2

(2.11)

22 Quantum mechanics: principles and relativistic e¬ects

we can show (see Chapter 12) that

σx σk ≥ 1 , (2.12)

2

which shows that two conjugated variables as x and k (that appear as prod-

uct ikx in the exponent of the Fourier transform) cannot be simultaneously

sharp. This is Heisenberg™s uncertainty relation, which also applies to t

and ω. Only for the Gaussian function exp(’±x2 ) the product of variances

reaches the minimal value.

As shown in Chapter 12, averages over k and powers of k can be rewritten

in terms of the spatial wave function Ψ:

dr Ψ— (’i∇)Ψ(r, t),

k (t) = (2.13)

dr Ψ— (’∇2 )Ψ(r, t).

k 2 (t) = (2.14)

Thus, the expectation of some observable A, being either a function of r

only, or being proportional to k or to k 2 , can be obtained from

dr Ψ— (r, t)AΨ(r, t),

ˆ

A (t) = (2.15)

ˆ

where A is an operator acting on Ψ, and

ˆ

k = ’i∇, (2.16)

ˆ

k 2 = ’∇2 . (2.17)

Similarly (but with opposite sign due to the opposite sign in ωt), the expec-

tation value of the angular frequency ω is found from equation by using the

operator

‚

ω=i

ˆ . (2.18)

‚t

The identi¬cations p = k (2.1) and E = ω (2.2) allow the following

operator de¬nitions:

p = ’i ∇,

ˆ (2.19)

ˆ

p2 = ’ 2 ∇2 , (2.20)

‚

ˆ

E=i . (2.21)

‚t

From these relations and expression of the energy as a function of momenta

and positions, the equations of motion for the wave function follow.

2.2 Non-relativistic single free particle 23

2.2 Non-relativistic single free particle

In principle we need the relativistic relations between energy, momentum

and external ¬elds, but for clarity we shall ¬rst look at the simple non-

relativistic case of a single particle in one dimension without external inter-

actions. This will allow us to look at some basic propagation properties of

wave functions.

Using the relation

p2

E= , (2.22)

2m

then (2.21) and (2.20) give the following equations of motion for the wave

function:

2 ‚2Ψ

‚Ψ(x, t)

=’

i , (2.23)

2m ‚x2

‚t

or

i ‚2Ψ

‚Ψ

= . (2.24)

2m ‚x2

‚t

This is in fact the time-dependent Schr¨dinger equation. This equation looks

o

much like Fick™s di¬usion equation, with the di¬erence that the di¬usion

constant is now imaginary (or, equivalently, that the di¬usion takes place

in imaginary time). If you don™t know what that means, you are in good

company.

If we choose an initial wave function Ψ(x, 0), with Fourier transform

g(k, 0), then the solution of (2.24) is simply

∞

1

Ψ(x, t) = √ dk g(k, 0) exp[ikx ’ iω(k)t], (2.25)

2π ’∞

k2

ω(k) = (2.26)

2m

The angular frequency corresponds to the energy:

( k)2 p2

E= ω= = , (2.27)

2m 2m

as it should. If ω had been just proportional to k (and not to k 2 ) then

(2.25) would represent a wave packet traveling at constant velocity without

any change in the form of the packet. But because of the k 2 -dependence

the wave packet slowly broadens as it proceeds in time. Let us assume that

g(k, 0) is a narrow distribution around a constant k0 , and write

k 2 = k0 + 2k0 ”k + (”k)2 ,

2

(2.28)

k0 = mv. (2.29)

24 Quantum mechanics: principles and relativistic e¬ects

In these terms the wave function can be written as

1

Ψ(x, t) = √ exp[ik0 (x ’ 1 vt)]

2

2π

∞

d”k g(”k, t) exp[i”k(x ’ vt)],

= (2.30)

’∞

(”k)2 t

g(k, t) = g(k, 0) exp[’i ]. (2.31)

2m

The factor in front of the integral is a time-dependent phase factor that

is irrelevant for the shape of the density distribution since it cancels in

Ψ— Ψ. The packet shape (in space) depends on x = x ’ vt and thus the

packet travels with the group velocity v = dω/dk. However, the packet

changes shape with time. In fact, the package will always broaden unless it

represents a stationary state (as a standing wave), but the latter requires

an external con¬ning potential.

Let us take a Gaussian packet with initial variance (of Ψ— |Ψ) of σ0 and

2

with velocity v (i.e., k = k0 ) as an example. Its initial description (disre-

garding normalizing factors) is

x2

Ψ(x, 0) ∝ exp ’ 2 + ik0 x , (2.32)

4σ0

g(k, 0) ∝ exp[’σ0 (”k)2 ].

2

(2.33)

The wave function Ψ(x = x ’ vt, t) is, apart from the phase factor, equal

to the inverse Fourier transform in ”k of g(k, t) of (2.31):

t

g(k, t) ∝ exp ’ σ0 + i

2