High-energy theory

Where an approximation starts being good

About 20 min if you have seen this material before, 50 min or so if it is new to you.

Background

Exact expressions are routinely replaced by approximate ones, because the exact expression may be unwieldy in a given context. The is

E = \sqrt{p^2 + m^2}

, in units where

c = 1

, so energy, momentum and mass are all measured in the same units. Two approximations to it are in common use, one for particles moving fast and one for particles moving slowly.

A particle of mass

m

and momentum

p

has energy

E = \sqrt{p^2 + m^2}

. A paper could carry an expression like

E \approx p + \frac{m^2}{2p}

without explicitly stating its regime of validity. Call the approximation accurate to

1\%

when it differs from

E

by less than

1\%

E

. Parts (a) to (c) locate that range two ways, once from the size of the first term thrown away and once from the exact expression, and the two do not quite agree. Part (d) does the same for the slow-particle approximation, and the two thresholds turn out to be related.

(a)

Continue the expansion one more term:

E = p + \dfrac{m^2}{2p} + c\,\dfrac{m^4}{p^3} + \dots

. What is

c

, including its sign?

(b)

Estimate the error by the size of that first neglected term, as a fraction of

E

. Above what momentum, in units of

m

, does that estimate put the approximation inside

1\%

(c)

Now do it exactly, with no expansion: solve

\left(p + \dfrac{m^2}{2p}\right) = 1.01\,E

for

p

in units of

m

(d)

For slow particles the other approximation is

E \approx m + p^2/2m

. Find exactly the momentum below which it is inside

1\%

, in units of

m

. That threshold and the one in (c) sit on either side of

p = m

, and the product of the two is the quickest way to see how they are related. Give the product.

Past 1.3 hr on this? That usually means a missing prerequisite rather than a lack of effort — worth reading up before pushing further, starting with Energy-momentum relation (Wikipedia).

Want to go deeper?