Wave packet and wave packet dynamics
How does one represent a freely moving particle in quantum mechanics?
where and can be found out from the relations and . There is a problem though. then turns out to be , which is constant. This would mean that there is equal probability of the particle to be present anywhere along the -axis, which is unacceptable. Moreover, a wave represented by the above wave function travels in the direction with the phase velocity . Now, since
(the so called dispersion relation)
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 values):
is the amplitude of the plane-wave having wave number or momentum . This superposition, at each instant of time, leads to constructive interference in a small region along -axis and destructive interference outside the region leading to being localized in and being spread out in or . 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:
which agrees perfectly with the classical velocity of the free particle. In order to obtain we find :
is the inverse Fourier transform (IFT) of the function . Therefore = Fourier transform (FT) of .
We may start with a given , find from it and then use it to find which would then give us the complete temporal evolution of the wave-packet. To illustrate this technique we take as follows:
The probability density then turns out to be:
which represents a Gaussian distribution. mean of the distribution at and standard deviation (square root of the variance) of the distribution at . also equals the initial position of the center of the wave packet and hence the position of the free particle. corresponds to the initial momentum of the wave-packet. Then we find and then substitute it in the expression for , perform the relevant integration to find and hence . The important observations are as follows: turns out to be a Gaussian distribution too:
Thus the wave-packet localized around at spreads out (broadens or disperses) as it moves with time with group velocity . The position of its centre changes with time from to . For For Thus for i.e. Heisenberg's uncertaity relation is satisfied with our choice of initial wave-packet. Moreover the uncertainty product is minimum. For For i.e. independent of . Thus for . (by making the substitutions) Hence, Heisenberg's uncertaity relation is again satisfied for . However, the uncertainty product is no more the minimum. and 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 ( ). Hence such waves travel without any dispersion. For such plane light waves phase velocity and for light pulse wave packets group-velocity is given by
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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