# Tagged Questions

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

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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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### 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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### 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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### 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 ...
1answer
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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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### 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 ...
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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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### 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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### Why are general wave functions expressed in terms of energy eigenfunctions?

I have read that the eigenfunctions of any hermitian operator can be used as a basis to express any function, but I have only ever really seen the eigenfunctions of the Hamiltonian used. Why is this? ...
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### Finding the initial state in the power method for Hamiltonian diagonalization

In section III of the lecture note Chapter 1: Exact Diagonalization, Weimer has described the Power method for Hamiltonian diagonalization. The process requires the choice of an random initial state ...
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### how many can we build a set of eigenbasis which describes arbitrary physical system?

Suppose Hamiltonian $H\phi = E\phi$. we can choose eigenstates of Hamiltonian by finding operator $A$ which is $[A,H] = 0$. Does it means that every operator which commutes with $H$ can have same ...
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### Why does time-independent Hamiltonian not depend on angle variable?

In Landau and Lifshitz Mechanics, $\S50$ Canonical variables a time-independent Hamiltonian is considered, and a canonical transformation is done such that adiabatic invariant $I$ becomes the new ...
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