Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
1,086
questions
3
votes
0answers
39 views
Physical meaning of gapped path between Hamiltonians in the same phase
I'm reading this famous paper about the classification of quantum phases, and I'm wondering about the physical meaning of the definition of phases the authors use.
They say that two Hamiltonians $H_0$ ...
22
votes
5answers
1k views
Why is the ground state important in condensed matter physics?
This might be a very trivial question, but in condensed matter or many body physics, often one is dealing with some Hamiltonian and main goal is to find, or describe the physics of, the ground state ...
0
votes
0answers
124 views
Dispersion law for a tight binding Hamiltonian and particle states at $t$$\rightarrow$$\infty$
A spinless fermion (possessing an electric charge) can move across the sites of
the discrete (translationally-invariant) lattice. The structure
features three kinds of sites: $α_n$, $β_n$, $γ_n$ with ...
-1
votes
1answer
38 views
Hamiltonian with identity operator: how to visualize the (time-evolution) rotation?
For the Hadarmard Hamiltonian, $\hat H = (\hat X+\hat Z)/\sqrt 2$, where $\hat X$ and $\hat Z$ are Pauli matrices. The time evolution of a state under this Hamiltonian could be visualized by a ...
2
votes
1answer
41 views
Interpreting Hamiltonian of single-mode squeezing
Hamiltonian represents energy. I can understand this when considering about harmonic oscillator, whose Hamiltonian is expressed as:
$$ \hat{H} = \frac{1}{2m}\hat{p}^2 + \frac{m\omega^2}{2}\hat{q}^2$$
...
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votes
2answers
88 views
Translation operator and parity operator
(This is taken from Introduction to Quantum Mechanics by D. Griffiths, 3rd edition, Problem 6.18 .)
If a system has inverse symmetry, we know that [$\hat{H},\hat{\Pi}] =0$ where $\hat{\Pi}$ is the ...
2
votes
0answers
37 views
How to make a $2\times 2$ Hamiltonian using any $2$ levels of an $N$-level Hamiltonian?
Is there a standard way for me to isolate 2 of N bands of a general $N\times N$ Hamiltonian? That is, I want to make a $2\times 2$ Hamiltonian given a larger one. I was told that there is a general ...
4
votes
4answers
151 views
Deriving Klein-Gordon from Hamilton's equations for fields using functional derivatives
I have found two different ways of doing this and I am seeking commentary on the fine nuance. Suppose there is a Hamiltonian
$$ H=\frac{1}{2}\int\!d^3x \left[ \pi^2+(\nabla\varphi)^{\!2}+m^2\varphi^...
2
votes
0answers
40 views
A derivation of an equation involving singular Hamiltonian
I was trying to follow the derivation of an adiabatic theorem in the Appendix F.1 of Jordan, S. P. (2008). Quantum computation beyond the circuit model.
The author claims that, for the sake of this ...
0
votes
3answers
70 views
Finding matrix representation of Hamiltonian operator
Let the quantum system composed by an orthonormal base with the states $|1\rangle, |2\rangle$ and $|3\rangle$ with all being degenereted states of the observable D with eingenvalue $\delta$. So, being ...
0
votes
0answers
15 views
Commutator of Hamiltonian and momentum [duplicate]
I was solving an assignment on the Galilean group and we were ask to compute the commutators of its generators. So, the Hamiltonian is the generator of time translations and momentum is the generator ...
2
votes
2answers
49 views
How do we perform the time derivative of the perturbation series for the time-evolution operator?
The following image is from Greiner's book, Field Quantisation, where he carried out the derivative in question. The only way I could make sense of it, was that the derivative acts only on the last ...
0
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1answer
51 views
Energy in hamiltonian formalism from phase space evolution
The hamiltonian for a free falling body is $$H = \dfrac{p^2}{2m} + mgy$$ and since we are using cartesian coordinates that do not depend on time and the potential only depends on the position, we know ...
1
vote
1answer
61 views
Finding the probability of measuring a particular eigenvalue of an operator for a system after time evolution
Consider a quantum system with Hamiltonian H and consider the measurement
of an observable $a_n$ associated with a different operator A.
Initially the system is an eigenstate $|\phi_n \rangle$ with ...
0
votes
0answers
9 views
Phase density representation two-site Hubbard Hamiltonian with Fermions
I'm looking for the two-site Fermi-Hubbard Hamiltonian in phase density representation in linearized form and I don't know how to derive it. I then want to derive the equations of motion from that.
...
0
votes
1answer
50 views
Can the hamiltonian be derived from phase space evolution?
Given the phase space evolution of a system, $x(t)$ and $p(t)$, is there any way of getting the hamiltonian to make a later study of the system under the hamiltonian formalism?
My first thought was ...
0
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0answers
30 views
Effective Hamiltonian in Heisenberg model
How can we divide the whole matrix into submatrices that we can write effective Hamiltonian on the Heisenberg model Based on Fundamentals of the Physics of Solids book (Volume I) written by Jen˝o ...
2
votes
1answer
61 views
Proving the Feynman-Hellmann Theorem in quantum mechanics
Concerning the Feynman-Hellmann theorem can someone point me on how solve this:
If $H E = E |E\rangle$ and assuming $H$ is depending on a variable $\lambda$ eg., $H = H(\lambda)$ then
$\langle \frac{\...
0
votes
0answers
31 views
Particle in a potential well in mathematica
For a three qubit-chain in a connected state barrier tunneling captured in the Hamiltonian below with J=1, U=3.
How do I fin J' when sites 1 and 3 decouple from each other?
0
votes
0answers
59 views
Pertubation Theory
For long-range barrier tunneling, consider three qubits governed by the Hamiltonian $H$.
$$
H = -J(\sigma_1^+\sigma_2^-+\sigma_1^-\sigma_2^++\sigma_3^+\sigma_2^-+\sigma_3^-\sigma_2^+)+\frac{U}2\sigma^...
0
votes
1answer
28 views
Energy measures and probability of measuring them
We have the following Hamiltonian
$\hat{H}=a|u_{1}\rangle\langle u_{2}|+a|u_{2}\rangle\langle u_{1}|$
with a $\in \mathbb{R}$ and $|u_{1}\rangle,|u_{2}\rangle$ an orthonormal system
The matrix ...
0
votes
1answer
39 views
Expanding the quantum mechanical propagator in terms of the (non-degenerate) eigenvalues of the Hamiltonian
Could anyone please help me with this derivation? I am struggling to see how the Propagator
Can be expanded out into the form
This is a non-degenerate two-level system.
Any help would be greatly ...
1
vote
4answers
159 views
Is $U^\dagger(R)\hat{H}U(R)=\hat{H}$ always true?
Consider a Rotation transformation on momentum state,
$$U^\dagger(R)\hat{\mathbf{p}}U(R)=R\hat{\mathbf{p}}$$
Now the question is whether,
$$U^\dagger(R)\hat{H}U(R)=\hat{H}\,?$$
Here, $\hat{H}$ is the ...
0
votes
0answers
16 views
Diagonalising the Hamiltonian phonon
I've been trying to derive the eigenvalues of the two mass atomic chain to get out both the acoustic and optic phonon dispersion curves. This is easy in classical physics but I wanted to see if I ...
0
votes
1answer
39 views
Why is the energy function not always equal to total energy? [duplicate]
Why is the energy function $h = \dot{q_i}\frac{\partial L}{\partial \dot{q_i}} - L $ not always equal to total energy $E = T + V$? Here $T$ is Kinetic Energy and $V$ is Potential Energy. I've read ...
1
vote
1answer
45 views
A quadratic Hamiltonian is a model of independent particles - why?
I'm reading some notes on the Anderson Hamiltonian:
$$ H=\sum h_i c_i^\dagger c_i -q\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i)$$
Where the $c_i/c_i^\dagger$ are fermionic annihilation/...
3
votes
1answer
58 views
Expectation value of time-evolved number operator for ground state Coulomb system
I'm going through "Advanced Quantum Mechanics" by Franz Schwabl, and he calculates the electron energy levels from the Coulomb interaction in a perturbative way (section 2.2.3). In the ...
1
vote
2answers
94 views
Can spin operator expectation value be time-independent while commutator with Hamiltonian is non-zero?
Considering the following (magnetic field) hamiltonian: $\hat{H}=-\gamma B_z \hat{S}_z$ ($\gamma$ is a constant). Suppose an electron is in an eigenstate of $S_x$, and we ask ourselves the question ...
1
vote
0answers
23 views
Regularization choices and problems in effective Lagrangian (Hamiltonian) derivation
I am trying to derive the explicit form of the effective interaction Lagrangian (Hamiltonian) for fermions interacting via a scalar particle (Yukawa's potential). For that, I am using the Lagrangian
$$...
0
votes
0answers
49 views
Two spin block Ising-Hamiltonian eigenvalue in 1D (transverse ising model)
This problem is regarding to the SDRG (strong disorder renormalization group) around the critical point considering the transverse Ising model. The Hamiltonian is given, but I can't find a way to ...
1
vote
1answer
49 views
Tilting a water glass so that you can run faster without spilling water (counter-diabatic driving Hamiltonian)
In this paper, there is an interesting figure:
Every attempt I've made to search online to confirm whether or not waiters/waitresses actually do this, has been unsuccessful.
Is there really an ...
2
votes
1answer
60 views
Any known way to diagonalize the Hamiltonian of a charge particle in a EM field?
Consider the Hamiltonian of a free (charged) particle, i.e.,
$$
H = \frac{p^2}{2m}.
$$
This is easily "diagonalized" by wave functions $e^{ikx}$ (where I'm speaking loosely of ...
1
vote
1answer
59 views
Combination of 'transposition operators': do they commute?
Suppose I have the Hamiltonian defined as $H =\hat A\hat B+\hat C\hat D$, where the operator $A,B,C and D$ are square matrices. If I label the positions of $A,B,C,D$ as $1,2,3,4$. Now I want to apply ...
-1
votes
2answers
56 views
Adding λI to the Hamiltonian has no impact? [closed]
Show that If we add λΙ in H, where I is the identical operator and $λ\in\mathbb{R}$, it won't affect any measurement.
0
votes
2answers
47 views
Simplifying a Bra-Ket Expression
Consider the following relations
$$H_0|\psi_a\rangle = E_a|\psi_a\rangle$$
$$H_0|\psi_b\rangle = E_b|\psi_b\rangle$$
I am struggling then to understand why the following identity holds (its probably ...
0
votes
1answer
71 views
Hamiltonian of relativistic particle in Coulomb field
In my problem we look at a relativistic particle of charge $q$ and mass $m$ in the presence of a second particle of unit charge and mass $M\gg m$, which is fixed at the origin. I need to find an ...
0
votes
1answer
91 views
Shared eigenbasis of commuting Operators
Suppose I have two Hamiltonian pieces $H_1$ and $H_2$ such that $[H_1,H_2]=0$. Then we know that the two pieces have shared eigenbasis. Assume both $H_1$ and $H_2$ have eigenvalues 2 and -2. Let $|\...
2
votes
2answers
55 views
Hamiltonian of a quantum circuit including a diode?
The LC circuit has a Hamiltonian:
$$\hat{H}={E_L\over2} \hat{\varphi}^2 + 4E_C \hat{n}^2$$
where $\hat{\varphi}$ is the magnetic flux and $\hat{n}$ is the number of charge.
What is the Hamiltonian ...
1
vote
1answer
56 views
How the matrix representation of a Hamiltonian affects the eigenvalues?
Suppose we're given the following Hamiltonian: $$\hat{H}=\frac{\omega}{\hbar} \left(\hat{S}_+^2+\hat{S}_-^2\right)$$ Suppose also that we measure $\vec{S}^2$ and get $6\hbar^2$, i.e. reduced to the $s=...
2
votes
1answer
93 views
Quantum Mechanics Spectral decomposition misunderstanding
My notes state that the spectral decomposition formula is of the form:
$$ \hat{A} = \hat{A}\hat{1} = \sum{\hat{A} } |A_i\rangle\langle A_i | = \sum{A_i } |A_i\rangle\langle A_i | $$
Now consider the ...
0
votes
1answer
51 views
Magnetization subspace and Hamiltonian representation
A follow-up question of the subspaces of 4-electrons: assume the magnetization of the system is conserved (the number of total spin-up $(\uparrow)$ particles is conserved), say 1, for example. Then ...
0
votes
0answers
12 views
Diagonalization of Time dependent Hamiltonian using ZHEEVR
I need to calculate all the eigen values of the time dependent Hamiltonian using ZHEEVR, but I don't know how to define the time dependent Hamiltonian matrix. Please help in this regard
0
votes
0answers
18 views
Continuous non-relativistic bound states
Consider a group of charged point(at least considered as such in this non-relativistic limit) particles such electrons,protons
, nucleii alone in an empty infinite universe and NOT considering any ...
1
vote
1answer
86 views
Commutators with Hamiltonian of the form $H=\frac{p^2}{2m} + V(x)$ [closed]
Consider a one-dimensional problem with a Hamiltonian
\begin{equation*}
H=\frac{p^2}{2m} + V(x)
\end{equation*}
where $x$ and $p$ are the position and momentum operators, $m$ is the mass and $V(x)$ ...
0
votes
1answer
154 views
Heisenberg equation of motion — why is $\vec{\sigma}_H=\vec{\sigma}$?
Trying to obtaining the Heisenberg EOM ( "for $\vec{\sigma}$" ) for the following Hamiltonian
$$
H = - \mu \vec{B}\cdot\vec{\sigma}
$$
where the magnetic field $\vec{B}$ is generic for now, $...
5
votes
2answers
75 views
Time ordering operator if commutator is $c$-number function
I have a question concerning the time ordering operator. Let's suppose we have a time evolution generated by some Hamiltonian $H(t)$ given by
$$
U(t)=T_\leftarrow\exp\left(-\mathrm{i}\int_0^t\mathrm{d}...
2
votes
2answers
87 views
From where does the Ising Hamiltonian come?
So in my Stat Mech course, we were introduced to the classical Ising Model:
$$H = -J\Sigma _{<ij>}S_iS_j - K\Sigma_i S_i$$
But from where does this come from? Is there any rationale behind this? ...
0
votes
1answer
75 views
Heisenberg Hamiltonian matrix and subspaces
I'm dealing with a 4-site Heisenberg's model with no external field:
$$
\begin{align*}
H = \sum_{i<j=0}^3h_{ij}, \quad where\ h_{ij} \equiv J_{ij}(\vec{\sigma_i}\cdot\vec{\sigma_j}) = J_{ij}(X_i\...
2
votes
0answers
76 views
Is the tensor product structure $\hat{H}_0 = \hat{h}_0 \otimes \mathbb{I} + \mathbb{I} \otimes \hat{h}_0$ wrong when interactions are included?
First, consider two uncoupled harmonic oscillators $x_1(t)$ and $x_2(t)$ with classical Lagrangian
$$
L_0 = \frac{1}{2} m_1 \dot{x}_1^2 - \frac{1}{2} m_1 \omega_1^2 x_1^2 + \frac{1}{2} m_2 \dot{x}_2^...
0
votes
0answers
34 views
Many-worlds interpretation and the Hamiltonian
In Everett's Many-worlds Interpretation of QM, Schrodinger's equation is never violated (unlike in Copenhagen Interpretation). But how is this even possible? When you measure a system with respect to ...
