# Questions tagged [hamiltonian]

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

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### Discretization of Hamiltonian with first derivative

In a particular 1D system, the Hamiltonian can be writen as $$H=\mathrm{i}\left(f(r)\frac{\partial}{\partial r}+\frac{1}{2}f'(r)\right)\; ,$$ wher $\mathrm{i}$ is the imaginary unit, and $f(r)$ is a ...
0answers
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### How to interpret overlap in Hamiltonian if it is not a degeneracy?

In Fruchart et al.'s An Introduction to Topological Insulators, the Bloch Hamiltonian for a two-band insulator is given in the general form $H(k)=$ \begin{bmatrix} h_0+h_z & h_x-i h_y \\ ...
1answer
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### Eigenvalues of the Hamiltonian

Is every eigenvalue of the Hamiltonian a form of energy? If not are there values of the Hamiltonian that do not correspond to the energy of the system?
1answer
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### Is the tight binding model an effective free fermion model?

The tight-binding Hamiltionian has the form $$H=-t\sum_i\left(c_i^\dagger c_{i+1} + c_{i}c_{i+1}^\dagger\right)$$ But does this mean that it can be represented in the form of free fermion modes?
2answers
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### How does a Hamiltonian 'generate' a unitary?

I know that the unitary (propagator) is given by $$U=e^{iHt}\tag{1}.$$ But I actually never saw how a Hamiltonian translates into a unitary. For example when I consider a two-level rotation in a ...
1answer
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### Hamiltonian of a quantum heat bath

I have seen the Hamiltonian for a heat bath written as: $$H_B = \hbar \int_0^\infty \omega b(\omega)^\dagger b(\omega) d\omega$$ I was hoping to understand this equation better. This suggests that ...
0answers
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### What is the 4x4 matrix for the charge inversion operator and how do you construct it?

I have a 4x4 Hamiltonian describing a part of my system. To get a holistic view of the situation I need to do a charge inversion on the matrix. What is the 4x4 charge inversion operator? And what is ...
2answers
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### Wilson Sommerfeld Method to solve for Energy

I have an example in my notes to find the quantum energy levels when the Hamiltonian is $H(p,q)={p^2}/{2m}+(mw^2q^2)/2$. However when given the Hamiltonian $H(p,q)={p^2}/{2m}$, I'm having ...
0answers
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### Mathematical representation of Symmetry Transformation

Consider a general Hamiltonian that is made up of three terms $\mathcal{H}$ = term I + term II + term III . Suppose the combination of charge conjugation and parity (CP) is a symmetry of this ...
1answer
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### What people mean by “state evolving with the interacting/free theory”?

This is a quite basic question but I confess it is something I didn't get up to this point. When defining the Moller operators and hence the $\cal{S}$-matrix one usually considers "states $\Psi$ ...
1answer
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### Potential must be real for Hamiltonian to be Hermitian?

I have seen a few proofs specify for finite wells, step functions, and harmonic oscillators, that $V$ must be real for $H$ to be Hermitian. Why is that? If we're solving the Schrodinger equation, we ...
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1answer
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### Hamiltonian time-independent, partial derivative always zero?

For conceptual simplicity, let's restrict the discussion to systems with a two-dimensional phase space $\mathcal P$ with generalized coordinates $(q,p)$. Hamiltonian is a function that maps a pair ...
0answers
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### Quantizing the double pendulum hamiltonian

So, just for kicks, I thought it might be fun to try to see what happens if you have a "quantum double pendulum". Take a simple point pendulum with mass $m$ and length $l$, and hang a second identical ...
0answers
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### Time dependence of the momentum operator for a free particle

I was studying Modern Quantum Mechanics by Sakurai, and at the page 85, it is given the analysis of a free particle. There, the author assumes that Hamiltonian is $$\hat H = \frac{ \hat p ^2}{ 2m},$$...
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1answer
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### Action of complex conjugation on Hamiltonian

Consider a finite-dimensional non-relativistic QM system with hamiltonian $H$. Let $K$ denote the complex conjugation operator. What does $K H K$ simplify to, if the system is: (a) spin-zero; (b) spin-...
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1answer
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### Can we abstractly deduce the $L^2$ is conserved assuming only rotational symmetry of Hamiltonian?

Here $L^2$ is defined as $$L^2=L_x^2+L_y^2+L_z^2$$ representing the observable of the magnitude of the angular momentum. There are a lot of proofs showing the $z$-projection of the angular ...
1answer
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### How to define the Hamiltonian properly in quantum field theory

In a rigorous fashion, how does one define the Hamiltonian of QFT as $$\hat{H}(t) = \int d^3x \hat{\mathcal{H}}(x, t)$$ For now I'm ignoring the fact that $\hat{\mathcal{H}}$ itself may be ill-...
1answer
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### Exponential of a Hamiltonian Matrix

I am trying to understand a problem which involves a two-level system given by: $$\begin{pmatrix} C1(t) \\ C2(t) \\ \end{pmatrix}=e^{-i\hat{H}t/2} \begin{pmatrix} 0 \\ 1\\ \end{pmatrix}$$ I am ...
3answers
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### Domains of $H$ and $U(t) = \exp(-iH t )$

I am not so familiar with functional analysis. But in my impression, the Hamiltonian $H$ is often not defined everywhere on the Hilbert space. On the other hand, the time evolution operator $U(t)$, ...
0answers
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### A short question regarding Hamiltonian [duplicate]

Can any one please tell me in what cases the hamiltonian is not Equal to Total energy. My guess, albeit educated, is if the potential is either a function of time explicitly or a function of velocity, ...
0answers
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### Can the Dirac Hamiltonian accommodate a variable speed of light?

The Dirac Hamiltonian has the form1 $$\left[\beta m c^2+c\sum_{n=1}^3\alpha_np_n\right]$$ where $\alpha_n$ and $\beta$ are Hermitian matrices, and $c$ is the speed of light. My question: Is there a ...