Welcome to the user documentation for QBox, simulation software for solving Schrödinger’s equation in two dimensions in real time. In this documentation I will assume you are familiar with standard concepts and terminology in quantum mechanics. For a brief introduction to the quantum mechanics used in QBox and for further reading see [Coming soon].

## 1. What is QBox

As stated above QBox solves Schrödinger’s equation in two dimensions. In particular this software is based on the eigenvalue problem formulation of Schrödinger’s equation. The energy eigenvalues and eigenstates have been precomputed for a fixed set of potentials which are then used to solve for the time evolution of arbitrary initial states defined by the user. In this app for each of the included potentials, the user is able to view the energy eigenstates, construct the initial wave function and simulate the time evolution of said initial state.

## 2. Display of Quantum States

As the primary purpose of this app is to view time dependent states, let us start with the display of quantum states. While the states being displayed can be energy eigenstates or an arbitrary state evolving according to the time dependent Schrödinger’s equation, how the states are displayed and the accompanying controls remain the same. A quantum state has a complex value on each point of the simulation domain. This is represented visually by one of three display modes that show some aspect of the state. The display modes are “Probability Density”, “Real Part”, “Imaginary Part” and “Phase Angle”, all of which display the state against a grayscale representation of the potential in the background.

### 2.1. Display Modes

In the “Probability Density” display mode the probability density of the state is represented by the brightness in the red channel. In the “Real Part” and “Imaginary Part” display modes the corresponding of the complex state is represented, where positive values are shown as brightness in the red channel and negative values are shown as brightness in the green channel. In the “Phase Angle” display mode both the magnitude and the phase angle of the state is represented. The phase angle is shown as the hue and the magnitude is shown as the brightness.

### 2.2. Display Controls

The display mode can be changed pressing the tab key. This will cycle through the modes in the order given above. The brightness of the modes can be changed by holding down the + or - keys, this can be used to view details in a low intensity region or to avoid clipping.

## 3. Program Modes

QBox has two modes of operation, “Steady States” and “Simulation”. The buttons to switch the program mode are on the left side of the screen and the current program mode is highlighted in green. Note that in simulation mode the user is first asked to set the initial wave function before running the simulation. Pressing the “Simulation” mode button will reset the simulation and return to the initial wave function selection screen.

This mode allows the user to view all of the energy eigenstates that have been loaded for the current potential. When this mode is activated it starts out on the ground state, then pressing the up arrow key switches the state being displayed to the next excited state and pressing the down arrow key will switch it to the next lower state. Note that pressing the up arrow key when at the last loaded state will do nothing, likewise for pressing the down arrow key when at the ground state.

### 3.2. Mode: Simulation

This mode has two parts, first the user is presented with an option for setting the initial state to simulate. Second, after pressing the “Run” button the initial state will evolve according to the time dependent Schrödinger’s equation.

#### 3.2.1 Initial State

When defining the initial state, the user is presented with some options for constructing the state “Gaussian” and “Color Wheel”. When finished with one of these tools, the user can keep using these tools to add more components to the initial state, and when finished the user can press the “Run” button to continue to the simulation. When using the tools, the current initial state is hidden until the user is finished with the tool.

##### Gaussian

This tool is used to add a Gaussian function to the initial state. The width of the Gaussian and the speed multiplier are set with the sliders. To change the position and velocity of the Gaussian press the “Set Position” button, then left click and drag on the screen. The location of the Gaussian is set to where the mouse button is first pressed and the velocity vector starts at where the user releases the mouse button and ends at the center of the Gaussian. All else aside when running the simulation the Gaussian will move away from the direction in which the mouse was dragged with a speed proportional to the distance that it was dragged. When changing any of these settings a low resolution version of the Gaussian is shown temporarily so the controls can be more responsive. The following are a few tips:

• Avoid placing the Gaussian where the potential is changing rapidly. These configurations require high energy states and will introduce lots of noise

• Use either the “Real Part” or “Phase Angle” display mode when setting the velocity. The periodic changes in phase angle visible with these modes makes it easier to judge direction and velocity

• It can be useful to set the position and velocity direction first, and then use the “Speed” slider to fine tune the speed afterwards.

##### Color Wheel

This tool is used to add a superposition of energy eigenstates to the initial state. The superposition is constructed by getting the user to select the probability amplitude of the individual energy eigenstates. The selection starts with the probability amplitude for the ground state and then after each selection it moves to the next excited state. The interface for selecting a probability amplitude consists of a color wheel where clicking on a point in the wheel determines the phase and magnitude of the probability amplitude. The differences in hue around the color wheel indicate the phase while the distance out from the center indicates the magnitude. One catch is that when the user has finished selecting the states it should be normalized, so in the process of selecting the states the sum of squared amplitudes must not exceed a value of one. To enforce this condition the wheel splits into two portions by radius. Out side of this radius the wheel is switched to grayscale and clicking here does nothing. The radius of the selectable region of the color wheel indicates the maximum magnitude that is allowed given the previous selections. If the user keeps selecting states until the colored region goes to zero, then the set of selections is necessarily normalized. However, if the user exits the editor before this point, then the set of probability amplitudes will be normalized before adding it to the initial state.

#### 3.2.2 Simulation

After pressing the “Run” button the simulation will begin. Pressing “Run Settings” provides options to restart the simulation from the beginning and a slider to adjust the simulation speed.

From the “Menu” button the user can access the potential selection menu, the options menu and open the documentation. From the “Select Potential” menu the user can select from one of the precomputed potentials. Clicking on one of the options will load the selected potential along with the energy eigenvalues and eigenstates. From the “Options” menu the user can reset the dialogue boxes and set the number of states to be loaded. Clicking the “Exit” button will close QBox.

## 5. Info button

Pressing the “Info” button opens a window on the lower portion of the screen with some information about the display, simulation settings and the current state shown on screen. This information includes:

• Potential: Name of the currently selected potential
• Display Mode: Name of the active display mode
• Expected Energy: The expected value when measuring the energy of the state
• Time: The time elapsed since the beginning of the simulation
• Potential Energy range: Range of the potential energy in the simulation region
• Length: Length of the simulated region
• Mass: Mass of the particle being simulated

All units are given in Hartree atomic units. So length is given in terms of the Bohr radius. Mass is given in terms of the electron mass. Energy is given in terms of the Hartree energy, which is approximately twice the ionization energy of the ground state of atomic hydrogen. Time is given in terms of Planks reduced constant divided by the Hartree energy.