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Wave packet and wave packet dynamics

How does one represent a freely moving particle in quantum mechanics?

Intuitively, one may consider a travelling plane wave represented by a wave-function, which in one dimension is the solution of the Schrodinger time dependent equation with $V=0$ :

\begin{displaymath} \psi(x,t) = Ae^{i(kx-\omega t)} \end{displaymath}

where $k$ and $\omega$ can be found out from the relations $p=\hbar k$ and $E=\hbar \omega$. There is a problem though. ${\vert\psi(x,t)\vert}^{2}$ then turns out to be $\vert A\vert^{2}$, which is constant. This would mean that there is equal probability of the particle to be present anywhere along the $x$-axis, which is unacceptable. Moreover, a wave represented by the above wave function travels in the $+x$ direction with the phase velocity $v_{p}=\omega/k$. Now, since
\begin{displaymath} E = \frac{p^{2}}{2m} \;\;\;\; \hbar\omega = \frac{\hbar^{2... ...; \omega = \frac{\hbar k^{2}}{2m} = constant\; k^{2} \;\;\;\; \end{displaymath}

(the so called dispersion relation)
\begin{displaymath} v_{p} = \frac{\omega}{k} = \frac{p}{2m} \end{displaymath}

which implies that the phase velocity is one-haf of the classical particle velocity. Thus the above plane wave wave-function does not have any physical significance and as such cannot truely represent a freely moving particle. Instead, one may take the linear combination of different plane wave wave-functions (with different wave numbers or $k$ values):
\begin{displaymath} \psi(x,t) = \int_{-inf}^{+inf} A(k)e^{i(kx-\omega t)} dk \end{displaymath}

$A(k)$ is the amplitude of the plane-wave having wave number $k$ or momentum $p=\hbar k$. This superposition, at each instant of time, leads to constructive interference in a small region along $x$-axis and destructive interference outside the region leading to $\psi(x,t)$ being localized in $x$ and being spread out in $k$ or $p$. The above wave-function represents what is known as a wave-packet, which in turn truely represents a classically free particle in quantum mechanics. With the change of time, the centre of the wave packet moves with a group velocity:
\begin{displaymath} v_{g} = \frac{d\omega}{dk} \;\;\;\; = \frac{d}{dk} \frac{\h... ...} \;\;\;\; = \frac{\hbar k}{m} \;\;\;\; = \frac{p}{m} \;\;\;\; \end{displaymath}

which agrees perfectly with the classical velocity of the free particle. In order to obtain $A(k)$ we find $\psi(x,0)$:
\begin{displaymath} \psi(x,0) = \int_{-inf}^{+inf} A(k)e^{i kx} dk \end{displaymath}

$\psi(x,0)$ is the inverse Fourier transform (IFT) of the function $A(k)$. Therefore $A(k))$ = Fourier transform (FT) of $\psi(x,0)$.
\begin{displaymath} A(k) = \int_{-inf}^{+inf} \psi(x,0) e^ {i kx} dx \end{displaymath}

We may start with a given $\psi(x,0)$, find $A(k)$ from it and then use it to find $\psi(x,t)$ which would then give us the complete temporal evolution of the wave-packet. To illustrate this technique we take $\psi(x,0)$ as follows:
\begin{displaymath} \psi(x,0) = const \;\; e^{-(x-x_{0})^{2}/(4\sigma_{0}^{2})} e^{i k_{0} x} \end{displaymath}

The probability density then turns out to be:
\begin{displaymath} \vert\psi(x,0)\vert^{2} = const \;\; e^{-(x-x_{0})^{2}/(2\sigma_{0}^{2})} \end{displaymath}

which represents a Gaussian distribution. $x_{0}=<x>=$mean of the distribution at $t=0$ and $\sigma_{0}=\sqrt{<(x-<x>)^{2}>}=\sqrt{<x^{2}>-<x>^{2}}=$standard deviation (square root of the variance) of the distribution at $t=0$. $x_{0}$ also equals the initial position of the center of the wave packet and hence the position of the free particle. $k_{0}$ corresponds to the initial momentum of the wave-packet. Then we find $A(k)$ and then substitute it in the expression for $\psi(x,t)$, perform the relevant integration to find $\psi(x,t)$ and hence $\vert\psi(x,t)\vert^{2}$. The important observations are as follows: $\vert\psi(x,t)\vert^{2}$ turns out to be a Gaussian distribution too:
\begin{displaymath} \vert\psi(x,t)\vert^{2} = (const/\sigma_{t}^{2})\;\;e^{-(x-x_{0}-p_{0}t)^{2}/(2 \sigma_{t}^{2})} \end{displaymath}


\begin{displaymath} \sigma_{t} = \sigma_{0}\;\;\sqrt{1+t^{2}/(4\sigma_{0}^{4})} \end{displaymath}

Thus the wave-packet localized around $x=x_{0}$ at $t=0$ spreads out (broadens or disperses) as it moves with time with group velocity $v_{g}=p_{0}/m=p_{0}$. The position of its centre changes with time from $x_{0}$ to $x_{0}+p_{0}t$. For $t=0:\;\;\Delta x=\sigma_{0}/\sqrt{2}$ For $t=0:\;\;\Delta p=\hbar /(\sqrt{2}\sigma_{0})$ Thus for $t=0:\;\; \Delta p\;\;\Delta x=\hbar/2$ i.e. Heisenberg's uncertaity relation is satisfied with our choice of initial wave-packet. Moreover the uncertainty product is minimum. For $t>0:\;\;\Delta x=\sigma_{t}/\sqrt{2}$ For $t>0:\;\;\Delta p=\hbar /(\sqrt{2}\sigma_{0})$ i.e. independent of $t$. Thus for $t>0:\;\; \Delta p\;\;\Delta x>\hbar/2$. (by making the substitutions) Hence, Heisenberg's uncertaity relation is again satisfied for $t > 0$. However, the uncertainty product is no more the minimum. $\hbar = 1$ and $m = 1$ throughout if not explicitly mentioned above (atomic units employed). One may note that plane electromagnetic or light waves and also light pulses (which are wave packets) travelling through vacuum have a linear dispersion relation in $\omega=c*k$ ( $c=constant=3\times 10^{8}\;m/s$). Hence such waves travel without any dispersion. For such plane light waves phase velocity $=v_{p}=c=\omega/k$ and for light pulse wave packets group-velocity is given by
\begin{displaymath} v_{g}=\frac{d\omega}{dk}=c. \end{displaymath}

However, since dispersion relation of particle wave-packets is non-linear, particle wave packets disperse or broaden with time.

An interactive wave-packet dynamics applet has been created by me to facilitate the study of the different aspects of the dynamics of a free particle wave packet as well as that of a wave packet tunnelling through a rectangular potential barrier or rectangular potential well. A Gaussian wave packet as described above has been used to launch the simulation.

Instead of following any analytical approach, I have used a numerical approach (The Crank Nicolson implicit method) to solve the relevant time dependent Schrodinger equation.

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