De-Broglie equation for other particles except photon

Question

If E=hf is applicable for electron and other particles, the De Broglie wavelength should be λ=hv/pc. Because, mc^2=hf which implies mc^2=hc/λ which implies m=h/λc and thus λ=hv/pc. But I have found in my text book that λ=h/p is applicable not only for photon but also for all particle. But how can λ=h/p=h/mv be applicable for all particle?

Vadim · Answer

The De Broglie relations
$$
E=hnu, p=frac{h}{lambda}
$$
are applicable to all particles.
However the relation between the frequency/energy and the wave length/momentum, is not the same. Thus, for photons $f=frac{c}{lambda}$, for non-relativistic electrons $E=frac{p^2}{2m}$, for relativistic electrons $E^2=m^2c^4 + p^2c^2$.

Alon Shoshan · Answer

For your last step you just introduce a v in both the numerator and denominator. That is meaningless as it is unity.
In the expression $lambda = frac{h}{mc}$, you just have to set $c=v$ to account for particles with different momenta and thus different velocities, not equal to the speed of light.
Do it in this way:
$$E = hf = mc^2 $$
$$frac{hc}{lambda} = mc^2$$
$$frac{h}{lambda} = mc$$
$$lambda = frac{h}{mc}$$
$$lambda = frac{h}{mv}$$

De-Broglie equation for other particles except photon

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