De Broglie wavelength (relativistic e-)

In summary, De Broglie's postulate states that the relationship between wavelength (λ) and momentum (p) is valid for relativistic particles. When applied to an electron with a kinetic energy of 3.00 MeV, this equation results in a wavelength of 3.58E-13 m. The equation E=hc/λ, which is an approximation for extremely relativistic speeds, is not valid for this problem. Instead, the equation E2=(mc2)2+(pc)2 should be used, where E is the total energy, including both kinetic and rest mass energy.
  • #1
Jules18
102
0
Wavelength of an electron

Homework Statement



De Broglie postulated that the relationship λ=h/p is valid for relativistic particles. What is the de Broglie wavelength for a (relativistic) electron whose kinetic energy is 3.00 MeV?

-Electron has 3.00 MeV (or 4.8*10^-13 Joules)
-it's relativistic
-finding λ.

Homework Equations



h=6.63*10^-34

λ=h/p (obviously)

And I'm not sure if they're needed, but the relativistic eq's are:

KE = mc^2/sqrt(1-(v/c)^2)
p = mv/sqrt(1-(v/c)^2)

I'm not sure if this one applies to relativistic speeds:

E = hc/λ

The Attempt at a Solution



Attempt 1:

E = hc/λ

4.8E-13 = (6.63E-34)(3E8)/λ
λ = (6.63E-34)(3E8)/(4.8E-13)
λ = 4.14E-13 m

BUT answer key says 3.58E-13

If you could help, that would be great.
Sorry if it's too long, and I'm a little unfamiliar with relativistic eqn's so forgive me if I screwed up on them.
 
Last edited:
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  • #2


Jules18 said:

Homework Equations



h=6.63*10^-34

λ=h/p (obviously)

And I'm not sure if they're needed, but the relativistic eq's are:

KE = mc^2/sqrt(1-(v/c)^2)
p = mv/sqrt(1-(v/c)^2)
Actually, this "KE" expresson is giving the total energy, kinetic + rest mass energy, so

KE + mc2 = mc2/sqrt(1-(v/c)2)

I'm not sure if this one applies to relativistic speeds:

E = hc/λ
That's an approximation that applies at extremely relativistic speeds (say v>0.99c), and is strictly true only when v=c, i.e. for photons and other massless particles.

Since this is a moderately relativistic situation, E=hc/λ is not valid.

You could try using the KE + mc2 equation instead, but many problems like this one make use of this:
E2 = (mc2)2 + (pc)2​
where, again, E is the total energy,
E = KE + mc2​
 
  • #3


Wait, I just realized this is close to an extreme relativistic situation.

Jules18 said:

The Attempt at a Solution



Attempt 1:

E = hc/λ

Yes, that will work. However, E is the total energy, kinetic + rest mass energy. Just using the kinetic energy for E is wrong.
 
  • #4
oookay that makes a lot more sense. Thanks so much, redbelly. :)
 

Related to De Broglie wavelength (relativistic e-)

1. What is the De Broglie wavelength of a relativistic electron?

The De Broglie wavelength of a relativistic electron is given by the equation λ = h/mv, where h is Planck's constant, m is the mass of the electron, and v is its velocity. This wavelength represents the quantum mechanical wavelength associated with the electron's motion.

2. How does the De Broglie wavelength change with increasing velocity?

The De Broglie wavelength decreases as the velocity of the electron increases. This is because as the electron's velocity approaches the speed of light, its mass increases and thus its momentum increases, resulting in a shorter De Broglie wavelength.

3. Can the De Broglie wavelength of a relativistic electron be observed?

Yes, the De Broglie wavelength of a relativistic electron can be observed using modern experimental techniques. For example, in electron diffraction experiments, the interference pattern observed is directly related to the wavelength of the electrons passing through the diffraction grating.

4. How does the De Broglie wavelength relate to the uncertainty principle?

The De Broglie wavelength is one of the fundamental principles of quantum mechanics and is closely related to the uncertainty principle. The uncertainty principle states that the position and momentum of a particle cannot be known simultaneously with absolute certainty. The De Broglie wavelength is a manifestation of this principle, as it represents the uncertainty in the momentum of a particle due to its wave-like nature.

5. Can the De Broglie wavelength be applied to particles other than electrons?

Yes, the De Broglie wavelength can be applied to all particles, not just electrons. This includes larger particles such as atoms and molecules, as well as subatomic particles like protons and neutrons. However, the De Broglie wavelength is most commonly associated with electrons due to their wave-like behavior and small mass.

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