How to propagate Heller's model of the Gaussian Wave Packet? I'm working on an undergraduate research project on gaussian wave packets, but am a pretty big noob in terms of theoretical chemistry and the like. I'm having a lot of difficulty grasping this concept and I know it may be oddly specific and unrelated to the other things people talk about here, but any help would be greatly appreciated.
The obstacle I'm currently facing is on understanding the math involved behind the propagation of gaussian wave packets. The paper that introduced this concept (I believe) is here: https://aip.scitation.org/doi/10.1063/1.430620
I'll mention my rough understanding of what is going on, followed by what in particular I don't understand:
The purpose (I believe) of these wave packets is that they can represent wave functions, but their evolution with time is much simpler to calculate than a more typical (?) approach to the time dependent Schrodinger equation. This is because the equations of motion for the involved parameters follow a classical trajectory rather than a quantum one. (a little iffy on that)
This imgur album has segments of the paper in order that I thought were important: https://imgur.com/a/v5ZMrOC Some information between the equations is missing (and possibly important), but the structure of the paper appears that anything beyond this is not required for calculating wave packet propagation in the most simplest sense along the most simplest quadratic potential.
In img 1 (eq. 1.2.), I am in some sense already lost. A wave packet form of the wave function is provided. I am unsure what $q_2$, $q_1$, and $q_0$ are. I assume just arbitrary functions of time.
Regardless, a new form is given in img 2 (eq. 2.1). Again, I am very unsure what these parameters even are. What is $\alpha_t$, or $p_t$, or $\gamma_t$? What do they signify?
I assume that knowledge would help greatly in understanding img 3 (eq. 2.3). I take it that if $x_t$ is some constant, then its rate of change with time is equal to the negative of the rate of the change of the hamiltonian with respect to $p$ (whatever that is) at the value $p = p_t$.
They go on to evaluate the Hamiltonian for "at most a quadratic function of $x$". I assume this is referencing $V(x)$, some potential function like a harmonic oscillator. In my head I just pictured $V(x) = x^2$. Then $V_0 = x^2$ , $V_x = 2x$, and $V_{xx} = 2$.
Another question that arises is that, if $V_x$ is $dV/dx$, is $\alpha_t$ equal to $d(\alpha_t)/dt$ ?
By some mathematical magic (that is not necessarily important to me), Heller arrives at the equations 2.11 a to d. The equations of motion that I imagine are key to propagating the wave packet with time given some initial parameters.
My goal here (in this stage of my project) is to understand how to propagate this form of the wave packet in Python. While I don't need to understand the derivation of the relevant equations, I should at least understand the relevant equations themselves.
For Heller's model of the propagating gaussian wave packet, what do the parameters mean? what starting parameters can I use? What do I need to solve or understand before I'm ready to get what is happening here? I have found so many resources online on gaussian wave packets and all of them explain things either using strange, different forms of the wave packet or with language that is way too technical and difficult for me to follow (usually both). If anybody could explain this at the level of somebody who really just has an okay understanding of derivatives from basic calculus, I would be eternally grateful.
 A: The 1D time dependent Schrodinger equation is a partial differential equation in $x$ and $t$. Suppose we assume that the wavefunction will always take the form of a Gaussian function of $x$. (It will turn out that this is only true if the potential is quadratic in $x$, such as a simple harmonic oscillator, or an upside-down simple harmonic oscillator). Then we will need to write this Gaussian in terms of some parameters, which will need to be time dependent. The most general form will be something like eqn (1.2) involving three arbitrary complex functions of time $q_0(t)$, $q_1(t)$, $q_2(t)$. They are simply the coefficients of the corresponding power of $x$ in the exponent of the Gaussian (apart from a prefactor of $i/\hbar$ taken out for convenience). If one plugs this into the Schrodinger equation, and if the potential is quadratic in $x$, then the spatial derivatives can be done analytically, all the explicit $x$-dependence will drop out, and the result will be three ordinary differential equations governing the time evolution of those three (complex) parameters.
It's more convenient to replace these three complex parameters with an equivalent set: $x_t$, $p_t$ (which are both real), $\alpha_t$, and $\gamma_t$ (both complex). Imagine expressing a real Gaussian function in terms of its position, its width, and its amplitude, instead of the coefficients of $x$ in the exponent. It is a fairly straightforward change of variables. The subscript $t$ just reminds us of the time dependence: it does not indicate a time derivative. The ordinary differential equations for these parameters are (2.11). (In passing, the normalization condition for the wavefunction has been introduced). The first two equations, in $x_t$ and $p_t$, govern the motion of the centre of the wavepacket, and correspond to classical motion. They do not involve $\alpha_t$ and $\gamma_t$. Those two parameters contain information about the width of the Gaussian, the phase of the wave function, and the amplitude (or normalization factor). Beyond that, I'm not sure that there is any deep "meaning" to be given to them.
You haven't quite understood the meanings of $V_0$, $V_x$, and $V_{xx}$. They are defined in eqn (2.5). They are not functions of $x$. They are, however, determined by the value of the parameter $x_t$. Note that the quantity $V_{xx}$ is not the second derivative of the potential in an expansion taken at $x=0$, but at $x=x_t$. This may seem a bit odd, but really it isn't. The potential $V(x)$ does not change in time, but the value of $V_{xx}$ will change as the wave packet moves. Think of this in the same way as you think of the first derivative, $V_x$, which determines the force in the classical equations of motion. To evaluate the force on a particle, you would calculate $-V_x = -\partial V(x)/\partial x$ at $x=x_t$.
All this is to set up the main part of Heller's paper, which is to propose a locally quadratic approximation to the potential, by Taylor-expanding it around the current centre of the wave packet $x_t$. This, of course, will be inaccurate in general. It will completely miss some important features of quantum mechanics. For example, one expects a wave packet incident on a finite barrier to be partially reflected and partially transmitted, so it would have to split into two wavepackets: the single Gaussian simply can't do that.
I'm concentrating here on the notational issues which seem to have confused you, and on some of the ideas behind this approach. I don't think it's appropriate to get into details such as how you choose your starting conditions. I'll mention, though, that it would be sensible to choose the initial wavepacket to be normalized: the imaginary part of $\gamma_t$ at $t=0$ can be chosen to ensure that this is the case, and I believe that the differential equations preserve this normalization. No doubt, once you have a working program, you will be able to experiment with the initial parameters.
