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 x₀ into 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 Mechanics 2nd edn (New York: Wiley)