# Questions tagged [hamiltonian]

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

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### Prove conservation law in quantum mechanics

I major in Math, and I am studying Quantum Mechanics (QM). I see the conservation law in QM as a mathematical theorem. Please check if my understanding is right, and help me to prove the theorem? ...
2answers
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### Solution of Time-dependent Schrodinger Equation for Unitary Operator

While reading Quantum Mechanics Book by Sakurai, I found the time-dependent Schrodinger equation for Unitary Operator. $$i\hbar \frac{\partial}{\partial t}\mathcal{U}(t,t_0)=H\mathcal{U}(t,t_0).$$ ...
0answers
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### Hamiltonian for a mode-shift operator

I have a discrete multi-level degree of freedom in my quantum system (for photons, for example this), which I write as $|l\rangle$. The degree of freedom is unbounded, i.e. $l$ can take ever positive ...
1answer
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### Symmetry of the hamiltonian $H = \frac{1}{2m}p^2 + V(r) + a \, \vec{s} \cdot \vec{l}$

Consider the hamiltonian \begin{align} H& = H_0 + a\, \vec{s} \cdot \vec{l} \\& = \frac{1}{2m}p^2+ V(r) + a\, \vec{s} \cdot \vec{l}, \end{align} where $V(r)$ denotes an arbitrary central ...
0answers
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### Does all measured quantities by Hamiltonian operator should be real valued? [duplicate]

am not familiar with QM , and i have checked web many times to know wether all measured Quantities of any arbitray system should be real since it is Hermitian ?
2answers
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### Commutation with unspecified potential function

Instead of a potential given like $V(r) = k r^2$ or $V(r) = y^2$ , if the potential is given like in the form a function but not clearly specified, can we tell that if that commutes with the ...
1answer
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1answer
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### How to diagonalize the BCS Hubbard Hamiltonian using the Bogoliubov transformation?

How do I diagonalize the following BCS (Bardeen-Cooper-Schrieffer) Hubbard Hamiltonian: \begin{equation} H= \sum\limits_{k \in [-\frac{π}{2}, +\frac{π}{2}[} \begin{bmatrix}c^\dagger_k & c^\...
1answer
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### How can we prove that the Hamiltonian for any quantum system is Hermitian? [closed]

By applying partial time derivative to $$\psi_t \rightarrow U \psi_{t_0}$$ we end up with an expression for the Hamiltonian $$H = i\hbar\frac{\partial U_{t}}{\partial t}U^{\dagger}_{t}$$ where $U$ is ...
3answers
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2answers
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### Differentiation of a ket vector with respect to a spatial dimension

Consider a state $|\psi\rangle$. While discussing the Schroedinger equation, we say $$\hat{H}|\Psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle$$ We also define the hamiltonian operator ...
0answers
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### Simple way to modify the diagonal elements of the hamiltonian after adding a strong interaction

Let's say we have a hamiltonian of a non-interactive system of two particles. We have correctly worked out the matrix form of the hamiltonian. Now, if we add a very strong attractive interaction ...
1answer
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### Dyson Series Iteration - Gives Exact Solution?

When we derive the Dyson series for usage as the time evolution operator in the case of a time dependent Hamiltonian, we start with the equation: \begin{align}\hat{U}_I(t,t_i) = 1 - \frac{i}{\hbar}\...
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### Tight Binding Hamiltonian for graphene

The TB Hamiltonian for the tetragonal lattice is $\hat H_0 = -J\sum_{m,n} (\hat a_{m+1,n}^\dagger \hat a_{m,n}+\hat a_{m,n}^\dagger \hat a_{m,n+1}+h.c.)$ How can this be derived for the hexagonal ...
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### Why are transformations $(q,p)\to (Q,P)$ that are canonical, more useful than any $(q,p)\to(Q,P)$?

I am facing a difficulty in understanding canonical transformations. It can be defined as a transformation $$(q_i,p_i)\to \big( Q_i(q_i,p_i,t),P_i(q_i,p_i,t)\big) \tag{1}$$ under which the Hamilton'...
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### What is the easiest system to take the matrix representation of a Hamiltonian?

To understand how the unitary operator, preserve the inner products, I wanted to explore the unitary operator as a matrix. Now the equation for the unitary operator (time evolution operator) has a ...
1answer
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### Unitary Transformation of an Interfering Beam Splitter

I was reading this research paper Quantum interference enables constant time quantum information processing and was confused by one particular expression involving the Hamiltonian of a beam splitter. ...
1answer
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### Time-dependence of free and interacting Hamiltonians

Consider an interacting field theory with Hamiltonian $$H=H_0+V$$ where $H_0$ is the Hamiltonian of the free theory and $V$ is the added interaction. Now, I know the full Hamiltonian $H$ should be ...
1answer
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### Why does the Hamiltonian of a Lagrangian that consists of only a coupled term become zero?

Let's say our Lagrangian looks something like this: $$L = \int dz\, Q\cdot \dot{A},\tag{1}$$ where $Q$ and $A$ are two generalized coordinates and $\dot{Q}$ and $\dot{A}$ would be the respective ...
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### Good quantum numbers from a given hamiltonian

The primary reason asking this question to understand good quantum number from a giver Hamiltonian. Is there any good approach that we can identify them? For example: We have a square and in that ...
4answers
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### Is energy $E$ in Schrödinger equation an observable/ Can $E$ be measured?

Take this quantum approach to estimate mean energy of a molecule: $$\langle\psi|H|\psi\rangle=\overline E$$ Question: Is $E$ an observable? How we can compare it to an experimental value? i.e how ...
1answer
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### PDE from Hamiltonian density

For the wave equation Hamiltonian density is $2H=\phi_t^2+\phi_x^2$ while the Lagrangian density is $2L=\phi_t^2-\phi_x^2$. I can easily compute the pde from the Lagrangian density but how does one do ...
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### Why is the generalized momentum replaced by the momentum operator but not the ordinary momentum?

I was trying to understand the derivation of the Hamiltonian for a charged particle in an electromagnetic field. https://en.wikipedia.org/wiki/Hamiltonian_mechanics#...
1answer
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### Significance of energy in a time dependent quantum box

The Hamiltonian for a particle in a finite box is $$H = \frac{p^2}{2m} + V(x)$$ which will give time evolution as $$i\hbar d/dt|{\psi(t)}\rangle = H|{\psi(t)}\rangle \, .$$ However, if I do a ...