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Chapter 4: Q32E (page 136)

Nonrelativistically, the energyEof a free massive particle is just kinetic energy, and its momentumis. of course,mv. Combining these with fundamental relationships (4-4) and (4-5), derive a formula relating (a) particle momentumto matter-wave frequency fand (b) particle energyEto the wavelengthλof a matter wave.

Short Answer

Expert verified

a) The derivative formula by relating particle momentum to matter-wave frequency is p=2mhf.

b) The derivative formula by relating particle energy to the wavelength of a matter wave is E=h22mλ2

Step by step solution

01

Relation of wave-particle momentum and energy.

The following equations describe the fundamental wave-particle momentum andenergy relationships, respectively.

p=hλ…………………(1)

E=hf……………….. (2)

02

Relationship between momentum and frequency.

The relationship between momentum and frequency is described by an equation.

The energy can be expressed asE=p22m

Putting this expression and equation(2), we have:

E=p22m

E=hf

p22m=hf

03

Derivative equation for  

(a)

The equation forpis then as follows:

p22m=hf

p2=2mhf

p=2mhf

As a result, the equation for momentum and frequency is p=2mhf.

04

Relation between energy and wavelength.

After that, we create an equation that connects energy and wavelength.

We know that,E=p22m.

Now, relate this to the equation (1), then the equation is as follows:

E=p22m

p=hλ

.

05

Derivative equation for energy-wavelength(b)

The equation forEis then as follows:

E=h22mλ2

As a result, the energy-wavelength equation is E=h22mλ2

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Most popular questions from this chapter

A crack between two walls is 10 cmwide. What is the angular width of the central diffraction maximum when

(a) an electron moving at 50 m/spasses through?

(b) A baseball of mass0.145 kg and speed 50 m/spasses through?

(c) In each case, an uncertainty in momentum is introduced by the “experiment” (i.e., passing through the slit). Specifically, what aspect of the momentum becomes uncertain, and how does this uncertainty compare with the initial momentum of each?

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