130_notes.dvi

5.7 Examples

5.7.1 The Square Wave Packet

Given the following one dimensional probability amplitude in the position variable x, compute the
probability distribution in momentum space. Show that the uncertainty principle is roughly satisfied.

ψ(x) =

1

√

2 a

for−a < x < a, otherwiseψ(x) = 0.

Its normalized.
∫∞

−∞

ψ∗ψdx=

∫a

−a

1

2 a

dx= 1

Take the Fourier Transform.

φ(k) =

1

√

2 π

∫a

−a

1

√

2 a

e−ikxdx

φ(k) =

1

√

4 πa

[

− 1

ik

e−ikx

]a

−a

φ(k) =

1

ik

√

4 πa

[

e−ika−eika

]

φ(k) =

1

ik

√

4 πa

[− 2 isinka] =−

√

1

πa

sin(ka)

k

Now estimate the width of the two probability distributions.

∆x= 2a

|φ(k)|^2 =

1

πa

sin^2 (ka)

k^2

∆k=

2 π

a

∆x∆k= 4π

5.7.2 The Gaussian Wave Packet*

Given the following one dimensional probability amplitude in the position variable x, compute the
probability distribution in momentum space. Show that the uncertainty principle is roughly satisfied.
ψ(x) = (απ)

(^14)
e−αx
(^2) / 2