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How would the Schrödinger equation be solved for curved barriers which change as a function of time, e.g., a paraboloid potential barrier with maximum height, $V$ changing with time into a Hyperboloid potential barrier (with the same constant height, $V$, at its saddle point), which further changes into an ellipsoidal barrier. What would be the mathematical tools required for analysis? Are such systems practically found?

Mathematical formulation:

Consider a n-dimensional Schrödinger equation of the form: $$\left[\sum_{k=1}^{n}\frac{\partial^{n}}{\partial{x_{k}^{2}}}-V(x,t)\right]\psi(x,\alpha)=\lambda(\alpha)\psi(x,\alpha)$$ where the potential $V(x,t)$ depends on the column vector $x$ belonging to the n-dimensional complex space $C^{n}$

Now let the elliptic potential be: the 2-gap Lamé potential $$V_{e}(x,t)=2\wp(x-x_{1}(t))+2\wp(x-x_{2}(t))+2\wp(x-x_{3}(t))$$

Now this potential varies with time and changes into a hyperbolic potential of the form: $$V_{h}(x,t)=aV_{0}coth(\alpha x)+bV_{1}coth^{2}(\alpha x)-cV_{2}cosech(\alpha x)+d-cos(\alpha t)$$ where $a,b,c,d$ and $V_{0},V_{1},V_{2}$ are constants. Here is a picture of the graph of the potential of only time independent variables: enter image description here

How would I solve such a system of time varying potential?

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  • $\begingroup$ @DavidElm What are the softwares available for especially solving these type of wave equations numerically, I could not do this on Matlab. $\endgroup$ Nov 24, 2016 at 10:14
  • $\begingroup$ Comment moved to answer... $\endgroup$
    – David Elm
    Nov 24, 2016 at 10:31
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    $\begingroup$ I personally would use Mathematica, but that's just because I'm familiar with it. The last time I attempted to simulate these guys was about 20 years ago, in that case I wrote the code in Turbo Pascal. Either way, you'll have to write a little bit of code to get the simulation going. $\endgroup$
    – David Elm
    Nov 24, 2016 at 10:33
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    $\begingroup$ I don't understand what the 2-gap Lamé potential is, but if your potential is periodic in time [as is $V_h(x,t)$], then you could make some progress by looking into the Floquet formalism. Nevertheless, I don't expect an analytical solution to be available. $\endgroup$ Sep 4, 2020 at 13:24

4 Answers 4

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You may have to simulate this situation and have to settle for a numerical answer. This would involve storing the wave function and a function for its first derivative.

Each time step you calculate a the function and its time derivative.

$\frac{\partial \psi}{\partial t}=\left({\frac{\partial \psi}{\partial t}}\right)_0+\frac{\partial^2 \psi}{\partial t^2} \delta t$

$\psi=\psi_0+\frac{\partial \psi}{\partial t} \delta t$

Where $\delta t$ is your time step.

You then get the next value for $\frac{\partial^2 \psi}{\partial t^2}$ from the Schrödinger Equation.

And then you repeat.

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The time-dependent Schrödinger equation for a single particle with time-dependent potential $U(r,t)$ is:

$$i\hbar\frac{\partial}{\partial t}\Psi(r,t)=\Big[-\frac{\hbar^2}{2m}\nabla^2+U(r,t)\Big]\Psi(r,t)\tag{1}$$

If we replace $U(r,t)$ with a time-independent potential $U(r)$, we get:

$$i\hbar\frac{\partial}{\partial t}\Psi(r,t)=\Big[-\frac{\hbar^2}{2m}\nabla^2+U(r)\Big]\Psi(r,t)\tag{2}$$

$(2)$ can be separated into two ODEs, if we assume:

$$\Psi(r,t)=\psi(r)T(t)$$

Substituting into $(2)$ we get two ODEs:

$$i\hbar\frac{T'(t)}{T(t)}=E\tag{3}$$ $$-\frac{\hbar^2}{2m}\frac{d^2\psi(r)}{dr^2}+U(r)\psi(r)=E\psi(r)\tag{4}$$

Where $E$ is the particle's total energy.

$(4)$ is of course the time-independent Schrödinger equation and $(3)$ is the time evolution ODE for the wave function.

This method of separating variables only works for a time-independent potential $U(r)$. It does not work for a $U(r,t)$ potential.

How would the Schrödinger equation be solved for curved barriers which change as a function of time

For that reason the usual methods of computing the wave function $\Psi(r,t)$ don't really work with a time-dependent potential and you'll find far fewer examples of such derivations in most textbooks.

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  • $\begingroup$ Thanks! So how hard is it to solve for curved hypervolumes? Could you please state any text book that has such an example. The reason I'm asking is because in class we were being taught Gamov's alpha particle tunnelling and we used a Riemann approximation for graphically solving it; I questioned why should we approximate and why can't we solve for a curved potential well and what would happen if we did so. $\endgroup$ Oct 12, 2016 at 21:16
  • $\begingroup$ @NaveenBalaji: I don't know of a single example in any of the textbooks I know but I'm only a medium level student of QM. 'On paper' PDEs of the type $(1)$ can of course be solved by various numerical, iteration methods. I don't see huge applications for time-dependent potentials though. Could be wrong on that, of course... Do you know of any? :-) $\endgroup$
    – Gert
    Oct 12, 2016 at 21:23
  • $\begingroup$ Thanks a lot! But just imagine if we could describe the tunnelling of particles in strong field barriers as they move through in space! Any textbooks you would recomened for QM apart from landau and Griffiths. $\endgroup$ Oct 12, 2016 at 21:29
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    $\begingroup$ @NaveenBalaji: F.Mandl's book('Quantum Mechanics') is quite good. Here's a good example of tunnelling models for $\alpha$ particles: hyperphysics.phy-astr.gsu.edu/hbase/nuclear/alptun.html (slow loading site) $\endgroup$
    – Gert
    Oct 12, 2016 at 21:34
  • $\begingroup$ This doesn't really answer the question. $\endgroup$
    – Javier
    Oct 12, 2016 at 21:45
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You seem to have a very clear idea about the shape of the potential, but on the other hand you ask: Are such systems practically found?

For general time dependent potential you will not solve the Schrödinger equation analytically.

You can probably find approximate solutions if:

  1. the potential changes very slowly (adiabatic approximation) or very quickly

or

  1. if the typical dimensions of the potential are much smaller or much bigger than the wave length.
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  • $\begingroup$ So if I do consider the adiabatic approximation, then how would I go about solving this system. I'm having more trouble Solving for a wavefunction that is time independent but is subject to time dependent potentials; I tried to intergrate the time-dependent hypervolumes of the changing potential functions into one single function that is continuous and static for an approaching wave in its boundary, but is time dependent and varies when the wave enters the barrier. But I don't have the mathematical rigour to do such a feat. $\endgroup$ Nov 23, 2016 at 21:06
  • $\begingroup$ I don't understand what you mean by "I tried to intergrate the time-dependent hypervolumes ....". Also I don't see any time dependence in the potentials you have. Generally I suggest to start with an easier problem (e.g. time independent potential with simple shapes), before trying to calculate something like you have here. $\endgroup$ Nov 23, 2016 at 21:16
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There are methods for finding exact solutions of the Schrodinger equation with a time dependent potential in some simple cases, such as the Lewis and Rosenfield method:

https://arxiv.org/abs/quant-ph/0307084

There are also numerical methods for solving the Schrodinger equation with a time dependent potential, such as the Crank-Nicolson method:

https://arxiv.org/abs/1412.1802

There are other methods that you can find by looking at the references of these papers.

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