# Questions tagged [hamiltonian]

The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.

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### When is the Hamiltonian of a system not equal to its total energy?

I thought the Hamiltonian was always equal to the total energy of a system but have read that this isn't always true. Is there an example of this and does the Hamiltonian have a physical ...
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### The formal solution of the time-dependent Schrödinger equation

Consider the time-dependent Schrödinger equation (or some equation in Schrödinger form) written down as $$\tag 1 i\hbar \partial_{t} \Psi ~=~ \hat{H} \Psi .$$ Usually, one likes to write that it has ...
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### Example where Hamiltonian $H \neq T+V=E$, but $E=T+V$ is conserved

I'm looking for an example of a Hamiltonian $H$, where $H\neq T+V$, but the total energy in the system, $E=T+V$, is still conserved. While I'm at it, I might as well add that I'd be most interested ...
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### When is a quantum state stationary?

If a quantum state is an eigenstate of the Hamiltonian, then it is stationary. But can a state be stationary if it is not an eigenstate of the Hamiltonian? If yes, how can one prove whether a state is ...
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### Difference between the energy and the Hamiltonian in a specific example

The problem is the following: Consider a particle of mass $m$ confined in a long and thin hollow pipe, which rotates in the $xy$ plane with constant angular velocity $\omega$. The rotation axis ...
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### Why can the Schroedinger equation be used with a time-dependent Hamiltonian?

I have a puzzle about Schroedinger equation with time-dependent hamiltonian, which is usually used in time-dependent quantum systems. However, one of the axioms in quantum mechanics postulates that ...
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### Why do excited states decay if they are eigenstates of Hamiltonian and should not change in time?

Quantum mechanics says that if a system is in an eigenstate of the Hamiltonian, then the state ket representing the system will not evolve with time. So if the electron is in, say, the first excited ...
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### Detailed derivation and explanation of the AKLT Hamiltonian

I am trying to read the original paper for the AKLT model, Rigorous results on valence-bond ground states in antiferromagnets. I Affleck, T Kennedy, RH Lieb and H Tasaki. Phys. Rev. Lett. 59, 799 (...
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### Does Hamilton Mechanics give a general phase-space conserving flux?

Hamiltonian dynamics fulfil the Liouville's theorem, which means that one can imagine the flux of a phase space volume under a Hamiltonian theory like the flux of an ideal fluid, which doesn't change ...
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### Dealing with tensor products in an exponent

I am looking at the following problem and I am struggling to follow the steps involved. Consider the non-interacting Hamiltonian $$H_{AB}=H_A\otimes I_B+I_A\otimes H_B$$ So I'm trying to prove that ...
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### How can an inverted anharmonic potential $V(x)=-x^4$ have discrete bound states?
I've been watching the lectures on mathematical physics by Carl Bender on youtube where he uses the non-Hermitian Hamiltonian methods to prove that the inverted anharmonic potential $V(x)=-x^4$ has a ...