In quantum physics, the dimension of the Hamiltonian depends on the system being considered. The Hamiltonian is an operator that represents the total energy of a quantum system and governs its time evolution.
The dimension of the Hamiltonian operator is determined by the dimension of the Hilbert space associated with the system. The Hilbert space is the mathematical space that describes the states of the system.
In general, the dimension of the Hilbert space is related to the number of degrees of freedom of the system. For example, in a system with a single particle moving in three-dimensional space, the dimension of the Hilbert space is infinite, and the Hamiltonian operator would also have an infinite dimension.
However, if we consider a system with a finite number of particles, such as an ensemble of particles in a box, the dimension of the Hilbert space would be finite, and therefore the dimension of the Hamiltonian operator would also be finite.
It's important to note that the dimension of the Hamiltonian is not necessarily related to the physical dimension of space. The dimension of the Hamiltonian is determined by the properties of the system and the mathematical framework used to describe it, such as the number of particles and their associated quantum states.