130_notes.dvi

Use the fine structure constant to avoid CGS units which are used inthe textbook.

α=

e^2

̄hc

= 1/ 137

This combination saves a lot of work.

̄hc= 1973 eV ̊A = 197.3MeVF

1 ̊A= 1. 0 × 10 −^10 m

1Fermi = 1. 0 × 10 −^15 m

The Bohr radius gives the size of the Hydrogen atom.

a 0 =

̄h

αmec

= 0. 529 × 10 −^10 m

mp= 938.3MeV/c^2

mn= 939.6MeV/c^2

me= 0.511MeV/c^2

5.6.7 The Dirac Delta Function

TheDirac delta function is zero everywhere except at the point where its argument is
zero. At that point, it is just the right kind of infinity so that

∫∞

−∞

dx f(x)δ(x) =f(0).

This is the definition of the delta function. It picks of the value of thefunctionf(x) at the point
where the argument of the delta function vanishes. A simple extension of the definition gives.

∫∞

−∞

dx f(x)δ(x−a) =f(a)

The transformation of an integral allows us to compute

∫∞

−∞

dx f(x)δ(g(x)) =

[

1

|dgdx|

f(x)

]

g(x)=0

the effect of the argument being a function.

If we make a wave packet in p-space using the delta function, and wetransform to position space,

ψ(x) =

1

√

2 π ̄h

∫∞

−∞

δ(p−p 0 )eipx/ ̄hdp=

1

√

2 π ̄h

eip^0 x/ ̄h

we just get the state of definitep.

This is a state of definite momentum written in momentum space.δ(p−p 0 )
Its Fourier transform isψp(x,t) =√ 21 πh ̄ei(px−Et)/ ̄h

This is a state of definite position written in position space.δ(x−x 0 )