- #1

Simen

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## Homework Statement

A particle with charge e and mass m_0 is accelerated by a potential V to a relativistic speed.

Show that the de Broglie wavelength is:

λ = h / √(2 m_0 eV ) √(1+( eV / 2 m_0 c^2 ))

## Homework Equations

E = qV

λ = h/p

E = √(p^2 * c^2 + m^2 * c^4 )

p = (m_0 * v) / (1 - (v^2/c^2)

## The Attempt at a Solution

We do not have an expression for v, and so we use the formula:

E = √(p^2 * c^2 + m^2 * c^4 )

which gives us that:

E^2/c^2 = p^2 +m^2 *e^2

Which gives:

p = √(E^2/c^2 + m^2 * c^2)

Putting p into the expression for the de Broglie wavelenght gives us:

λ = h/ √(E^2 / c^2 + m^2*c^2)

We know that the potential energy of the particle in the Field before accelleration starts (and therefore the kinetic energy) is simply eV, and put that into the expression as well.

λ = h/ √(eV^2 / c^2 + m^2*c^2)

It has been a very long time since I last did any real maths, super basic algebra was sufficient for most of what I've done for the last two years. I am guessing the path to making the expressions the same lies in derivation, though why that should be an operation I have no idea.

I am well and truly lost on this, so if anyone have any insight whatsoever, I will be very grateful.

(Oh, and this is one of several attempts down different roads. I think this is the one that is closest to a proper solution, and also the one that is the least confusing for those of you who actually know this stuff...)

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