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Implement a semi-classical visualization of a quantum oscillator using GlowScript (VPython). Set up a 3D scene with a specified width and height of 700 pixels, a white background, and a black foreground. Title the scene "Quantum Oscillator".

Create functions and visual objects to model a quantum harmonic oscillator: define a potential energy function U(s), two masses represented by spheres, a spring (helix) connecting the masses, and energy level lines (levels) that the user can select. The potential well is visualized with a curve, and energy levels are radii-creatures positioned at different energies. The scene includes interactive elements where clicking on an energy level will put the oscillator into that energy state, visually indicated by changing the color of the level to red and restoring the previous level's color to white. The system's parameters like mass, spring constant, and energy levels are configurable, and the simulation updates dynamically, with the oscillator's position animated according to classical mechanics principles.

The code involves binding mouse click events to select energy levels, calculating the oscillator's amplitude based on the selected energy, and updating positions in a continuous loop with a small timestep. This setup provides a semi-classical approximation of quantum behavior in a visual, interactive manner suitable for educational demonstrations or research visualizations.

Paper For Above instruction

The visualization of quantum oscillators through semi-classical models is an engaging approach that bridges classical mechanics and quantum theory's conceptual understanding. Using GlowScript, a platform that allows for 3D visual programming in Python, provides an accessible way to simulate and animate such systems, making abstract quantum concepts more tangible.

In this implementation, the scene is initialized with a fixed size and color scheme to clearly delineate the elements. The potential energy curve, representing the quadratic form of a harmonic oscillator, is plotted across a range of displacements, illustrating the classical potential well where the oscillator resides. The energy levels are depicted as horizontal cylinders positioned at discrete energy values, serving as quantized states akin to those found in quantum systems. Users can interact with these levels via mouse clicks to select the desired energy state, which is visually highlighted by changing the color to red. This interaction mimics the quantum measurement process, where the system "collapses" into a specific energy eigenstate.

The model employs two masses connected by a spring to simulate oscillatory motion. One mass is fixed at one end of the spring, and the other oscillates based on classical physics equations, with its position updated iteratively. The amplitude of the oscillation is derived from the selected energy level, integrating quantum mechanical quantization into classical motion. The position update relies on basic harmonic motion principles, with angular frequency determined by the spring constant and the mass.

The visualization layer is enhanced with additional elements such as vertical lines and well curves to provide context and clarity. These elements help observers visualize the energy landscape and the oscillator's position within it. The program's event-driven structure facilitates real-time interaction, with mouse clicks triggering updates to the oscillator’s energy state, which then influences the subsequent motion in the simulation loop.

Overall, this semi-classical model serves as a pedagogical tool, illustrating how quantum states can be represented in a classical framework via quantized energy levels and oscillatory motion. It emphasizes the connection between energy quantization and classical amplitude, providing students and researchers with an intuitive understanding of quantum harmonic oscillators. Such simulations have broader applications in teaching quantum mechanics, exploring quantum-classical correspondence, and developing visual intuition about subatomic phenomena.

References

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