I have a finite harmonic potential where I trap an electron. The confinement length changes in size. Now, I'm interested in the ground state energy, so I have this 1D Poisson solver which gives me the ground state energy $E_0$ and wave function.

If we now add a confinement dimension, can we estimate how $E_0$ is going to change? Let's assume the second dimension has the same shape, so our new potential well is just a finite 2D parabola.

For the case on the left, we can say we're safely below the energy continuum, so the energy formula for a N-dimensional harmonic potential is a good estimate:

$$E_{n_x, n_y} \approx \left(n_x + n_y + \frac{N}{2} \right)\hbar \omega $$

Adding the additional confinement will therefore just double $E_0$. But how can we estimate the new $E_0$ for the case on the right where we're close to the continuum?

Finite harmonic potentials of different size with ground state energy and wavefunction

My guess is that for the situation on the right, the new $E_0$ is going to be roughly in the middle between the energy continuum and the old $E_0$. However, I don't have a justification for that.

How can we treat this problem? Is there some good physical argumentation for why this should be like that (or why it should be different)? Are there approximate formulas for this case?


For my case, calculating the confinement potential (which is not exactly but roughly harmonic) is computationally intensive, that's why I don't want to compute the whole 2D potential landscape. It would be nice to solve the 1D case and then have a rough estimate for what $E_0$ is going to be in 2D.


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