The ψ(x,t) wavefunction of a Gaussian wavepacket spreading in free space (V(x)≡0)
is expressed in a didactic form. The expression found is a product of pure real
factors and pure phase factors. This makes it very easy to derive the expression
for the probability density from the wavefunction. The physical meaning of each of
the factors is analysed.
Zusammenfassen. Die Wellenfunktion ψ(x,t) eines Gaußschen Wellenpaketes, welches sich bei der
Ausbreitung im freien Raum (V(x)≡0) verbreitert, wird in einer didaktischen Form
ausgedrückt. Der gefundene Ausdruck ist ein Produkt von reellen Faktoren und
reinen Phasen-Faktoren. Dies vereinfacht die Herleitung des Ausdruckes für die
Wahrscheinlichkeitsdichte aus der Wellenfunktion. Die physikalische Bedeutung von
jedem Faktor wird analysiert.
While writing a paper [1] about the time evolution of
different wavepackets I wanted to find a didactic expression for the ψ(x,t)
wavefunction of a Gaussian initial state. The particular expression that was
finally constructed is different from those found in quantum mechanics texts [2 ,3].
Figure 1. x- and t-dependent parts of the wavefunction. (a), (b) and (c)
show the time development of the real part of factor 1, factor 2 and
of the full ψ(x,t). The x, y scale is the same for all (a), (b) and (c). (d)
shows the time development of the probability density ρ(x,t). The x, y scale
is the same for all time instants. Atomic units
are used. a = 2.5
Bohr = 0.13 nm, 
nm. The atomic time unit is 2.41 × 10-17 s. See the
text for details.
Figure 2. t-dependent parts of the wavefunction. (a) shows the time
dependent prefactor of factor 2 as function of time. (b) and (c) show the
time dependence of the terms of factor 3. Their real (full curve) and
imaginary (broken curves) parts are plotted against time. The thin dashed horizontal
lines in (b) show the asymptotes for t=∞. Atomic units are used. See the text
for details.
Initial state
Our initial state is a simple Gaussian wavepacket of the form
This wavepacket is a product of three factors:
A normalization factor that makes the norm 
of the wavefunction unity.
A plane wave factor that accounts for the non-zero momentum p0 of
the wavepacket.
A bell-shaped localizing function with half width at half maximum 
.
Time evolution
The time development of the initial ψ0(x) state is given by [2]:
This Fourier integral can be calculated easily with Gaussian integrals and
leads to a wavefunction like
Transformation of into didactic form
Now we want to transform this into something more informative. First note that the
centre of the wavepacket is moving with the group velocity
. Hence it is worth writing
instead of x0into the first term of the numerator in the exponential. Working this out gives the
following result
It is getting clearer already! Now let us get rid of the complex denominators!
Utilizing this we get finally
where arg z is the phase of the complex number z, i.e. 
. Our ψ(x,t) has three main factors ((7), (8) and
(9)). The first factor (7) is a product of two pure real coefficients and a
plane wave. This plane wave part of factor 1 and the entire second (8) and
third (9) factors are pure phase factors, i.e. their magnitude is one. Hence
it is very easy to calculate the probability density 
; one has only to calculate the square of the pure real
coefficients of factor 1 which gives:
The three terms of ψ(x,t) are as follows.
Factor 1. (Cf (7)) A Gaussian of the form (1). This is an expression having
the same form as ψ0(x) but the centre of gravity of the Gaussian is
moving with speed 
and its
width is increased to 
. The maximum
value of the Gaussian is decreasing as its width increases making the area under
ρ(x,t) (total probability) constant (one). The time evolution of factor 1 is
shown in figure 1(a).
Factor 2. (Cf (8)) An x- and t-dependent phase factor that is quadratic in
x. One can see from figure 1(b) that this factor oscillates faster for larger|x| values. This accounts for the fact that the higher momentum components
of the initial Gaussian ψ0(x) move with higher velocities. The function
which describes the time-dependent prefactor of the phase is 
. This function (cf figure 2(a)) is not
monotonic in time. Its value is zero for t = 0 and t=∞ and has a maximum at 
.
Factor 3. (Cf (9)) An x-independent (but still t-dependent) phase factor.
This phase factor is a product of two terms. The first term is a monotonic function
of time while the second one is oscillating. The phase of the first term is zero for
t = 0 (a(t) is pure real) and -π/4 for t=∞ (a(t) is pure imaginary). The second term
is 
where 
and it accounts for the time
development of the plane wave component 
in factor 1. These two phase factors are plotted in
figures 2(b) and 2(c) against time.
Acknowledgment
This work was partially supported by the Hungarian OTKA grant No F 014236.
References
[1] Márk G I Influence of the wavepacket shape to its
time development to be published
[2] Cohen-Tannoudji C, Diu B and Laloë F
1977 Quantum Mechanics (New York: Wiley)
[3] Merzbacher E 1970 Quantum Mechanics2nd edn (New York: Wiley)
