130_notes.dvi

5 Wave Packets

Gasiorowicz Chapter 2

Rohlf Chapters 5

Griffiths Chapter 2

Cohen-Tannoudji et al. Chapter

5.1 Building a Localized Single-Particle Wave Packet

We now have a wave function for afree particle with a definite momentump

ψ(x,t) =ei(px−Et)/ ̄h=ei(kx−ωt)

where the wave numberkis defined byp= ̄hkand the angular frequencyωsatisfiesE= ̄hω. It is
not localized sinceP(x,t) =|ψ(x,t)|^2 = 1 everywhere.

We would like a state which is localized andnormalized to one particle.

∫∞

−∞

ψ∗(x,t)ψ(x,t)dx= 1

To make awave packet which is localized in space, we must add components of different wave
number. Recall that we can use a Fourier Series (See section 5.6.1) to compose any functionf(x)
when we limit the range to−L < x < L. We do not want to limit our states inx, so we will take the
limit thatL→ ∞. In that limit, every wave number is allowed so the sum turns into an integral.
The result is the very closely related Fourier Transform (See section 5.6.2)

f(x) =

1

√

2 π

∫∞

−∞

A(k)eikxdk

with coefficients which are computable,

A(k) =

1

√

2 π

∫∞

−∞

f(x)e−ikxdx.

The normalizations off(x) andA(k) are the same (with thissymmetric form) and both can
represent probability amplitudes.

∫∞

−∞

f∗(x)f(x)dx=

∫∞

−∞

A∗(k)A(k)dk

We understandf(x) as awave packetmade up of definite momentum termseikx. The coefficient
of each term isA(k). The probability for a particle to be found in a regiondxaround some value of