Dimension of a Hilbert Space

[Alan Grayson]

For a given system, how can we determine if its Hilbert Space dimension is finite or infinite? TY, AG

[Alan Grayson]

On Monday, November 18, 2024 at 10:30:46 PM UTC-7 Alan Grayson wrote:

For a given system, how can we determine if its Hilbert Space dimension is finite or infinite? TY, AG

My best guess is that it depends on the possible solutions of the observable of the system under consideration. So, for the Hydrogen atom, assuming we're solving for its energy states, since the number of energy states is countably infinite, so is the corresponding dimension of its Hilbert Space, and the same holds for other elements in the Periodic Table. If we're solving for the spin of a half-spin particle, the dimension of its corresponding Hilbert Space is two (2). But suppose we're solving for the x-position of a free particle? Will its corresponding Hilbert Space have an infinite dimension which is uncountable? TY, AG