Quantum mechanical transistors, particularly single electron transistors and quantum dots, operate by confining electrons in a quantum well, allowing for precise control over their energy levels through voltage applied to a gate. This gate voltage influences the occupancy of electron states, making these systems highly sensitive to even minor voltage changes, capable of detecting single electrons. The discussion also mentions the DELTT (Double Electron Layer Tunneling Transistor) and seeks additional technical information and diagrams related to its quantum mechanics. Overall, the conversation emphasizes the intricate relationship between quantum mechanics and transistor functionality. Further inquiries into specific technologies and their applications are encouraged.