De Borglie's wavelength equation

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In summary: If the obstacle is smaller than the wavelength of the wave, the wave will diffract and be bent more. If the obstacle is larger than the wavelength of the wave, the wave will be spread out and not diffracted. For waves to diffract, the opening (or distance from the obstacle) must be smaller than the wavelength of the wave. The width of a slit is approximately the wavelength of a wave.
  • #1
athymy
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Homework Statement



A molecule is 3 nanometers wide and weighs 5 x 10^-15kg. It is fired through a slit that is 5 nanometers wide. Approximately how slow does the molecule have to go so that it diffracts?

Homework Equations



I'm thinking that I could use De Borglie's wavelength equation: λ=h/m=h/mv and then solve for v but since I don't know that lambda is I don't see how I can use it.

Does anyone have a clue on how to solve this?
 
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  • #2


What is "diffraction'?
 
  • #3


voko said:
What is "diffraction'?

Quote from Wiki: "Diffraction refers to various phenomena which occur when a wave encounters an obstacle. In classical physics, the diffraction phenomenon is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings"
 
  • #4


The question is, is there any relation between the size of the obstacle and some characteristic of the wave for diffraction to become significant?
 
  • #5


voko said:
The question is, is there any relation between the size of the obstacle and some characteristic of the wave for diffraction to become significant?

If I remember correctly, doesn't the wavelength have to be pretty small in order for the particle to diffract? So could I try different values for v and see what my λ turns out to be?

I still don't understand what the slit width has to do with the question. Is it just there to show that the particle is able to go through it?
 
  • #6


You quoted Wiki: "spreading out of waves past small openings". How "small" must the opening be for spreading to become significant?
 
  • #7


Ahh, so the wavelength should be approximately the size of the slit right?
 
  • #8


Correct.
 
  • #9


voko said:
Correct.

Alright so since the width length 5nm, λ≈5nm or 5x10^-6m

So v=h/(5*10^(-15)*5x10^-6m)≈2.65*10^-14m/s

Is that reasonable? It seems like that is an extremely slow speed...
 
  • #10


Yes, it has to be slow, but not quite as slow. 1 nm is 10^-9 m, not 10^-6 m.
 
  • #11


voko said:
Yes, it has to be slow, but not quite as slow. 1 nm is 10^-9 m, not 10^-6 m.


Oh yeah, sorry about that. And thank you so much for the help, it was very informative.
 
  • #12


athymy said:
Alright so since the width length 5nm, λ≈5nm or 5x10^-6m

So v=h/(5*10^(-15)*5x10^-6m)≈2.65*10^-14m/s

Is that reasonable? It seems like that is an extremely slow speed...

Well, it was an extremely big molecule.

Think: water molecule, for example has molar mass of M=0.018 kg. The mass of one molecule is M/A (A is the Avogadro number, 6x1023) So the mass of a water molecule is about 3x10-26 kg.

ehild
 

FAQ: De Borglie's wavelength equation

What is De Borglie's wavelength equation?

De Borglie's wavelength equation is a fundamental equation in quantum mechanics that relates the wavelength of a particle to its momentum. It was proposed by French physicist Louis de Broglie in 1924 and is given by λ = h/mv, where λ is the wavelength, h is Planck's constant, m is the mass of the particle, and v is its velocity.

What is the significance of De Borglie's wavelength equation?

De Borglie's wavelength equation is significant because it demonstrates the wave-particle duality of matter, which is a fundamental concept in quantum mechanics. It shows that all particles, even those traditionally thought of as only having particle-like behavior, also exhibit wave-like properties.

How does De Borglie's wavelength equation relate to the Heisenberg uncertainty principle?

De Borglie's wavelength equation is closely related to the Heisenberg uncertainty principle, which states that it is impossible to know the exact position and momentum of a particle simultaneously. This principle is mathematically represented by the equation ΔxΔp ≥ h/4π, where Δx is the uncertainty in position and Δp is the uncertainty in momentum. De Borglie's wavelength equation shows that the wavelength of a particle is inversely proportional to its momentum, meaning that the more precisely we know the momentum of a particle, the less we know about its position and vice versa.

Can De Borglie's wavelength equation be applied to macroscopic objects?

No, De Borglie's wavelength equation is only applicable to microscopic particles, such as electrons and protons, due to the extremely small values of Planck's constant and the masses of these particles. For macroscopic objects, the wavelength would be too small to be measured and have any significant effect.

How does De Borglie's wavelength equation differ from other wave equations?

De Borglie's wavelength equation differs from other wave equations, such as the electromagnetic wave equation, in that it relates the wavelength of a particle to its momentum rather than its energy or frequency. This is because in quantum mechanics, particles are described by their momentum rather than their energy, unlike in classical physics. Additionally, De Borglie's wavelength equation is applicable to all particles, not just electromagnetic waves.

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