The Hamiltonian operator determines the Hilbert space of a quantum system
Physics discussions establish that the Hamiltonian is an operator defined on a pre-existing Hilbert space rather than determining the Hilbert space itself.
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Does the Hilbert space include states that are not solutions of the Hamiltonian?. https://physics.stackexchange.com/questions/280725/does-the-hilbert-space-include-states-that-are-not-solutions-of-the-hamiltonian
# Does the Hilbert space include states that are not solutions of the Hamiltonian? Tags: quantum-mechanics, wavefunction, hilbert-space, schroedinger-equation, hamiltonian - Score: 11 - Views: 2101 - Answers: 4 - Answered: yes - Asked by: P. C. Spaniel (5086 rep) - Asked: 2016-09-17 - Edited: 2016-09-17 - Site: physics ## Question I've studied Quantum Mechanics and I know the usual answer "The dimension of the Hilbert Space is the maximum number of linear independent states the system can be found in". There is something about this statement that bothers me, let me try to explain it. Imagine a particle whose dynamics satisfy Schrodinger equation. Before we give it a hamiltonian, in principle the particle can have any Square-integrable continuous function as a state. When we write a particular Hamiltonian we find the actual eigenstates of the particle and then every possible state is a linear combination of that eigenstates. Now, acording to the first definition of the hilbert space, it has all the eigenstates of the hamiltonian. Now, several questions arise: 1) Does the Hamiltonian determine the Hilbert Space? 2) What if I make two particles with diferent Hamiltonians intera
Why do we need both Hamiltonian and Hilbert Space to specify a .... https://physics.stackexchange.com/questions/246647/why-do-we-need-both-hamiltonian-and-hilbert-space-to-specify-a-quantum-system
# Why do we need both Hamiltonian and Hilbert Space to specify a Quantum System? Tags: quantum-mechanics, hilbert-space, hamiltonian - Score: 8 - Views: 1954 - Answers: 2 - Answered: yes - Asked by: Wei-Ting Kuo (91 rep) - Asked: 2016-03-31 - Edited: 2016-03-31 - Site: physics ## Question From my understanding, when we have the Hamiltonian, in principle we can know the eigenstates for our system of interest. Then, we can calculate everything we want. In addition, these eigenstates will form a Hilbert space of our quantum system. It seems that having Hamiltonian is enough to specify a quantum System. However, there are some textbooks which mention we need both Hamiltonian and Hilbert Space to specify a quantum System. Why do we really need Hilbert space to specify our quantum system? ## Answers ### Answer by Yly (score: 9 [ACCEPTED]) There is a mathematical reason and a (sort of) physical reason: Mathematical reason: The Hamiltonian is an operator on a Hilbert space to begin with. Without knowing a Hilbert space, it doesn't even make sense to speak of an operator on it. Physical reason (sort of): What looks like the same Hamiltonian, e.g. the free particle Hamiltonian
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