ψk,l(x)=kexp(ikx)+2κ(−x)sin(kx) 2π(κ2+k2)
, (3.60)
and incidence of particles from the right,
ψk,r(x)=kexp(−ikx)−2κ(x)sin(kx) 2π(κ2+k2)
. (3.61)
The completeness relation is ψκ(x)ψκ(x)+
∞
0
dk!
ψk,l(x)ψk,l(x)+ψk,r(x)ψk,r(x)"
=δ(x−x). (3.62) There is no bound state solution for a repulsiveδpotential
V (x)=Wδ(x)= ¯h2κ
m δ(x) (3.63)
and the even parity energy eigenstates become φk,+(x)= 1
√π
kcos(kx)+κsin(k|x|)
√κ2+k2 . (3.64)
The completeness relation for the eigenfunctions of the repulsive δ potential is therefore
∞
0
dk!
ψk,−(x)ψk,−(x)+φk,+(x)φk,+(x)"
=δ(x−x). (3.65)
3.4 Evolution of Free Schrửdinger Wave Packets
Another important model system for quantum behavior is provided by free wave packets. We will discuss in particular free Gaussian wave packets because they provide a simple analytic model for dispersion of free wave packets. This example will also demonstrate that the spatial and temporal range of free particle models is constrained in quantum physics. We will see that free wave packets of subatomic particles disperse on relatively short time scales, which are however too long to interfere with lab experiments involving free electrons or nucleons. Nevertheless, the discussion of the dispersion of free wave packets makes it also clear that simple interpretations of particles in quantum mechanics as highly localized free wave packets which every now and then get disturbed through interactions with other
wave packets are not feasible. Particles can exist in the form of not too small free wave packets for a little while, but atomic or nuclear size wave packets must be stabilized by interactions to avoid rapid dispersion. We will see examples of stable wave packets in Chaps.6and7.
The Free Schrửdinger Propagator
Substitution of a Fourieransatz ψ (x, t )= 1
2π ∞
−∞dk ∞
−∞dω ψ(k, ω)exp[i(kx−ωt )] (3.66) into the free Schrửdinger equation shows that the general solution of that equation in one dimension is given in terms of a wave packet
ψ(k, ω)=√
2π ψ(k)δ
ω− ¯hk2 2m
, (3.67)
ψ (x, t )= 1
√2π ∞
−∞
dk ψ(k)exp
i
kx− ¯hk2 2mt
. (3.68)
The amplitude ψ(k) is related to the initial condition ψ(x,0) through inverse Fourier transformation
ψ(k)= 1
√2π ∞
−∞dx ψ(x,0)exp(−ikx), (3.69) and substitution ofψ(k)into (3.68) leads to the expression
ψ (x, t )= ∞
−∞dxU (x−x, t )ψ(x,0) (3.70) with the freepropagator
U (x, t )= 1 2π
∞
−∞dkexp
i
kx− ¯hk2 2mt
. (3.71)
This is sometimes formally integrated as5
5The propagator is commonly denoted asK(x, t ). However, we prefer the notationU (x, t )because we will see in Chap.13that the propagator is nothing but thexrepresentation of the time evolution operatorU (t ).
3.4 Evolution of Free Schrửdinger Wave Packets 53
U (x, t )= m
2πiht¯ exp
imx2 2ht¯
. (3.72)
The propagator is the particular solution of the free Schrửdinger equation ih¯ ∂
∂tU (x, t )= − ¯h2 2m
∂2
∂x2U (x, t ) (3.73) with initial conditionU (x,0)=δ(x). It yields the correspondingretarded Green’s function
ih¯ ∂
∂tG(x, t )+ ¯h2 2m
∂2
∂x2G(x, t )=δ(t )δ(x), (3.74) G(x, t )
t <0=0, (3.75)
through
G(x, t )= (t )
ih¯ U (x, t ). (3.76) This can also be derived from the Fourier decomposition of Eq. (3.74), which yields
G(x, t )= 1 (2π )2h¯
∞
−∞dk ∞
−∞dω exp[i(kx−ωt )]
ω−(hk¯ 2/2m)+i. (3.77) The negative imaginary shift of the pole(hk¯ 2/2m)−i,→ +0, in the complexω plane ensures that the condition (3.75) is satisfied. We will encounter time evolution operators and Green’s functions in many places in this book. The designation propagator is often used both for the time evolution operator U (x, t ) and for the related Green’s function G(x, t ). U (x, t ) propagates initial conditions as in Eq. (3.70) whileG(x, t )propagates perturbations or source terms in the Schrửdinger equation.
Width of Gaussian Wave Packets
A wave packet ψ (x, t ) is denoted as a Gaussian wave packet if |ψ (x, t )|2 is a Gaussian function ofx. We will see below through direct Fourier transformation thatψ (x, t )is a Gaussian wave packet inxif and only ifψ (k, t )is a Gaussian wave packet ink.
Normalized Gaussian wave packets have the general form
ψ (x, t )= 2α(t )
π 1
4
exp
−α(t )[x−x0(t )]2+iϕ(x, t )
, (3.78)
and we will verify that the real coefficientα(t )is related to the variance through x2(t )=1/4α(t ). The expectation values ofxandx2are readily evaluated,
x(t )=
2α(t ) π
∞
−∞dx xexp
−2α(t )[x−x0(t )]2
=
2α(t ) π
∞
−∞dξ[ξ+x0(t )]exp
−2α(t )ξ2
=x0(t ), (3.79)
x2(t )=
2α(t ) π
∞
−∞dx x2exp
−2α(t )[x−x0(t )]2
=
2α(t ) π
∞
−∞dξ[ξ+x0(t )]2exp
−2α(t )ξ2
=
2α(t ) π
x02(t )−1 2
d dα(t )
∞
−∞
dξ exp
−2α(t )ξ2
=x02(t )+ 1
4α(t ), (3.80)
and therefore we find indeed
x2(t )= x2(t )− x2(t )= 1
4α(t ). (3.81)
Free Gaussian Wave Packets in Schrửdinger Theory
We assume that the wave packet of a free particle at timet = 0 was a Gaussian wave packet of widthx,
ψ(x,0)= 1
(2π x2)1/4exp
−(x−x0)2 4x2 +ik0x
. (3.82)
This yields a Gaussian wave packet of constant width
k= 1
2x (3.83)
3.4 Evolution of Free Schrửdinger Wave Packets 55 inkspace,
ψ(k)= 1
√2π ∞
−∞dx ψ(x,0)exp(−ikx)
= 1
(2π )3/4(x2)1/4 ∞
−∞dx exp
−(x−x0)2
4x2 +i(k0−k)x
= exp[i(k0−k)x0] (2π )3/4(x2)1/4
∞
−∞dξ exp
− ξ2
4x2 +i(k0−k)ξ
= 2x2
π 14
exp
−x2(k−k0)2−i(k−k0)x0
, (3.84)
ψ (k, t )=ψ(k)exp
−ihk¯ 2 2mt
. (3.85)
Substitution ofψ(k)into Eq. (3.68) then yields
ψ (x, t )= x2
2π3 14
exp
−x2k02+ik0x0
× ∞
−∞dk exp
−
x2+iht¯ 2m
k2+
2x2k0+i(x−x0)
k
= (2π x2)1/4
!2π x2+iπ(ht /m)¯ "1/2exp
−x2k20+ik0x0
×exp
!2x2k0+i(x−x0)"2
4x2+2i(ht /m)¯
= (2π x2)1/4
!2π x2+iπ(ht /m)¯ "1/2exp
− [x−x0−(hk¯ 0/m)t]2 4x2+(h¯2t2/m2x2)
×exp
i
k0x− ¯hk20 2mt+ ¯ht
8m
[x−x0−(hk¯ 0/m)t]2 (x2)2+(h¯2t2/4m2)
. (3.86)
Comparison of Eq. (3.86) with Eqs. (3.78) and (3.81) yields x2(t )=x2+ h¯2t2
4m2x2, (3.87)
i.e. a strongly localized packet at timet = 0 will disperse very fast, because the dispersion time scaleτ is proportional tox2. The reason for the fast dispersion is that a strongly localized packet att =0 comprises many different wavelengths.
However, each monochromatic component in a free wave packet travels with its own phase velocity
v(k)= ω k = ¯hk
2m, (3.88)
and a free strongly localized packet therefore had to emerge from rapid collapse and will disperse very fast. On the other hand, a poorly localized packet is almost monochromatic and therefore slowly changes in shape.
The relevant time scale for decay of the wave packet is τ = 2mx2
¯
h . (3.89)
Electron guns often have apertures in the millimeter range. Assumingx =1 mm for an electron wave packet yields τ 2 × 10−2s. This sounds like a short time scale for dispersion of the wave packet. However, on the time scales of a typical lab experiment involving free electrons, dispersion of electron wave packets is completely negligible, see e.g. Problem 3.17 as well as Problem 3.18 for a corresponding discussion for neutrons.
On the other hand, suppose we can produce a free electron wave packet with atomic scale localization,x = 1 Å. This wave packet would disperse with an extremely short time scaleτ 2×10−16s, which means that the wave function of that electron would be smeared across the planet within a minute. See also Problem3.19for another example of electron dispersion.
We will see in Chaps.6and7that wave packets can remain localized under the influence of forces, i.e. the notion of stable electrons in atoms makes sense, although the notion of highly localized free electrons governed by the free Schrửdinger equation is limited to small distance and time scales.
We can infer from the example of the free Gaussian wave packet that the kinetic term in the Schrửdinger equation drives wave packets apart. If there is no attractive potential term, the kinetic term decelerates any eventual initial contraction of a free wave packet and ultimately pushes the wave packet towards accelerated dispersion.
We will see that this action of the kinetic term can be compensated by attractive potential terms in the Schrửdinger equation. Balance between the collapsing force from attractive potentials and the dispersing force from the kinetic term can stabilize quantum systems.
Comparison of Eq. (3.85) with Eqs. (3.78) and (3.81) yields constant width of the wave packet inkspace and therefore
p= ¯hk= ¯h
2x, (3.90)
i.e. there is no dispersion in momentum. The product of uncertainties of momen- tum and location of the particle satisfies px(t ) ≥ ¯h/2, in agreement with