Please explain to me the derivition of de broglie wavelength of a moving body?

Question

The derivition is confusing.
E= mc^2= hf
Wavelength * f= speed
on equating, we get
wavelength = h/mc
then c is replaced by Velocity of the body.
Isn't equating the mass energy relationship, with frequency energy relationship, wrong. Please explain in detail.
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How can this be done?

Answer 1

Okay, this is actually an excellent question, because this concerns a fact that is rarely mentioned in introductory texts on relativity and quantum mechanics. The wavelength of a body is called the de Broglie wavelength, which does NOT travel at the same velocity of the body, but rather at the speed = c ² / v. That's right, it's always faster than the speed of light, even infinity when v = approaches 0. This is called the "phase velocity", which is distinct from the "group velocity", which is v, the velocity of the body. When you add a gaussian spectrum of de Broglie wavelengths, you will find the resulting envelope the sum makes travels at v, while the average velocity of the phase waves will be c ² / v. When using relativistic equations, you must use the group velocity v, not the phase velocity.

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

λ = 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:

λ = h / p

which is the correct relativistic expression for the de Broglie wavelength.

For a quick look at the correct formula for the de Broglie wavelength of an object, either relativistic or non-relativistic, check the Wolfram site:

Answer 2

Okay, let's consider some real life situations:

Let's say we want to work out the debroglie wavelength of a car (let's say the car is 1000 kg). Let's assume the car is travelling at 10 metres per second. If we use the formula for debroglie wavelength that you derived BEFORE you changed the speed to the speed of light we have:

wavelength = h v / m c^2

Now, c is about 3*10^8 and v is 10. v / c^2 is going to be an absolutely tiny number (almost zero).

You could give a debroglie wavelength of the car in this case as being about:

2*10^-44 m

Thats 0.000000000000000000000000000000000000000000002 metres. If you consider Heisenburg's uncertainty principal you'll see that this is an immeasurably small number!

In fact, because the debroglie wavelength for a fixed mass varies like v/c^2, only when the speed approaches the speed of light does the debroglie wavelength start becoming significant.

For this reason, it can be reasonable to approximate the debroglie wavelength by:

lambda = h / mc