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Using the definition of the de Broglie wavelength, explain how wavelike properties of matter increase with a decrease in mass or decrease in speed. Use as examples an electron (mass = 9.11 × 10 –31 kg) with a speed of 5.0 × 10 6 m/s and a proton (mass = 1.67 × 10 –27 kg) with a speed of 8.0 × 10 6 m/s.

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Test prep for ap courses

In a Davisson-Germer type of experiment, a crystal with a parallel-plane separation ( d ) of 9.1 × 10 –2 nm produces constructive interference with an electron beam at an angle of θ = 50°. Which of the following is the maximum de Broglie wavelength for these electrons?

  1. 0.07nm
  2. 0.09 nm
  3. 0.14 nm
  4. 0.21 nm

(c)

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In a Davisson-Germer experiment, electrons with a speed of 6.5 × 10 6 m/s exhibit third-order ( n = 3) constructive interference for a crystal with unknown plane separation, d . Given an angle of incidence of θ = 45°, compute the value for d . Compare the de Broglie wavelength to electromagnetic radiation with the same wavelength. (Recall that the mass of the electron is 9.11 × 10 –31 kg.)

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Section summary

  • Particles of matter also have a wavelength, called the de Broglie wavelength, given by λ = h p size 12{λ = { {h} over {p} } } {} , where p size 12{p} {} is momentum.
  • Matter is found to have the same interference characteristics as any other wave.

Conceptual questions

How does the interference of water waves differ from the interference of electrons? How are they analogous?

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Describe one type of evidence for the wave nature of matter.

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Describe one type of evidence for the particle nature of EM radiation.

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Problems&Exercises

At what velocity will an electron have a wavelength of 1.00 m?

7.28 × 10 –4 m size 12{7 "." "28" times "10" rSup { size 8{–4} } " m"} {}

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What is the wavelength of an electron moving at 3.00% of the speed of light?

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At what velocity does a proton have a 6.00-fm wavelength (about the size of a nucleus)? Assume the proton is nonrelativistic. (1 femtometer = 10 15 m. size 12{"10" rSup { size 8{ - "15"} } " m"} {} )

6.62 × 10 7 m/s size 12{6 "." "62" times "10" rSup { size 8{7} } " m/s"} {}

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What is the velocity of a 0.400-kg billiard ball if its wavelength is 7.50 cm (large enough for it to interfere with other billiard balls)?

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Find the wavelength of a proton moving at 1.00% of the speed of light.

1.32 × 10 –13 m size 12{6 "." "62" times "10" rSup { size 8{7} } " m/s"} {}

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Experiments are performed with ultracold neutrons having velocities as small as 1.00 m/s. (a) What is the wavelength of such a neutron? (b) What is its kinetic energy in eV?

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(a) Find the velocity of a neutron that has a 6.00-fm wavelength (about the size of a nucleus). Assume the neutron is nonrelativistic. (b) What is the neutron’s kinetic energy in MeV?

(a) 6.62 × 10 7 m/s size 12{6 "." "62" times "10" rSup { size 8{7} } " m/s"} {}

(b) 22.9 MeV

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What is the wavelength of an electron accelerated through a 30.0-kV potential, as in a TV tube?

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What is the kinetic energy of an electron in a TEM having a 0.0100-nm wavelength?

15.1 keV

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(a) Calculate the velocity of an electron that has a wavelength of 1 . 00 μm. size 12{1 "." "00 μm"} {} (b) Through what voltage must the electron be accelerated to have this velocity?

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The velocity of a proton emerging from a Van de Graaff accelerator is 25.0% of the speed of light. (a) What is the proton’s wavelength? (b) What is its kinetic energy, assuming it is nonrelativistic? (c) What was the equivalent voltage through which it was accelerated?

(a) 5.29 fm

(b) 4 . 70 × 10 12 J size 12{4 "." "70" times "10" rSup { size 8{ - "12"} } " J"} {}

(c) 29.4 MV

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The kinetic energy of an electron accelerated in an x-ray tube is 100 keV. Assuming it is nonrelativistic, what is its wavelength?

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Unreasonable Results

(a) Assuming it is nonrelativistic, calculate the velocity of an electron with a 0.100-fm wavelength (small enough to detect details of a nucleus). (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

(a) 7.28 × 10 12 m/s size 12{7 "." "28" times "10" rSup { size 8{"12"} } " m/s"} {}

(b) This is thousands of times the speed of light (an impossibility).

(c) The assumption that the electron is non-relativistic is unreasonable at this wavelength.

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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cm
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Can you compute that for me. Ty
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what is chemistry
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what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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what's motion
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what are the types of wave
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answer
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progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Reofrir Reply
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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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