Book edition 2nd Edition

Author(s) Randy Harris

Pages 633 pages

ISBN 9780805303087

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Short Answer

(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 $10^{- 15} m$ . 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 $7.63 \times 10^{- 16} m$

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

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given data

$Mass of proton, m = 1.67 \times 10^{- 27} kg Speed of light, c = 3.0 \times 10^{8} m / s Plank' s constant, h = 6.63 \times 10^{- 34} J . s$

Step 2: Relativistic effect in de Broglie's equation

The internal energy E of an object is

$E = m c^{2}$

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

$K E = (γ_{u} - 1) m c^{2}$

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

De Broglie's wavelength

$λ = \frac{h}{p}$

Where, p is the momentum.

Relativistic effect in de Broglie's equation

$λ = \frac{h}{γ_{u} m v}$

Where, is the velocity at which the protons kinetic energy equals its internal energy, and $γ$ is Lorentz factor.

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

$m c^{2} = (γ_{u} - 1) 1 = γ_{u} - 1 2 = γ_{u}$

Lorentz factor is given by a relation,

$γ_{u} = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} 2 = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} \sqrt{1 - \frac{v^{2}}{c^{2}}} = \frac{1}{2} 1 - \frac{v^{2}}{c^{2}} = \frac{1}{4} \frac{v^{2}}{c^{2}} = \frac{3}{4} v = \frac{\sqrt{3}}{2} c$

Step 3: Substitute the value of velocity in wavelength,

(a)

Substitute the value of velocity in wavelength $λ$ ,

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

$λ = \frac{h}{γ m (\frac{\sqrt{3 c}}{2})}$

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

$λ = \frac{6.63 \times 10^{- 34} J . s}{(2) (1.67 \times 10^{- 27} kg) (\sqrt{\frac{3 \times 3.0 \times 10^{8} m / s}{2}}} = 7.63 \times 10^{- 16} m$

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

Step 4: Explain proton behavior

(b)

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

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

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

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Modern Physics

Answers without the blur.

Short Answer

(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 $10^{- 15} m$ . Would this proton behave as a wave or as a particle?

Step by Step Solution

Step 1: Given data

Step 2: Relativistic effect in de Broglie's equation

Step 3: Substitute the value of velocity in wavelength,

Step 4: Explain proton behavior

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Modern Physics

Answers without the blur.

Short Answer

(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 10-15m. Would this proton behave as a wave or as a particle?

Step by Step Solution

Step 1: Given data

Step 2: Relativistic effect in de Broglie's equation

Step 3: Substitute the value of velocity in wavelength,

Step 4: Explain proton behavior

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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 $10^{- 15} m$ . Would this proton behave as a wave or as a particle?