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

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64 views

Energy of hydrogen atom - Schrodinger equation [closed]

The wavefunction of the electron in the hydrogen atom is $ k* exp(-r/a)$ (k is the normalization constant), but it does not take n into account, whereas the solution of Schrödinger's equation ($H(...
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82 views

Hamiltonian operator in spherical coordinates

I'm studying the hydrogen atom from a quantum mechanics perspective, but I'm having troubles understanding a step. Consider the stationary Schroedinger equation: $$\hat H \psi = E\psi$$ Let $M$ be ...
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55 views

Can we measure the energy of one of several identical particles?

Suppose we have a many-particle system described via a many-particle wavefunction that involves single-particle states $\lvert\lambda_{a}\rangle$, $\lvert\lambda_{b}\rangle$, $\lvert\lambda_{c}\rangle$...
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60 views

Broadband light term in a Hamiltonian

In atomic systems, for a two-level system, the Hamiltonian can be written in the form: $$H=\left( \begin{array}{cc} E_1 & C_{12} \\ C_{21} & E_2 \\ \end{array} \right)$$ where $E_1$ and $...
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72 views

Liouville's theorem for systems with dissipation described by a single hamiltonian

Following this link, one can treat dissipation in the lagrangian by using a factor $e^{\frac{t \beta}{ m}}$ in addition to the Lagrangian $L_0$ of a system without disspation: $ L_0[q, \dot{q}] = \...
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48 views

Hamiltonian average of energy of two stationary states

In quantum mechanics, the description of the infinite square well is given with the potential energy defined as $$V(x) = \begin{cases} 0 & \text{if } 0 \leq x \leq a,\\ \infty & \...
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72 views

Constructing a Hamiltonian from a mass matrix?

I was solving some questions regarding the Hamiltonian, which required a lot of algebra, but as I finished and looked professor answer I saw that he constructed a matrix from the kinetic energy and ...
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44 views

Energy Expectation Value

I had an assignment question in which I was asked to calculate the expectation value of energy, $\langle E\rangle (t),$ and in the solution to it, the following was stated: \begin{align*} \langle E\...
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37 views

Interaction Hamiltonian and shifts

When we quantize a free field theory, we set $\phi(x)$ to be the operators and we take the Fourier transform to determine the creation and annihilation operators $a_\omega,a^\dagger_\omega$ such that $...
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84 views

A mysterious conserved quantity for a central potential

In teaching a course in classical mechanics and I have come across (from my predecessor) a to me mysterious conserved quantity. We are considering a gravitational (or electric) potential with the ...
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56 views

Dispersion relation from Hamiltonian

Note: This is obviously for homework so I'm not asking for the answer to be spoon fed, I'm just not understanding the steps I have to take. I have a fairly simple Hamiltonian for a ring tight binding ...
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72 views

Commutation between angular momentum and Hamiltonian

Consider the following Hamiltonian of a 3-dimensional system: $$H=\frac{p^2}{2m}+V(r)$$ If the components of the angular momentum, $L_i$, commute with $H$, then: $$[H,L_i]=0$$ This condition can ...
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87 views

What happens to the wave function of a particle immediately after measuring its energy?

For this question, I will be adhering to the Copenhagen interpretation (since that's what I've learned in university so far). For the sake of brevity/clarity, also, assume the Hamiltonian here has ...
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53 views

Hamiltonian in commutator contradiction [duplicate]

Consider the following: $$[ \hat H, \hat x]=\left[-\frac{\hbar^2 \hat p^2}{2m}+V,\hat x\right]\ne0 \text{ in general}$$ But $$[ \hat H, \hat x]=\left[i\hbar \frac{\partial }{\partial t},\hat x\right]...
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40 views

What's the point of hamiltonian mathematical formalism of classical mechanics? [duplicate]

Just what the title asks. What are the applications of it?
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79 views

Operators for a Perturbed Hamiltonian: Heisenberg Picture ($\hat{x}$, $\hat{p}$)

Problem I am trying to calculate the Equations of Motion in the Heisenberg picture for $\hat{x}$ and $\hat{p}$ in a perturbed Hamiltonian, $$ \tag{1} \hat{H} = \hat{H}_0 + \hat{H}' $$ Assume ...
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23 views

how to reduce the order of the hamiltonian equation for electrical problem given below?

I am having my FYP in this Hamiltonian project to analysis the integrator for Hamiltonian system. Can anyone please guide me how to reduce the equations to first order using substitution method?
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21 views

How to reduce the order of Hamiltonian equation for electrical problem [duplicate]

I want to reduce the order of this Hamiltonian but I don't know how to proceed. The equations are given below: $$H(p,q) = \frac{1}{2} (kq^2) + \frac{p^2}{2m} $$ This is the Hamiltonian for a simple ...
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19 views

ray tracing through a stack of flat plates

Say I want to trace rays in 3 dimensions through a stack of flat plates of various refractive indices. My rays have canonical coordinates {Q,P}. The plates are normal to the z axis, all the rays start ...
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220 views

Real versus complex Hamiltonian

While a Hamiltonian must be a Hermitian matrix, it can either be real or complex. Is there a significance for having a real Hamiltonian? Does it have any additional physical symmetries? For example,...
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75 views

Time evolution of wave function in QM

Recently I've been studying quantum dynamics with Sakurai's modern quantum mechanics, but I am confused with why the time evolution operator is written as $$U(t,t_0)=\exp\left[\frac{-iH(t-t_0)}{\hbar}...
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1answer
82 views

What is the meaning of definite total energy of the wave function?

David J. Griffiths in Introduction to Quantum Mechanics asked: What's so great about separable solutions of time independent Schroedinger equation? His answer was They are states of ...
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205 views

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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133 views

Can all symplectic-form preserving canonical transformations generated by generating functions

This question is related to this fascinating post and this post and this post, but more limited in scope in discussing the practical definition canonical transformations. Canonical transformation ...
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344 views

Off-diagonal elements of Hamiltonian matrix $H_{12}$ & $H_{21}$: energy of transition from $|1\rangle$ to $|2\rangle$ or amplitude of transition?

$$ \newcommand{\k}[1]{\left| #1 \right\rangle} \newcommand{\dd}[1]{\frac{d #1}{dt}} $$ In a Hamiltonian Matrix like this: $$H = \begin{pmatrix} E_{11} & E_{12} \\ E_{21} & E_{22} \end{...
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175 views

How/Why did Feynman relate the element of Hamiltonian matrix $H_{12}$ to the amplitude to go from $|1\rangle$ to $| 2\rangle$?

$$ \newcommand{\bk}[2]{\left\langle #1 | #2 \right\rangle} \newcommand{\ket}[1]{\left| #1 \right\rangle} \newcommand{\bra}[1]{\left\langle #1 \right|} \newcommand{\biik}[3]{\left\langle #1 | #2| #3\...
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78 views

The second order scalar quartic field theory Feynman diagrams

I'm trying to find the second order Feynman diagrams for the scattering of a scalar phion, described by the Lagrangian $$ L = L_0 + L_I \\ L_0 = \frac{1}{2}[\partial_{\mu} \phi(x)]^2 - \frac{1}{2}m^2 ...
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103 views

Derivation of Schrodinger equation using unitary operators

I encountered the derivation of Schrodinger time dependent equation using expansion of a unitary time propagation operator into power series in a small quantity $\delta t$, working to first order in $\...
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2answers
487 views

Does the poisson bracket $\{f,g\}$ have any meaning if neither of $f$ or $g$ is the system's Hamiltonian?

Say one has a mechanical system with hamiltonian $H$, and two other arbitrary observables $f,g$. $H$ is super useful since $\{H, \cdot\} = \frac{d}{dt}$. But does $\{f,g\}$ give any useful information ...
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21 views

Multiple-scale analysis for 2D Hamiltonian?

I came across a technique called "multiple-scale analysis" https://en.wikipedia.org/wiki/Multiple-scale_analysis where the equation of motion involves a small parameter and it is possible to obtain an ...
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1answer
111 views

Finding the Hamiltonian for sound vibrations in a gas in the momentum representation

I am working on a problem where I have been given the following Lagrangian density for describing sound vibrations in a gas: $\mathcal{L}=\frac{1}{2}[\rho_0\dot{\eta}^2+2P_0\nabla\cdot\eta-\gamma P_0(...
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1answer
59 views

Complex Inner Product for Integral Expressions

I am currently working through some QFT derivations and running into conceptual problems. In particular, I am deriving the free field Hamiltonian of the form: $H_{k} = \frac{1}{2} \int d^{3}k \hspace{...
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26 views

Dirac equation in the presence of a defect

The 1D Dirac equation in the presence of a defect is described by a position dependent mass term known as a "kink" or "soliton". It is sign changing and tends to a constant at positive and negative ...
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53 views

Hamiltonian describing a 3D Weyl point: seperability

A 3D analogue of a Dirac point is a Weyl point, with first quantized Hamiltonian $H = \sigma_x p_x + \sigma_y p_y + \sigma_z p_z $ where $\sigma_i$ are Pauli's matrices and $p_i$ are momentum ...
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72 views

Hamiltonian in Majorana basis

I read (for example here: cond-mat/0010440) very often that if we transform the Hamiltonian from a fermionic basis to the basis of Majorana operators by expanding the fermionic operators in real and ...
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1answer
82 views

Independence of generalised coordinates and momenta in Hamiltonian mechanics [duplicate]

I am told that in Hamiltonian mechanics, we put the generalised coordinates $q_i$ and generalised momenta $p_i$ on equal footing, and treat them as being independent from one another. But I'm ...
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1answer
145 views

Poisson brackets and magnetic field [closed]

I'm a maths student trying to teach myself some physics so sorry if I'm missing something simple here. I think the main problem is lack of experience with the Levi-Cevita symbol. We have a particle ...
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1answer
359 views

Commutation between Dirac hamiltonian and angular momentum

In reading about angular momentum and spin, I came across a derivation showing that the Dirac Hamiltonian does not commute with orbital angular momentum, and hence L is not conserved. What is it about ...
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1answer
118 views

What is time evolution operator?

Could you explain to me (level 1 years undergrade) what is a time evolution operator? I read on Wikipedia, and it confuses me.
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1answer
67 views

Effective theories and unbounded operators

If you have two operators, one the true Hamiltonian $H$ and one we call an effective Hamiltonian $H_{eff}$ and say they agree on every eigenvector with eigenvalue up to $E_{eff}.$ Above that, they can ...
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1answer
115 views

Do time-invariant Hamiltonians define closed systems?

In classical mechanics, every time-invariant Hamiltonian represents a closed dynamical system? Can every closed dynamical system be represented as a time-invariant Hamiltonian? Or are there closed ...
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310 views

Time-ordering and Dyson series

In Dyson series we use a time-ordered exponential by arguing that a Hamiltonian at two different instants of time does not commute. Why is it that so? Can anyone explain with example why should the ...
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2answers
283 views

How to include Berry connection in Hamiltonian?

When we calculate Berry connection, $A(R)=i<\psi(x,y)|\frac{d}{dR}|\psi(x,y)>\hat{R}$ corresponding to the Berry phase of any system, the gauge potential is related to the $R$ of the parameter ...
2
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3answers
257 views

Why does Hamiltonian follow the property $H^*_{ij} = H_{ji} $?

I was reading Feynman's Lectures III's Hamiltonian Matrix. There I found this property of Hamiltonian Matrix: The Hamiltonian has one property that can be deduced right away, namely, that $$H^*_{...
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6answers
504 views

Why does time evolution operator have the form $U(t) = e^{-itH}$?

Let's denote by $|\psi(t)\rangle$ some wavefunction at time $t$. Then let's define the time evolution operator $U(t_1,t_2)$ through $$ U(t_2,t_1) |\psi(t_1)\rangle = |\psi(t_2)\rangle \tag{1}$$ and ...
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1answer
123 views

Complex scalar theory: annihilation and creation operators give wrong commutators with Hamiltonian

The theory of a real (hermitian) scalar field can be found in many books and everywhere online. On the other hand, if we take the field non-hermitian, then I can only find notes on path integrals. I ...
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1answer
147 views

How to get Heisenberg equation of motion? [closed]

A system Hamiltonian is given by $$ H=\hbar\omega_{1}\hat{a}^{\dagger}\hat{a}+\Sigma_{i=1}^{N}\left(\hbar\omega_{se}\hat{\sigma}_{ss}^{i}+\hbar\omega_{ge}\hat{\sigma}_{ee}^{i}\right)-\hbar\Sigma_{i=1}...
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1answer
170 views

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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42 views

Trapped ion Hamiltonian: simultaneous blue and red sidebands

In the paper "Spin-motion entanglement and state diagnosis..." (Home et al., Nature, 2015), the authors talk about a single trapped ion, optically pumped, simultaneously driving the red and blue ...
3
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1answer
156 views

Is the Klein-Gordon Hamiltonian unbounded below?

This question is about the Klein-Gordon Hamiltonian for simplicity, but the problem seems to remain when dealing with other fields (e.g. Dirac, photon...). One usually writes the Hamiltonian (density)...