Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
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Correspondence between quantum operators and classical formulas
Background
From what knowledge of quantum mechanics I have so far, it is a postulate that Hermitian operators corresponding to a certain observable act on a quantum state $\psi$ to produce a new ...
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Connection between Hamiltonian and density matrix
I found this equation in my notes from a seminar on BCS theory
$$
H_{m,n} = \frac{dE}{d\rho_{n,m}}
$$
where $H_{m,n}$ are elements of the Hamiltonian in matrix representation, $\rho_{n,m}$ are ...
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1answer
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Hamiltonian of charged particle in an EM field and magnetic field does no work on charged particles
I am trying to understand a part of I.E.P.'s answer here.
I.E.P. states that one can see from the following Hamiltonian,
$$
H = \frac{1}{2m}|{\bf p}+q{\bf A}|^2 +q \phi \tag{8.35}
$$
that the magnetic ...
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How is Hamilton's first equation useful in solving mechanics problems? [duplicate]
Here is the first Hamilton equation:
$\frac{\partial H}{\partial {p}_q} = \dot{q}$
Let's use it. Imagine a ball rolling down a frictionless hill (ignore the friction vector in the image). As time goes ...
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Hamiltonian of a particle in magnetic field squared
I'm trying to follow Tong lectures about Gauge Theories, but I think I'm doing some really stupid mistake.
At one point he takes the Hamiltonian for a spin $1/2$ particle in a potential as the usual
\...
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How do I determine the zero elements of a Hamiltonian in a 4 ket space?
The Hamiltonian matrix of particle subject to a central potential is described by
$$
H=\begin{pmatrix}
H_{11} & H_{12} & H_{13} & H_{14}\\
H_{21} & H_{22} & H_{23} & H_{24}\\
...
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Finding the unitary transformation associated with a symmetry
Suppose we have a Hamiltonian that has a symmetry. Let's consider a simple example where the Hamiltonian depends on a vector $ H(\vec r)$ such that a rotation $R \vec r$ is a symmetry.
So, $H(r)$ and $...
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Are there non-constant potentials that result in eigenstates of the Hamiltonian that are all plane waves?
It is commonly known that the eigenstates to the Hamiltonian of a constant potential are plane waves, aka
$$
V(r) = V_0 \Rightarrow H\psi = n \text{ with } \psi = \exp\left(\frac{ip}{\hbar}x\right)\...
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Transverse field Ising in 2 dimensional lattice - kronecker product
Assume we have a transverse field Ising chain (1D):
$\hat H =-J\sum_{i=1}^{N}\sigma^z_i\sigma^z_{i+1}-h\sum_{i=1}^{N}\sigma^x_i$,
where $\sigma^{\alpha}_i$ are the local spin operators at site i ...
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Decoupling theory by diagonalising the Hamiltonian
I have a Hamiltonian of the form
$H = 2k(\alpha \alpha^* -\beta \beta^*) -2\lambda (\alpha\beta^* + \beta \alpha^* )$
and I'd like to decouple the $\alpha$'s and $\beta$'s if possible. I know I need ...
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1answer
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Deriving the Hamiltonian of the Klein-Gordon field in terms of ladder operators (Peskin and Schroeder 2.31)
In Peskin and Schroeder's QFT book they give
\begin{align*}
H &= \int d^3x\int \frac{d^3p d^3 p'}{(2\pi)^6}e^{i(\mathbf{p+p'})\cdot \bf x}\left\{-\frac{\sqrt{\omega_{\bf p}\omega_{\bf p'}}}{4} (a_{...
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1answer
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Derivative of a potential when deriving Boltzmann equation
Consider a system with $N$ identical particles of mass $m$, whose coordinates and momenta are $(q_i,p_i)$, $i = 1,\ldots,N$, and with Hamiltonian
$$
H=\sum_{j=1}^N \frac{p_j^2}{2m} + \sum_{1\leq j <...
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1answer
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Construction of propagator for time-dependent hamiltonian
In deriving a general propagator to the time-dependent ($H = H(t)$) Hamiltonian problem, Shankar works to first order in $\Delta = T/N$ (a small time interval for large $N$) and argues that by ...
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Time dependence of ladder operators in QFT
I'm currently going through Matthew D. Schwartz book Quantum Field Theory and the Standard Model, p. 23. For free (non interacting) field theories we are able to quantise the field by expanding our ...
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1answer
31 views
Generator of Time Shift in Classical and Quantum Mechanics
The time evolution of a point in phase space in classical mechanics can be described as
\begin{equation}\label{eq:TmeShift}
( q_i(t + \Delta t),p_i(t + \Delta t) ) = \left( 1 - i\Delta t \hat{L}\...
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Why is the Legendre transformation the correct way to change variables from $(q,\dot{q},t)\to (q,p,t)$?
I always found Legendre transformation kind of mysterious. Given a Lagrangian $L(q,\dot{q},t)$, we can define a new function, the Hamiltonian, $$H(q,p,t)=p\dot{q}(p)-L(q,\dot{q}(q,p,t),t)$$ where $p=\...
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1answer
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A priori properties of classical Hamiltonian (unitarity in Classical Mechanics)
A canonical equation of motion has form:
\begin{equation}
\dot{p}_i = -\frac{\partial H}{\partial q_i} = \left\{ p,H\right\}, \quad \dot{q}_i = \frac{\partial H}{\partial p_i} = \left\{q,H\right\}....
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Is Hamiltonian a linear operator on Hilbert space?
I believe this question may seem silly, every student who has studied quantum mechanics in school must has been told that Hamiltonian is a linear operator on Hilbert space.
However, today I think this ...
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38 views
Do Hamiltonian operators preserve square integrability? [closed]
Is Hamiltonian operator an operator that preserves the property of square integrability?
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How does symplectic geometry relate to classical hamiltonian mechanics?
I just found out about symplectic geometry in the context about this question on volume preservation in phase space.
It seems somewhat complicated and I am not sure what to do with the notation $\...
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Help on Hamiltonian tensor equations in Einstein's original General Relativity papers
Studying Einstein's original Die Grundlage der allgemeinen Relativitätstheorie published in 1916's Annalen Der Physik, I came across Equations 47b) regarding the gravity contribution to the stress-...
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Getting the eigenvalues of a quadratic boson Hamiltonian numerically
I have the following quadratic Hamiltonian (of boson type):
$$\hat{H}=\epsilon b^\dagger b -v(b^\dagger b^\dagger + b b )$$, where both $\epsilon$ and $v$ are real parameters. The operators $(b,b^\...
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1answer
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Is this called an operator?
Consider the Hamiltonian:
$$H=D\bigg(S_z-\frac{1}{3}S(S+1)\bigg)$$
Where $S_z$ is the spin-$z$ operator (one half the Pauli matrix for a doublet state) and the matrix representation of $S$ is the unit-...
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2answers
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How the Hamiltonian of a classical system expressed in quantum mechanics?
I was dealing with a problem, which said that,
Supposedly Hamiltonian of a conservative system in classical mechanics is $\omega xp$, where $\omega$ is a constant, and $x$ and $p$ are the position ...
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1answer
67 views
Why are time derivatives of states in QFT equal to zero?
In equation 6-38 on page 176 of the book "Student Friendly QFT" by Robert D. Klauber it is said that the partial derivative w.r.t. time of a multi-particle state is equal to zero since we ...
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Eliminating an eigenvalue from the Hamiltonian
I have a momentum space Hamiltonian $H(\vec k)$ for a Kagome lattice and I want to find its eigenvalues which may be dependent on $\vec k$. Now, I'm told that one of the eigenvalues for such ...
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Predictability in decoherence theory to find the classical states: at which time must we evaluate?
I have read Decoherence, einselection, and the quantum origins of the classical, end a way to quantify the classicality of states is the following.
We have the system $S$ and its environment $E$. The ...
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Book about the measurement energy of the Hamiltonian
I am searching for a book about the measurement energy of hamiltonian in adiabatic quantum computing. Have you ever seen good resources?
I need good references for my work.
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Identifying the relevant directions in the Ising model renormalization
I'm reading the chapter about the renormalization group in Yeoman's book "Statistical mechanics of phase transitions" and I'm puzzled about how the author relates the scaling of the RG with ...
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1answer
58 views
Bethe ansatz wavefunction vs plane waves
I am reading Negele & Orland's "Quantum many-particle systems". In problem 1.9 you show that the (Bethe ansatz) wave function
$$ \psi(\{x \}) = \exp \left( - \alpha \sum_{i < j}^N |...
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1answer
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From spins to fields
In statistical field theory, one usually considers the so-called Landau Hamiltonian:
$$\beta H = \int d^{d}x\bigg{[}\frac{t}{2}m^{2}(x) + \alpha m^{4}(x)+\frac{\beta}{2}(\nabla m)^{2}+\cdots+ \vec{h}\...
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Treat stochastically non-Hamiltonian perturbations
Let us consider a classical dynamical system whose obserbvables $A$ evolve according to the following equation of motion
\begin{align}
\dot A &= -\{H,A\}+f(q)
\end{align}
$f(q)$ is a non-...
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1answer
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$T$-odd vs $T$-violation
I am a bit confused by the difference between $T$-odd and $T$-violation. For example, I read that the existence of a fundamental particle EDM is a violation of time symmetry. However, placing an ...
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Physical interpretation of a Hamiltonian [closed]
In natural basis $| 0 \rangle = \begin{pmatrix} 1 \\0 \end{pmatrix}$, $| 1 \rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, what physical situation/model does the following Hamiltonian represent: $H = ...
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Derivation of transition amplitude probability between two adjacent space points for general time independent hamiltonian
I am studying Srednicki book of Quantum Field theory. In chapter 6 regarding the path integral there was derived equation of transition probability for hamiltonian of type:
$$H(\hat{P},\hat{Q})= \frac{...
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The “basic hamiltonian” of topological systems
I am currently studying topological insulators and repeatedly found the claim (e.g. here), that the "basic hamiltonian" of a topological system in $d$ spatial dimensions can be written using ...
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1answer
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Canonical ensemble: Why do I lose dependency on the number of particles N here?
I have a problem understanding the solution of an exercise that deals with a gas in the framework of the canonical ensemble. Because I'm not a native english speaker some sentences might sound a bit ...
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1answer
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Can Jacobi's formulation of Maupertuis' principle be derived in Riemannian geometry?
i want to arrive to hamilton-jacobi equation using the riemannian geometry.
So let $\textbf{X}\in \mathfrak{X}(M)$, where $M$ is Riemannian manifold whose metric is $g:\textbf{T}M \times \textbf{T}M \...
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Instrinsic spin orbit coupling in tight-binding Hamiltonian
I'm looking to write down a second quantized Hamiltonian to include the intrinsic spin-orbit coupling term in addition to the hopping spin-orbit coupling Rashba effect.
How would I construct the term ...
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1answer
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How to solve differential equation involving commutator and anti-commutator?
In one of my exercise, I got following differential equation for density matrix $\rho$,
$$
\frac{d\rho}{dt}=-i[H_1,\rho]+\{H_2,\rho\}
$$
where $H_1$ and $H_2$ are the Hermitian Hamiltonian, and $[.,.]$...
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1answer
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Particle moving in Morse potential
I'm solving for a particle moving in the Morse's potential
$$
H=\frac{p^2}{2m}+A\left( e^{-2\alpha x}-2e^{-\alpha x}\right)
$$
Now, considering an operator $B=-\partial_x +C e^{-\alpha x}-D$ and its ...
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2answers
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Deriving the $0$-component of 4-momentum using the relativistic Lagrangian
My question arises from Susskinds book on Special Relativity and Classical Field Theory. (page 102 equation 3.29 to 3.30 and page 105 equation 3.34 to 3.36.)
The relativistic Lagrangian for a free ...
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1answer
56 views
Out-of-equilibrium correlation functions
Let us consider a classical thermal ensemble
\begin{equation}
\rho_\beta = \frac {1}{Z_\beta} e^{-\beta H}.
\end{equation}
The Hamiltonian generates a mixing dynamics if
\begin{equation}
C_\beta(t) = \...
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2answers
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Deriving the quantum Hamiltonian from the expression of classical energy
I am currently learning about the Dirac formalism in quantum mechanics, but don't quite understand how we derive the expression of the quantum Hamiltonian, given the value of energy in classical ...
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1answer
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What exactly is meant by a “time-reversed Hamiltonian”
For context, I am reading this paper.
Basically, the paper makes reference to "evolving with respect to the time-reversed Hamiltonian". I'm slightly unclear as to what this actually means. Here is my ...
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2answers
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Is symmetrization $xp-px$ required for commutation $[H,x]=0$?
Given a Quantum Hamiltonian: $$\hat{H}=ax^2+bp^2$$ It does not commute with either $x$ or $p$. Suppose we have a Hamiltonian :$$H = k \hat{p}\hat{x}$$ why do we need it to be: $$H = k (\hat{p}\hat{x} -...
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Hamiltonian in real and reciprocal space
I found that sometimes people mentioned that Hamiltonian in real space or Hamiltonian in reciprocal/$k$-space. I wonder what difference of Hamiltonian in real and reciprocal spaces are?
For example, ...
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2answers
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Is $H = T + U$ for a pendulum on a circle movement?
I have this problem:
Obtain Hamilton's equations of motion for a plane pendulum of length $l$ with mass point $m$ whose radius of suspension rotates uniformally on the circumference of a vertical ...
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2answers
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How do you know if a operator commutes with the hamiltonian?
In the question there is a central potential within a Hamiltonian, and I have to find the appropriate quantum numbers. They say that $j, m, s$, $\ell$ are the appropriate quantum numbers to describe ...
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1answer
84 views
Is the Hamiltonian fully defined by a quantum state (vector)? [duplicate]
From what I have read, the evolution of a quantum state is determined by the Hamiltonian (Schrodinger equation). However, I'm trying to understand if the Hamiltonian itself can be fully derived from ...
