130_notes.dvi

∆p∆x≥

̄h

2

It says we cannot know the position of a particle and its momentum atthe same time and tells us
the limit of how well we can know them.

If we try to localize a particle to a very small region of space, its momentum becomes uncertain. If
we try to make a particle with a definite momentum, its probability distribution spreads out over
space.

5.4 Position Space and Momentum Space

We can represent a state with eitherψ(x) or withφ(p). We can (Fourier) transform from one to the
other.

We have the symmetric Fourier Transform.

f(x) =

1

√

2 π

∫∞

−∞

A(k)eikxdk

A(k) =

1

√

2 π

∫∞

−∞

f(x)e−ikxdx

When we change variable fromktop, we get theFourier Transforms in terms ofxandp.

ψ(x) =

1

√

2 π ̄h

∫∞

−∞

φ(p)eipx/ ̄hdp

φ(p) =

1

√

2 π ̄h

∫∞

−∞

ψ(x)e−ipx/ ̄hdx

These formulas are worth a little study. If we defineup(x) to be thestate with definite momen-
tump, (in position space) our formula for it is

up(x) =

1

√

2 π ̄h

eipx/ ̄h.

Similarly, the state (in momentum space) withdefinite positionxis

vx(p) =

1

√

2 π ̄h

e−ipx/ ̄h

These states cannot be normalized to 1 but they do have a normalization convention which is satisfied
due to the constant shown.