130_notes.dvi

also a Gaussian. We will show later that a Gaussian is the best one can do to localize a particle in
position and momentum at the same time.

In both of these cases off(x) (transformed from a normalizedA(k) localized in momentum space)
we see

• A coefficient whichcorrectly normalizesthe state to 1,


• eik^0 x– awave corresponding to momentum ̄hk 0 ,


• and a packet function which islocalizedinx.

We have achieved our goal of findingstates that represent one free particle. We see that we can
have states which are localized both in position space and momentum space. We achieved this by
making wave packets which are superpositions of states with definite momentum. The wave packets,
while localized, have some width inxand inp.

5.3 The Heisenberg Uncertainty Principle

The wave packets we tried above satisfy anuncertainty principle which is a property of waves.
That is ∆k∆x≥^12.

For the “square” packet the full width inkis ∆k=a. The width inxis a little hard to define, but,
lets use the first node in the probability found atax 2 =πorx=^2 aπ. So the width is twice this or
∆x=^4 aπ. This gives us
∆k∆x= 4π

which certainly satisfies the limit above. Note that if we change the width ofA(k), the width of
f(x) changes to keep the uncertainty product constant.

For the Gaussian wave packet, we can rigorously read theRMS width of the probability distri-
butionas was done at the end of the section on the Fourier Transform of aGaussian (See section
5.6.4).

σx =

√

α

σk =

1

√

4 α

We can again see that as we vary the width in k-space, the width in x-space varies to keep the
product constant.

σxσk=

1

2

TheGaussian wave packet gives the minimum uncertainty. We will prove this later.

If we translate into momentump= ̄hk, then

∆p= ̄h∆k.

So theHeisenberg Uncertainty Principlestates.