velocities is

${\mathbit{K}}{\mathbit{E}}{\mathbf{=}}{\mathbf{\left(}}{{\mathbit{\gamma }}}_{{\mathbf{u}}}{\mathbf{-}}{\mathbf{1}}{\mathbf{\right)}}{\mathbit{m}}{{\mathbit{c}}}^{{\mathbf{2}}}$

Where, m is rest mass, and c is the speed of light.

De Broglie's wavelength

${\mathbit{\lambda }}{\mathbf{=}}\frac{\mathbf{h}}{\mathbf{p}}$

Where, p is the momentum.

Relativistic effect in de Broglie's equation

${\mathbit{\lambda }}{\mathbf{=}}\frac{\mathbf{h}}{{\mathbf{\gamma }}_{\mathbf{u}}\mathbf{m}\mathbf{v}}$

Where, is the velocity at which the protons kinetic energy equals its internal energy, and ${\mathbit{\gamma }}$ is Lorentz factor.

Speed at which the internal energy become equal to the kinetic energy

${\mathbit{m}}{{\mathbit{c}}}^{{\mathbf{2}}}{\mathbf{=}}\left({\gamma }_{u}-1\right)\phantom{\rule{0ex}{0ex}}{\mathbf{1}}{\mathbf{=}}{{\mathbit{\gamma }}}_{{\mathbf{u}}}{\mathbf{-}}{\mathbf{1}}\phantom{\rule{0ex}{0ex}}{\mathbf{2}}{\mathbf{=}}{{\mathbit{\gamma }}}_{{\mathbf{u}}}$

Lorentz factor is given by a relation,

## Step 3: Substitute the value of velocity in wavelength,

(a)

Substitute the value of velocity in wavelength $\lambda$,

The wavelength of a proton that has kinetic energy equal to its internal energy is given by,

Substitute $6.63×{10}^{-34}\mathrm{J}.\mathrm{s}\mathrm{for}\mathrm{h},1.67×{10}^{-27}\mathrm{kg}\mathrm{for}\mathrm{m}\mathrm{and}3.0×{10}^{8}\mathrm{m}/\mathrm{s}$for c in the above equation to solve for $\lambda$

Hence, the wavelength of a proton that has kinetic energy equal to its internal energy is $\lambda =7.63×{10}^{-16}\mathrm{m}$

## Step 4: Explain proton behavior

(b)

The wavelength of a proton that has kinetic energy equal to its internal energy is $\mathrm{\lambda }=7.63×{10}^{-16}\mathrm{m}$

As the wavelength of the proton is smaller than the roughly size of the proton .

$\left({10}^{-15}\mathrm{m}\right)$So, the moving proton would behave like as a particle in nature.

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