Interactive simulations and visualizations augment the teaching and learning of quantum mechanics by making equations and concepts come to life. However, graphical visualizations are nearly always limited to a set of hard-coded functionalities. Text-based codes can offer a higher degree of flexibility, but only at the expense of steep learning curves or time investments. We introduce Quantum Composer, which allows the user to build, expand, or explore quantum mechanical simulations by interacting with graphically connectable nodes, each corresponding to a physical concept, mathematical operation, or visualization. Quantum Composer eliminates numerical and programming details while retaining accessibility, emphasis on understanding, and rapid feedback mechanisms. We illustrate its open-ended applicability through a series of examples in introductory and advanced quantum mechanics courses, student projects, and research environments.

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