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Chapter 38: Q9Q (page 1180)

(a) If you double the kinetic energy of a nonrelativistic particle, how does its de Broglie wavelength change? (b) What if you double the speed of the particle?

Short Answer

Expert verified

a) The wavelength decreases by a factor of $\frac{1}{\sqrt{2}}$ .

b) The wavelength is decreases by a factor $\frac{1}{2}$ .

Step by step solution

01

Describe the de Broglie wavelength

The de Broglie wavelength is given by,

$位 = \frac{h}{p}$

Here, h is the Planck鈥檚 constant, and p is the momentum of the particle.

02

Find the change in de Broglie wavelength when the kinetic energy is doubled

(a)

The momentum of the particle and kinetic energy are related as follows.

$p = \sqrt{2 m E}$

It is known that $位 = \frac{h}{\sqrt{2 m E}}$ therefore,

$位鈭� \frac{1}{\sqrt{E}} \frac{位_{1}}{位_{2}} = \sqrt{\frac{E_{2}}{E_{1}}}$

Here, $E_{2} = 2 E_{1}$ and $位_{1} = 位$ .

$位_{2} = 位_{1} \sqrt{\frac{E_{1}}{E_{2}}} = 位 \sqrt{\frac{E_{1}}{2 E_{1}}} = \frac{位}{\sqrt{2}}$
Therefore, the wavelength decreases by a factor of $\frac{1}{\sqrt{2}}$ .

03

Find the change in de Broglie wavelength when the speed of the particle is doubled

(b)

The momentum of the particle is given by,

$p = m v$

Here, v is velocity of the particle.

The wavelength of the particle is given by,

$位 = \frac{h}{m v} 位鈭� \frac{1}{v} \frac{位_{1}}{位_{2}} = \frac{v_{2}}{v_{1}}$

Here, $v_{2} = 2 v_{1}$ , and $位_{1} = 位$ .

$位_{2} = 位_{1} \frac{v_{1}}{v_{2}} = \frac{v_{1}}{2 v_{1}} 位 = \frac{位}{2}$

Therefore, the wavelength is decreases by a factor $\frac{1}{2}$ .

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Most popular questions from this chapter

The highest achievable resolving power of a microscope is limited only by the wavelength used; that is, the smallest item that can be distinguished has dimensions about equal to the wavelength. Suppose one wishes to 鈥渟ee鈥� inside an atom. Assuming the atom to have a diameter of $100pm$ , this means that one must be able to resolve a width of, say, $10 pm$ .

(a) If an electron microscope is used, what minimum photon energy is required?

(b) If a light microscope is used, what minimum photon energy is required?

(c) Which microscope seems more practical? Why?

What are (a) the energy of a photon corresponding to wavelength $1 .00 nm$ , (b) The kinetic energy of an electron with de Broglie wavelength $1 .00 nm$ , (c) the energy of a photon corresponding to wavelength $1 .00 fm$ , and (d) the kinetic energy of an electron with de Broglie wavelength $1 .00 fm$ ?

If the de Broglie wavelength of a proton is $100fm$ , (a) what is the speed of the proton and (b) through what electric potential would the proton have to be accelerated to acquire this speed?

X rays of wavelength 0.0100 nm are directed in the positive direction of an axis onto a target containing loosely bound electrons. For Compton scattering from one of those electrons, at an angle of $180 掳$ ,what are (a) the Compton shift, (b) the corresponding change in photon energy, (c) the kinetic energy of the recoiling electron, and (d) the angle between the positive direction of the axis and the electron鈥檚 direction of motion?

Calculate the de Broglie wavelength of (a) a $1.00keV$ electron, (b) a $1.00keV$ photon, and (c) a $1.00keV$ neutron.

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