This the lowest allowed energy the particle can have

This simulation solves the Schrodinger equation for “One Dimensional Potential Wells” of various configurations. In a real atom, the electrons are bound within a three dimensional potential well. The 1-D wells of this simulation are much more mathematically straightforward to solve and display, but still give a good feel for the discrete nature of allowed energy levels and wavefunctions that we find in more complex 3-D systems such as atoms.

1.Square well
Start with the one “square” well.

b. Describe the stationary state probability density and wavefunction φ for the next two lowest allowed energy levels.

For the next energy level allowed, the probability density resembles a sine graph, it has two "bumps", and it does not move as it increases from 0.00 fs and onward. The wave function behaves in a way that is similar to the lowest allowed energy level in part a. It keeps fluctuating. The two "bumps" on the graph look like they are alternating orientations.

This is like a finite potential well where a particle is confined to a box that has finite potential walls. Since it is finite, that means that there is a probability that the particle can be found outside of the box.This effect might represent the nanoscale world or maybe how to calculate/predict a system's behavior.

e.Click the “Superposition State” button. This box allows you to construct a wavefunction by adding together allowed wavefunction solutions (like the ones you have been looking at). For example, if you enter 0.25 in each of the first 5 coefficient boxes (c1 ……c4), you create a single wavefunction that is composed of ¼ of each of the first four lowest allowed wavefunctions. This is how quantum mechanics works in real life. Our wavefunctions are composed of many allowed solutions to Schrodinger’s equation, all superposed together.

On the right hand side, first click “reset all”, which resets the app back to its original state.

Then select “harmonic oscillator” from the potential well drop-down menu.

c.Based on what you see in (b), where is the most likely (highest probability) position within the well to actually find the particle at a given moment? Give your answer in terms of location along the horizontal axis, which is labeled Position (nm).

Yes, the particles cannot go past where the last waves end. This is because it is the waves are limited to that position. The particle could be anywhere if it was -infinity to +infinity.