- #91
IRobot
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Could we say that in the path integral formalism we don't need a Hilbert space, at least to write down the functional integral, not to calculate with it.
Quantum Mechanics without Hilbert Space
- Thread starter Varon
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The discussion explores the relationship between Hilbert space and the Schrödinger equation in quantum mechanics, questioning the necessity of Hilbert space for understanding quantum states. It highlights that while the Schrödinger equation describes system dynamics, Hilbert space provides a framework for representing states, with wavefunctions serving as shorthand for these states. The conversation emphasizes that Fourier components can be considered a basis for Hilbert space, but they represent just one of many possible bases. It also notes that while single-particle quantum mechanics can be approached without Hilbert space, more complex interactions and superpositions require its formalism. Ultimately, the dialogue illustrates the foundational role of Hilbert space in modern quantum mechanics, despite its absence in earlier formulations.
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