Homework Help: De Broglie wavelength (relativistic e-)

1. Jun 9, 2009

Jules18

Wavelength of an electron

1. The problem statement, all variables and given/known data

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

2. Relevant 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

3. 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

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: Jun 9, 2009
2. Jun 9, 2009

Redbelly98

Staff Emeritus
Re: Wavelength of an electron

Actually, this "KE" expresson is giving the total energy, kinetic + rest mass energy, so

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

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. Jun 9, 2009

Redbelly98

Staff Emeritus
Re: Wavelength of an electron

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

Yes, that will work. However, E is the total energy, kinetic + rest mass energy. Just using the kinetic energy for E is wrong.

4. Jun 9, 2009

Jules18

oookay that makes a lot more sense. Thanks so much, redbelly. :)