Novel Schrodinger equation examples for 1D

[learn.steadfast]

I've been studying the 1D schrodinger equation, and getting a feel for solutions in the harmonic oscillator, or potentials of inverse radius (atomic/hydrogen), and many versions of stair-step/ square potentials (square wells.)

But, I've noticed that there are very few exact 1D potentials in the literature to study. I'm not satisfied with the conclusions I'm coming to based on these limited 1D cases.

Is anyone aware of exact/analytical solutions for Schrodinger's equation in the case where energy ramps linearly, or where momentum ramps linearly with distance?

For example, given a well with finite walls at +-a, E(particle) < V(wall) and a linear potential V(x)=x everywhere else, is there already a published exact solution?

Or, perhaps instead of V(x)=x^2 - C , as in the harmonic well; is there a known solution for the case where V(x)=C - x^2 with finite walls at some arbitrary location +-a (so as to normalise the solution)?

I'm interested in studying the effects on stationary states (purely real psi) and especially the phase of psi, where the change in momentum with distance is an easily expressed non-constant function. eg: I'm hoping for cases where either V or $\hbar k$ is linear in X over a region of the problem. ( I don't care much what the walls/ boundaries look like. )

I'm not seeing pre-worked problems on the internet or in my textbooks, and am wondering if someone has come across examples they could point me to as a time saver.

Thanks!

 

[stevendaryl]

I believe that the potential $V(x) = Ax$ can be solved using Airy functions:

https://en.wikipedia.org/wiki/Airy_function

 

[stevendaryl]

You can also reverse-engineer a solution. Pick any old square-integrable wave function $\psi_0(x)$ and then let $V(x) = \frac{\hbar^2}{2m} \frac{\psi_0''}{\psi_0}$. That potential has $\psi_0(x)$ as one solution (with energy eigenvalue 0). It's not guaranteed that there will be other exact solutions, though.

 

[learn.steadfast]

You can also reverse-engineer a solution. Pick any old square-integrable wave function $\psi_0(x)$ and then let $V(x) = \frac{\hbar^2}{2m} \frac{\psi_0''}{\psi_0}$. That potential has $\psi_0(x)$ as one solution (with energy eigenvalue 0). It's not guaranteed that there will be other exact solutions, though.

Excellent observation. Using that ratio is, in fact, the way I've been verifying Scrhodinger's equation solutions for psi, for years! I never thought about plugging in arbitrary equations to "come up" with an energy.

I doubt arbitrary functions will really give me what I need, which is a way to examine how the imaginary part of Schrodinger's solutions (psi) is related to energy stored in the spin of the electron (and therefore the "phase"); because only the simplest of momentum distributions (linear) will let me easily guess averages of translational momentum over an entire phase change of pi or 2pi radians. ( I will look at the Airy equations later, as those might give me what I need.) Linear changes in Energy vs distance (AKA constant force) aren't as simple as constant changes in translational momentum versus distance, but I might be able to guess the radius function with constant energy changes ...

Here's how I'm thinking:
Classical Rotational momentum is proportional to mass and the "radii" it is spinning at.
$p_r = mvr$

The uncertainty relationship means that there is some effetive <r> which *changes* depending on the particle's translational momentum changing vs an axis. (<r> does NOT change if translational momentum is constant, and there is no energy or momentum exchange in that case. )

We don't know exactly "how" the mass is spinning / rotating / or having effects similar to these classical concepts, but there has to be an average of radius and mass which ought to *correspond* to the classical calculations of circular rotation which is why we call instrinsic angular momentum, "spin."

This change in effective <r> means that the balanced force found in a circularly rotating mass would be unbalanced during translational acceleration of the particle (or ensemble).

Hence, energy ( force times change in radius ) is stored or retrieved from the spin of a particle during acceleration *because* the particle has a *fixed* momentum of $\hbar \over 2$ and a force is required to change the radius (Force x Distance=energy) that spin momentum effectively exists at. AKA: Fixed angular momentum does NOT mean the spinning particle contains a fixed amount of energy.

In the case of the harmonic oscillator, the "probable" particle distribution is experiencing a net force at every point in the oscillator, except at the neutral position of X=0 in the exact center of the harmonic well.

So, when I try to integrate classical momentum vs. spatial distance (from the Hamiltonian) ... and compare that to the phase rate of change predicted by actual Hermitian solutions to the harmonic well, what I find is that the rate is exactly correct at the center of the harmonic oscillator (as expected); but that the phase change is faster than computed naiievely as the particle gets farther away from the center.

The calculation is consistent with energy being extracted from the "spin" of the particle because the radii of "spin" is increasing in some uncertain way; Therefore, the translational momentum of the particle must increase in order to conserve energy extracted from the spin.

Doing some simple math, I predict that the effective radius is likely proprotional to 2/k(x) Where k is the localized momentum $\hbar k$, that is really a changing function of x.

But, when I attempt to see if that interpretation is consistent with the harmonic well ... there are frame of reference issues that make it impossible for me to guess how to correlate the change in average radius with a change in axis, x.

Note: I'm only interested in the total change (integral of translational momentum extracted from rotational momentum) over a phase change of pi or 2pi radians. Computing the change in spin's <r> vs. x (at every point), would be affected by the exact geometry and distribution of mass, which we don't know (and perhaps can't know). So, I'm only wanting to find out what the average of the energy extraction and conversion to translational momentum has to be in order for Ehrnhast's theorem to be valid over a full rotation of "spin" (phase over pi or 2pi radians.)

Thank you for your suggestions.
I will explore the Airy functions, as that might allow me to simplify the problem to one where a classical analogue can easily be made to compare the results and check the consistency of Schrodinger's equations with average classical rotational momentum and energy that must be stored in "spin" on average (classically).

 

[vancouver_water]

This is not exactly what you are asking for but I think this is also a good opportunity to practice youre numerical computing skills (which you will need whether you do experiment or theory). What I did when I took my first quantum class was to just look at some numerical solutions to interesting potentials, such periodic potentials and different symmetric and antisymmetric potentials. It gives you a good feel of what the states are doing. You could write something to solve for time dependent solutions or only the eigenstates. Both are interesting to look at and will be useful in the future when you want to get a rough idea of what is happening when you are working on analytic solutions and approximations.

 

[learn.steadfast]

Numerical methods are fine, I'm just looking for analytical examples because numerical software has to be tested and checked.
What I have now is not very stable.

I'll probably post some gnu-plot scripts, later, when I get the problem worked out with Airy equations.
( Gnuplot has the airy function, but not it's derivatives ... so, I need to work out what the energy profile is by hand. )

I do have numerical integrators that I use to solve Schrodinger's equation for stationary states ; but they aren't very stable.

I've done lots of periodic potentials for solid state semiconductor modelling, but that's generally done with step potentials models ( like Krnoig-Penny ).
Those models don't handle acceleration of individual electrons and the subtle magnetic effects caused by them, but they are easy to do with great numerical accuracy.

However, I'm wanting to get into some of the subtle details, such as the fine constant of hydrogen, Zeeman orbital splitting, and finally numerical modeling of helium (two electron, three body problem).
Knowing the potential profile is important in these cases.

I'm hoping to run an experiment (Monte carlo kind of analysis) where I track two electrons with minimum uncertainty (one wavelength) around a helium atom.
I'd like to see how the spectrum of helium is affected by doing that type of numerical simulation. I tried analytical Hartree-Fock analysis, some years ago, but I really didn't understand it and the results weren't particularly accurate.

Right now, Numerical convergence is a problem with the raw Schrodinger equation using my tools; for example, when solving the harmonic oscillator for a stationary state, the exponential decay in the classically forbidden zone will be accurate to about 4 decimal places out to a distance of ~3 wavelengths (eg: wavelengths of free particle, as the wavelength of exponential decay is infinite.) However, after tunnelling a around four to five wavelengths, the integrator will suddenly change from exponential decay , to exponential growth ... which is wrong.

In general, if I start the numerical solver on a peak location of a time invariant $\phi(x)$ in the harmonic well; The integrator *will* produce the stationary state profile all the way down to the zero crossing of phi with about 6 digits of accuracy; but it slowly looses accuracy at each zero crossing.
As a fix, I've been guessing where two adjacent peaks are, numerically integrate down to the same zero crossing, and then make a correction for whether the peaks are really closer together or farther apart. The same issue especially affects the exponential decay profile, as $| \psi(x) |$ gets close to zero ... the accuracy is compromised.

I think I have a way to counteract the numerical instability, by accounting for spin; but I need a slightly simpler 1D analytical model to test the integrator with and calibrate it for numerical round off correction.