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

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How can one determine whether two given Hamiltonians are supersymmetric partners?

Given two Hamiltonians in a general form of second order differential equations, how do I find out if they are SUSY partners or not? Given the factorisation of a Hamiltonian in the form of $a^\dagger ...
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35 views

How to plot numerically the wave functions according to the Hamiltonian?

It is often difficult to analytically solve the Schrodinger equation, and so we need to obtain a solution numerically. An example plot is shown below. Here, the wave functions for a three junction ...
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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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21 views

Zeeman effect - eigenstates and its degeneracy - with and without magnetic field

Consider a hydrogen atom in a homogeneous magnetic field $\vec{B}=B\vec{e_z}$. Using the coulomb gauge ($\nabla \vec{A}=0$) we can take $\vec{A}=\frac{1}{2}\vec{B}\times \vec{r}$ as a vector potential....
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30 views

Bohr frequency of an expectation value?

Consider a two-state system with a Hamiltonian defined as \begin{bmatrix} E_1 &0 \\ 0 & E_2 \end{bmatrix} Another observable, $A$, is given (in the same basis) by \begin{bmatrix} 0 &a \...
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595 views

Hubbard Model Hamitonian

$H = -\sum\limits_{i,j} A_{ij} c_i^{\dagger} c_j + \frac{U}{2} \sum\limits_i(c_i^\dagger c_i)(c_i^\dagger c_i -1)$ is defined to be a Hamiltonian for modeling quantum random walk of identical ...
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186 views

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 ...
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35 views

Relationship Between Magnetic Dipole Moment and Spin Angular Momentum

I am reading Introduction to Quantum Mechanics 1st edition by David J. Griffiths and I have a couple questions about this section on page 160. A spinning charged particle constitutes a magnetic ...
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86 views

Is there any Hamiltonian that contains time derivative? [duplicate]

Quantum mechanics is governed by Schrodinger's equation: $$\hat{H}\psi=i\hbar\partial_t \psi$$ It seems that Hamiltonian acts on wave functions like a time derivative. Just out of curiosity, is ...
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57 views

Hamiltonian - Fourier transform of order parameter [closed]

I have a rather simple task, but it seems I can't move forward with the solution. I have a Hamiltonian as seen in the picture. I have to use the Fourier transform of the order parameter $\phi(x)$ and ...
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1answer
60 views

Does the Hamiltonian time-evolution operator actually change the state of the system?

According to my understanding of things, the time evolution operator in QM looks something like this, $$U = \exp(-iHt/\hbar)$$ Which acts on the state vector / wave-function of the system to ...
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31 views

Does an odd potential commute with parity operator?

I can prove when a Hamiltonian commute with the partity operator if the potential is even. But what about an odd potential? my understanding is that the parity operator mirrors the coordinate system, ...
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45 views

Derivation involving finite unitary transformation [closed]

Hi I just want to confirm a short derivation involving a particular finite unitary transformation which is important in QM. My working is as follows: Given the finite unitary transformation defined ...
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1answer
48 views

Energy conservation and time translations

The time translation is given by a finite unitary transformation $$\hat{U_{\tau}}(\hat{H}) = e^{\big(\frac{i}{\hbar}\tau \bar{H}\big)}.$$ Where $$\hat{U_{\tau}}(\hat{H})|\psi(t) \rangle = |\psi(t-\tau)...
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28 views

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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38 views

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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1answer
70 views

Hamiltonian flow?

I was wondering what the Hamiltonian flow actually is? Here is my idea, I just wanted to know if I am correct about this. So let $(x(t),p(t))' = X_{H}(x(t),p(t))$ are the Hamilton's equations and $...
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1answer
51 views

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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52 views

Representing Hamiltonian in discrete position basis

I am trying to numerically find eigenstates of an Hamiltonian. Let $V(x)$ be some potential. Suppose my space is between $-L,L$ and that the allowed positions are every $\Delta x$, such that $\frac{1}{...
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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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225 views

Is there a physical interpretation to invariant random matrix ensembles?

Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
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118 views

Ground state energy of spin 1 particle

So I have this Hamiltonian for a particle with spin 1: $$ H=aS_{z}^2+\frac{\hbar\omega}{\sqrt2}S_{x}$$ where ($a$ and $\omega$ both real constants): $$ S_{z}=\hbar\begin{pmatrix} 1 & 0 & 0 \...
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402 views

Schrödinger equation for many body systems

$$H_{tot}=\sum \dfrac{p_i^2}{2m}+\sum\dfrac{p_I^2}{2M_I}+\sum V_{nucl}(r_i)+\dfrac{1}{2}\sum_{i\ne j} \dfrac{e^2}{|r_i-r_j|}+\dfrac{1}{2}\sum_{I\ne J}\dfrac{z_Iz_Je^2}{|R_I-R_J|} $$ with: $$V_{nuc}...
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39 views

Why the constancy of an observable w.r.t time depends on whether it commutes with $H$ or not?

I have been reading Modern Quantum Mechanics by J.J.Sakurai. Under the chapter Quantum Dynamics, the author says if an observable $A$ initially commutes with the Hamiltonian operator $H$, then it ...
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36 views

How to check if a Hamiltonian is PT symmetric or not?

Consider the Hamiltonian $$H=p^2+ix^3+ix.$$ This paper by Carl M bender claims this is a $PT$ symmetric Hamiltonian. In this he describes $PT$ symmetry as parity $P$, whose effect is to make ...
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46 views

Gaining intuition over Hamiltonian for qubit systems

A typical Hamiltonian for a two state system with some driving field can be written as $$H=J(t)\sigma_z+h\sigma_x$$ This represents a qubit system driven along a single axis. On the other hand we ...
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52 views

Why is the energy operator special?

Only the energy operator controls the time dependence of a quantum system, but not the others, why is that?
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30 views

Meaning of Hamiltonian between two different states

If we have states $\left | 1 \right>$ and $\left | 2 \right>$, and the Hamiltonian operator $\hat{H}$, what is the meaning of the expression $$\left< 1 \right | \hat{H} \left | 2 \right >$$...
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46 views

Momentum operator in effective mass approximation

When we calculate the band structure of some solid then we often find that in the bottom of the conduction band the dispersion looks approximately quadratic with some new effective mass: $$E(k) = \...
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20 views

What is Loewdin downfolding method?

I am a student in solid state physics. I wonder if somebody could explain the mathematical background of downfolding method that is often used by Ole K. Anderson. Which restriction (conditions) to ...
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35 views

Is expectation value of the Hamiltonian always the energy? [duplicate]

There are time dependent & space dependent systems (magnetic fields) and time independent (particle in a box or harmonic oscillator). In the latter the expectation value is the 'average' energy ...
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106 views

Is the eigenvalue of Hamiltonian invariant under linear transformation of momentum operator?

It is given The dynamics of a particle moving one-dimensionally in a potential V(x) is governed by the Hamiltonian $H_0 = p^2 /2m + V(x) $, where $p = -i\hbar d/dx$ is the momentum operator. ...
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70 views

Bogoliubov transformation with two pairing terms

Let us assume that we have a Hamiltonian of the form: $$ H = \sum_{k,\sigma,s}\epsilon_{\sigma s}\left(k\right)c_{k\sigma s}^{\dagger}c_{k\sigma s} + \sum_{k,s}\Delta_{0}\left(k\right)c_{k\uparrow s}^...
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41 views

Build Hamiltonian function

Suppose we have three-point system Points A and B are connected with rod of fixed length $r_0$. Point C rotates around rod, vector R begins at rod's centre of mass. There is a potential of general ...
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16 views

Fermi-level of lattice model

I have a chain of $N$ lattice points with periodic boundary conditions. Every lattice point only has one orbital for an electron to occupy and spin is not included for simplicity. The lattice points ...
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49 views

Quantization of the Hamiltonian of a particle in a uniform magnetic field

If a particle of mass $m$ and charge $q$ is subject to a uniform magnetic field and if we have a vector potential $\mathbf{A}$ then we know that classically the dynamics of the particle will be ...
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1answer
67 views

Going to the interaction picture in the Jaynes–Cummings model [closed]

In the Jaynes–Cummings model for a two level atom, the Hamiltonian for the atom is defined as (I let $\bar{h}=1$) $$H_a=\omega_a\frac{\sigma_z}{2}$$ and the field Hamiltonian is $$H_f=\omega_ca^{\...
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1answer
75 views

Single particle tunneling Hamiltonian

In reference to Problem 9, Chapter 2 in Modern Quantum Mechanics by JJ Sakurai, For a single particle tunneling in a 1D double well potential, with position eigenkets $\mid R\rangle$, $\mid L\rangle$....
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830 views

Is there a Hamiltonian for the (classical) electromagnetic field? If so, how can it be derived from the Lagrangian?

The classical Lagrangian for the electromagnetic field is $$\mathcal{L} = -\frac{1}{4\mu_0} F^{\mu \nu} F_{\mu \nu} - J^\mu A_\mu.$$ Is there also a Hamiltonian? If so, how to derive it? I know how ...
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220 views

What is $\langle \phi | H | \psi \rangle$ in QM?

I know that $\langle \phi | \psi \rangle$ is the probability of going from the $\psi$-state to the $\phi$-state, and that $\langle \phi | H | \phi \rangle$ is the expectation value of the energy for ...
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1answer
1k views

How does a state in quantum mechanics evolve?

I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as $$ i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle $$ I am self-...
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Can you help me solve this using the current value Hamiltonian? [closed]

Okay, so I am getting a little stuck on this question, I will post it and then tell you how far I get. $$ max - \int_0^2 (x^2 + u^2)e^{-0.03t}dt\, $$ $$ x' = x-2u $$ $$ x(0) = 3 $$ $$ x(2)free $$ ...
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2answers
120 views

Why do we need both Hamiltonian and Hilbert Space to specify a Quantum System?

From my understanding, when we have the Hamiltonian, in principle we can know the eigenstates for our system of interest. Then, we can calculate everything we want. In addition, these eigenstates ...
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522 views

Hamiltonians, density of state, BECs

When working with Bose-Einstein condensates trapped in potentials, how can one tell what the density of state of a system of identical bosons given the Hamiltonian, $H$? (I have been told that it is ...
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108 views

Hamiltonian and Energy of a charged particle in an Electromagnetic field

The Lagrangian of a charged particle of charge $e$ moving in an electromagnetic field is given by $$L=\frac{1}{2}m\dot{\textbf{r}}^2-e\phi-e\textbf{A}\cdot \textbf{v}$$ where $\phi(\textbf{r},t)$ is ...
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Gauge Invariance of the Hamiltonian of the electromagnetic field

The Hamiltonian for an electron of mass $m$ and charge $e$ in an exterior electromagnetic field is $$H=\frac{1}{2m}(p-(e/c)A)^2+e\varphi.$$ The corresponding (via canonical quantization) quantum ...
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270 views

Ostrogradski’s theorem's proof

I am looking for a proof of the next theorem: "If the higher order time derivative Lagrangian is non-degenerate, there is at least one linear instability in the Hamiltonian of this system." Where ...
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58 views

Fourier transform of Hamiltonian for scalar field

In the Srednicki notes (http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf) page 36 he goes from $$H = \int d^{3}x a^{\dagger}(x)\left( \frac{- \nabla^{2}}{2m}\right) a(x) $$ to $$H = \int d^{3}p\...
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57 views

What is the deBroglie Wavelength in a region with potential energy?

This question was on a physics GRE: A free particle with initial kinetic energy $T$ and deBroglie wavelength $\lambda$ enters a region in which it has potential energy $V$. what is the particle's new ...