De Broglie wavelength (relativistic e-)

[Jules18]

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.

 

[Redbelly98]

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​

 

[Redbelly98]

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

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.

 

[Jules18]

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