How Do You Calculate the Energy of Particles Using De Broglie's Equation?

[Von Neumann]

Problem:

a. Calculate the energy in eV of an electron with a wavelength of 1 fm.

b. Make the same calculation for a neutron.

Solution (so far):

a. λ=h/p=(hc)/(pc)=(1240 MeV fm)/(pc)=1fm

so, pc=1240 MeV

E=√[(pc)^2+E_0^2]

=√[(1240 MeV)^2+(.511MeV)^2]

∴E=1.24 GeV

This is the same answer as the back of my book, so I'm assuming this is the correct method of solution. However, I do the same thing for the neutron and my answer does not agree.

b. E=√[(1240 MeV)^2+(940 MeV)^2]

∴E=1560 Mev

My book says the correct answer is 616 MeV.

I don't see how an energy like that is even possible. Solving the following for pc,

E^2=(pc)^2+E_0^2

pc=√[E^2-E_0^2]

When you plug in the "correct" answer of E=616 MeV you get,

pc=√[(616 MeV)^2-(940 MeV)^2]

You certainly cannot take a square root of a negative number and get a meaningful answer. Any suggestions?

 

[rude man]

Fact: when we talk of a particle with x eV of energy we are talking about its KINETIC energy, not its total energy.

So - compute the neutron's total energy E, subtract rest energy E_0, which gets you the K.E. , and you get what?
(Hint: the advertised answer).

 

[Von Neumann]

Rude man,

You will always catch my stupid mistakes, and for that I thank you.

It hadn't occurred to me that the kinetic energy is what was being asked for.

 

[rude man]

Rude man,

You will always catch my stupid mistakes, and for that I thank you.

It hadn't occurred to me that the kinetic energy is what was being asked for.

Don't worry about it, it's just the convention!

 

[Nickie]

Upon further research, I have found that the correct calculation for the energy of a neutron using De Broglie's equation is:

E=√[(pc)^2+(mnc^2)^2]

Where mn is the rest mass of the neutron (939.565 MeV/c^2).

Substituting the values into the equation:

E=√[(1240 MeV)^2+(939.565 MeV)^2]

E=√[(1537600 MeV^2)+(883876.522225 MeV^2)]

E=√[2421476.522225 MeV^2]

E=1556.5 MeV

This is closer to the answer given in the book (1560 MeV). The slight difference may be due to rounding errors or the use of different values for the rest mass of the neutron.

In conclusion, the correct method for calculating the energy of a particle using De Broglie's equation is to use the rest mass of the particle in addition to its momentum. It is also important to use consistent units (MeV) throughout the calculation.