# Tagged Questions

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

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### Temperature in the Hamiltonian limit

There is a well known connection between statistical mechanics in D spatial dimensions and quantum field theory in D-1 spatial dimensions. Changing the temperature in statistical mechanics corresponds ...
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### Is it possible to reconstruct the Hamiltonian from knowledge of its ground state wave function?

Is it possible to "construct" the Hamiltonian of a system if its ground state wave function (or functional) is known? I understand one should not expect this to be generically true since the ...
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### Minimum gap between consecutive energy levels?

Assume a standard one-particle, non-relativistic Hamiltonian of the form $$H=\frac{p^2}{2m}+V(r)$$ and denote its eigenvalues as $E_{n,\tau}$, where $n$ is the principal ...
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### What is the form of the kinetic energy operator on a one-dimensional (real space) lattice? (In second quantization)

I'm trying to figure out how one would write down the Hamiltonian of a free fermion system (eventually in second quantization) on a one dimensional lattice and I'm having trouble both coming up with ...
679 views

### Increasing a potential causes increase in energy levels

Suppose a potential $V(x)$, and suppose a bound particle so the allowed energy levels are discrete. Suppose a second potential $\widetilde{V}(x)$ such that $\widetilde{V}(x) \geq V(x)$ for all $x$ (...
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### Systems with extensive ground state degeneracy

This is sort of a follow up to this question: What does it mean for a Hamiltonian or system to be gapped or gapless? There it is stated in one of the answers that a system is gapped if it fulfills ...
661 views

### The formal solution of the Schrödinger equation

Consider the Schrödinger equation (or some equation in Schrödinger form) written down as $$\tag 1 i \partial_{0} \Psi ~=~ \hat{ H}~ \Psi .$$ Usually, one likes to write that it has a formal solution ...
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### Does the unboundedness of the potential mean necessarily there is no normalizable state? [closed]

Consider the Hamiltonian $H = p^2 + V(x)$. Suppose the potential $V$ is unbounded from below in at least one direction ($x \rightarrow \pm \infty$). Does this necessarily mean that there exists no ...
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### Normalizable eigenvectors of the inverted harmonic oscillator

Consider the inverted harmonic potential $V(x) = - x^2$. Does the corresponding Hamiltonian $$H = p^2 - x^2$$ have any normalizable eigenstate? How about $$H = p^2 - x^4 ?$$ Any good ...
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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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### Quantum Mechanics: Relate solutions for two dual hamiltonians?

Consider a Hamiltonian in quantum mechanics: $$H_x=-\frac{d^2}{dx^2}+V(x,c)$$ where $x\in\mathbb{R}$ and the potential $V(x,c)$ depends on position $x$ and a continuous parameter $c$. Furthermore, ...
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### Using Dyson formula in Schrodinger picture

From Time-ordering and Dyson series and what I learnt, Dyson formula is used in the situation of interaction picture: $$i\frac{dU_I}{dt} = H_{I}(t)U_I$$ where $H_I(t)$ is interaction Hamiltonian ...