Quantum Spin Visualizer - Readme

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Preface

To understand the mathematical notations in this document, you may want to have a look at the summary of mathematical notations and conventions before reading on.

Introduction

The Quantum Spin Visualizer is an interactive application, which visualizes the spin state of quantum objects. It can visualize different types of objects, including spin 1 objects like photons and spin 1/2 objects like electrons.

Technically, a spin state is represented as a multi-dimensional array of complex numbers, in the simplest case as a two-dimensional array. These arrays are then visualized as 2D images, where the individual complex numbers correspond to the pixels of the image. The phase angle of a complex number is encoded as the pixel color, and the magnitude of the complex number is encoded as color intensity.

These visualizations are very helpful to understand the spin of quantum objects, because they mirror many mathematical properties of spin states, in particular:

Feature Overview

The application not only visualizes a single spin state. Instead, it visualizes the process to construct a particular spin state by superimposing scaled and phase-shifted basis states according to the following formula:

\[ {\ \color{blue} \ket{\psi}} = a_1 {\ \color{blue} \ket{B_1}} + a_2 {\ \color{blue} \ket{B_2}} \]

Here \({\ \color{blue} \ket{B_1}}\) and \({\ \color{blue} \ket{B_2}}\) are basis states which are scaled and phase-shifted by the complex numbers \(a_1\) and \(a_2\) respectively, and then superimposed to get the result state \({\ \color{blue} \ket{\psi}}\). In the case of a two-level quantum system, you can get all possible system states by choosing the appropriate values for \(a_1\) and \(a_2\).

The application visualizes all of the following states: The basis states \({\ \color{blue} \ket{B_1}}\) and \({\ \color{blue} \ket{B_2}}\), the intermediate states \(a_1 {\ \color{blue} \ket{B_1}}\) and \(a_2 {\ \color{blue} \ket{B_2}}\), and the result state \({\ \color{blue} \ket{\psi}}\). In addition, it also visualizes the values of the complex numbers \(a_1\) and \(a_2\).

The user interface is organized as a grid, which has five distinct rows:

Configuration Controls

The configuration controls in Row 1 are used to configure the following properties:

There is one hidden feature: If you press the keyboard button X, an additional drop-down field “visualization” is shown. It can be used to visualize the individual array layers of 3D spin 1/2 representations. For details, see section State Representation and Visualization.

Buttons to Manipulate the Amplitudes

The buttons in Row 2 are used to manipulate the amplitudes in row 4.

The buttons in the left/right columns control the left/right amplitudes as follows:

The buttons in the middle column affect both the left and the right amplitude:

By default, the step widths of the buttons are as follows:

By activating the check box small steps, the step widths are divided by two.

There is one hidden feature: If you press one of the phase rotation buttons for individual amplitudes while holding the shift key, the other phase will be rotated in the opposite direction. This is helpful to rotate a 3D spinor state.

State Representation and Visualization

Each state (basis state, scaled basis state, and result state) is internally represented as a complex-valued array. In case of the 2D object types, a 200x200 array is used. In case of the 3D spinor type, a 200x200x2 array is used.

Visualization of a 200x200 array: A 200x200 array is visualized by converting each complex number into a color. The state is then visualized as an image with 200x200 pixels with the corresponding colors.

Each complex number is mapped to a color as follows:

Visualization of a 200x200x2 array: In this case, the last dimension encodes a local spinor structure. By default, a local spinor structure with components \((c_1, c_2)\) is mapped to a color as follows: First, the complex value \(\pm \sqrt{c_1 c_2}\) is computed. Then this value is mapped to a color like in the 200x200 array case.

You can switch to a different visualization, which shows one of the individual spinor structure layers, as follows. Press X on the keyboard and then choose layer 1 or layer 2 in the appearing drop-down field “visualization”. If you choose layer 1, the \(c_1\) values are visualized as colors. If you choose layer 2, the \(c_2\) values are visualized as colors.

Adding and scaling state vectors: The operations to add and scale complex-valued arrays are straightforward:

Object type: 2D spin 1 (photon)

This mode visualizes a spin-1 model in two dimensions. Such a model represents spin-1 particles with zero rest mass. Examples include photons.

The application provides the following sets of basis states:

This object type has a close relation to the EM spin visualizer. The EM spin visualizer also represents photon spin states, but in a completely different way. However, both representations show the same high-level behavior. In mathematical terms: They are isomorphic.

Object type: 2D spin 1/2 (neutrino?)

The modes 2D spin 1/2 v1 and 2D spin 1/2 v2 both visualize a spin 1/2 object in two dimensions. Compared to the spin-1 mode, the phase structure has a square-root-like character: when traversing a full spatial loop, the internal phase behaves like a half-turn object.

This model represents spin 1/2 particles with zero rest mass. It is unclear whether such particles really exist in nature. It is possible (but not proven) that there is a neutrino flavor with zero rest mass. This is why the neutrino is given as an example with a question mark. But it is not important whether such a particle really exists. The main purpose of this model is educational: It presents the simplest-possible spin-1/2 representation one can think of.

There are two variants of this model:

Both variants, v1 and v2, represent the same information. They are mainly distinguished for educational reasons.

Object type: 3D spin 1/2 (electron)

This mode visualizes a three-dimensional spin-1/2 object. Such a model represents spin-1/2 particles with non-zero rest mass. Examples include electrons.

Conceptually, the visualization depicts a 3D sphere. Each point on the 3D sphere is a local spinor structure, whose state is represented as a color.

Technically, the basis states are computed by projecting the 3D sphere onto the 2D plane, resulting in a 200x200x2 array of complex numbers. This array is then visualized as described in section State Representation and Visualization. All further operations like scaling and adding are performed by scaling and adding the 200x200x2 arrays.

The following sets of basis states are supported:

Here are some important properties of this object type and its visualization:

This object type has a close relation to the Bloch sphere representation. The flag direction of the Bloch sphere representation corresponds to the rotation axis of the 3D spinor representation.

Hidden features

The application has the following non-obvious features: