Is there a significant error in using De Broglie's equation for an electron at really high speed? I

Annabel Sullivan

Annabel Sullivan

Answered question

2022-05-08

Is there a significant error in using De Broglie's equation for an electron at really high speed?
I was wondering if using the De Broglie equation
λ = h p
for object traveling at really high speeds would result in a significant error. For example if an object travelled at 0.02 c would the error be negligible? How can I calculate the uncertainty in the result?

Answer & Explanation

priffEmipsy4i37m

priffEmipsy4i37m

Beginner2022-05-09Added 17 answers

The formula is still correct for speeds close to c, provided you use the relativistic definition of momentum for p
Using natural units ( = c = 1), the wave function of a particle of momentum p is given by
ψ ( x , t ) = ψ 0 exp ( i E t + i p x ) .
The formula is the same, both in non-relativistic and relativistic mechanics. The only difference is the definition of energy, which in relativity is
E = m 2 + p 2 .
Also the dependence of momentum on speed changes. Since you are interested in a particle traveling slower than c, it must be massive, and the momentum is
p = m γ v .
The ratio of the relativistic and nonrelativistic wavelengths is just
λ r e l λ n r = p n r p r e l = 1 γ = 1 v 2 1 v 2 2
So, for v = 0.02, the relativistic wavelength would be around 0.04 % smaller than the non-relativistic one.

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