I came across this question from the book THE CONCEPTS OF MODERN PHYSICS(6thedit

Question

I came across this question from the book THE CONCEPTS OF MODERN PHYSICS(6thedition) by ARTHUR BEISER..... The chapter i am referring to is chapter no 3:WAVE PROPERTIES OF PARTICLES Q.By what percentage will a non relativisticcalculation of the de broglie wavelength of a 100 keV electron bein error? my doubt is regarding the non relativistic relation it refersto.how is the comparison being done...kindly help me through thisquestion. I came across this question from the book THE CONCEPTS OF MODERN PHYSICS(6thedition) by ARTHUR BEISER..... The chapter i am referring to is chapter no 3:WAVE PROPERTIES OF PARTICLES Q.By what percentage will a non relativisticcalculation of the de broglie wavelength of a 100 keV electron bein error? my doubt is regarding the non relativistic relation it refersto.how is the comparison being done...kindly help me through thisquestion.

Explanation / Answer

The wavelength of a body is called the de Broglie wavelength,which does NOT travel at the same velocity of the body, but ratherat the speed = c ² / v. That's right, it's always faster thanthe speed of light, even infinity when v = approaches 0. This iscalled the "phase velocity", which is distinct from the "groupvelocity", which is v, the velocity of the body. When you add agaussian spectrum of de Broglie wavelengths, you will find theresulting envelope the sum makes travels at v, while the averagevelocity of the phase waves will be c ² / v. When usingrelativistic equations, you must use the group velocity v, not thephase velocity.

The correct expression is f = c ² / v, not f =v. Using this, we have

= c ² / (v f)

Since relativistically, we have E = h f = (p c ²) / v, or

f = (p c ²) / (v h)

we make a substituton and arrive at:

= h / p
It gives the non-relativistic relation

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