All Questions
Tagged with galilean-relativity quantum-mechanics
29 questions
3
votes
0
answers
32
views
Galilean boost operator for quantum multi-particle system
If I have a two particle system with with a potential of form $V(x_1,x_2)$, is it possible to apply the galilean boost operator to only a single coordinate? Essentially, is it possible to move only a ...
- 331
2
votes
1
answer
168
views
Transformation of wavefunction
While learning QM, I was wondering how would the wavefunction of a particle, suppose charged particle, look for different observers moving with respect to each other.
To begin with, let the electric ...
- 457
2
votes
1
answer
57
views
Are projective representiations of a Lie group a representation of the semi-direct product of the group with $U(1)$ if the norm is preserved?
Let's say we have a function $f(x_{\mu},t)$ that transforms under the action of an $N$-parameter group $G(a_{\nu})$. Then a projective representation of $G(a_\nu)$ in the $f(x_\mu,t)$ basis would ...
- 372
4
votes
0
answers
77
views
Using Galilean covariance to find conditions on physical observables
Let's suppose that coordinates have to transform accoring to the Inhomogenous Galilean Group. Then
$$ x' = x + a + v(t+b) $$
$$ t' = t + b $$
Let's use a funtion $\psi(x,t)$ of $x$ and $t$ as the ...
- 372
7
votes
2
answers
344
views
Energy levels of a translating quantum harmonic oscillator
It is well known that the energy levels
$$
E_n = \hbar \omega\left(n+\frac{1}{2}\right)
$$
of the quantum harmonic oscillator verify the eigenvalue problem
$$
\hat{H}|\psi_n\rangle = E_n |\psi_n \...
- 1,252
2
votes
0
answers
41
views
Is there a general methodology for causal nets of observables regardless of kinematics?
The typical definition of a causal net of observables in quantum theory is to consider, for the case of a (globally hyperbolic) spacetime $M$, the category of open sets $O(M)$ ordered by inclusion, in ...
- 16.7k
1
vote
1
answer
683
views
Differential Equation that combines QM with Galilean relativity
In Galilean Relativity if there are two objects, the initial positions of the objects, their masses, and the forces acting on the objects is not enough to uniquely determine where the objects will be ...
- 2,021
2
votes
1
answer
132
views
What is the symmetry group of a particle interacting with external fields?
I am following along with Ballentine's (in his Quantum Mechanics: A Modern Development) construction/identification of symmetry generators as operators representing the standard observables (...
- 1,261
3
votes
1
answer
328
views
Galilean covariance of the Schrödinger equation without choosing a representation
The most general form of Schrödinger equation is
$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$
where $\psi(t)$ is an element of a Hilbert space $\mathcal H$ (not necessarily $L^2$), and $H$ is a ...
- 757
8
votes
2
answers
506
views
Are there two different versions of non-relativistic quantum mechanics?
The first version is the usual one we're taught. But there's this other version too : A quantum non-relativistic field theory.
Take a non-relativistic classical field, like the non-relativistic limit ...
- 6,742
6
votes
2
answers
729
views
Why is the phase of a matter wave not Galilean invariant? And what does this say about the Schrödinger equation? [duplicate]
Matter waves are not Galilean Invariant
Consider a non-relativistic freely-propagating matter wave in an inertial frame $\Sigma'$ moving along the $x'$-direction with kinetic energy $E'=1/2m_0v'^2$, ...
- 1,826
2
votes
1
answer
912
views
Galilean invariance of Schrödinger equation [closed]
I'm trying to prove that if $\psi (\mathbf r, t)$ satisfies
$$
i\hbar \frac{\partial\psi}{\partial t}(\mathbf r, t) =
-\frac{\hbar^2}{2m} \left( \nabla-\frac{iq}{\hbar} \mathbf A \right)^2\psi(\...
- 1,999
4
votes
0
answers
103
views
Why do Galilean transformations change both states and operators?
When a Galilean transformation on a quantum system is performed, the states and the operators change:
$$|\phi\rangle \rightarrow |\phi\rangle'$$
$$\hat A \rightarrow \hat A'$$
I don't understand the ...
- 1,978
2
votes
2
answers
1k
views
Show the Galilean covariance of Schrödinger equation
I'm trying to show the Galilean covariance of the (time-dependent) Schrödinger equation by transforming as follows:
$$
\left\{\begin{eqnarray}\psi(\vec{r},t) &=& \psi(\vec{r}'-\vec{v}t,t),\\ \...
- 1,448
0
votes
1
answer
46
views
Galilei group and Constrained QM
Let's assume spin-0 for simplicity.
So far as I understand the issue, the Galilei simmetries constraints the possible hamiltonians of a quantum systems so that the only possible interactions of a ...
- 458
3
votes
1
answer
1k
views
Derive the Lagrangian that yields the free Schrödinger's equation from Galileian Invariance
The Lagrangian Density $$L(\Psi, \Psi^*)=i \hbar \dot{\Psi} \Psi^* + \frac{\hbar^2}{2m} \Psi \Delta \Psi^*$$ will yield the schroedinger equations for $\Psi$ and $\Psi^*$. Can we derive this ...
- 6,955
2
votes
0
answers
763
views
Galilean invariance of the free schroedinger equation [duplicate]
My question follows this question: Naive interpretation of Galilean invariance of the TDSE
Essentially, I'm not sure how to proceed mathematically.
We have the transformations:
$$\begin{cases}x'=x-...
- 454
14
votes
2
answers
5k
views
Why the Galileo transformation are written like this in Quantum Mechanics?
In Quantum Mechanics it is said that the Galileo transformation $$\hat{\mathbf{r}}\mapsto \hat{\mathbf{r}}-\mathbf{v}t\quad \text{and}\quad \hat{\mathbf{p}}\mapsto \hat{\mathbf{p}}-m\mathbf{v}\tag{1}...
- 37.4k
8
votes
1
answer
4k
views
Galilean transformation in non-relativistic quantum mechanics
I'm reading Weinberg's Lectures on Quantum Mechanics and in chapter 3 he discusses invariance under Galilean transformations in the general context of non-relativistic quantum mechanics. Being a ...
- 732
3
votes
1
answer
215
views
According to the Galileo Algebra, space translations commute with time translations. Does this mean that $[\vec P,H]=0$?
The Galileo Algebra is discussed in, for example, the wikipedia article Representation theory of the Galilean group. In that article, we can see that, for example,
$$
[E,P^i]=0
$$
which means that ...
0
votes
0
answers
871
views
Galilean transformation of Schrodinger equation and momentum operator [duplicate]
Let
$$
\left.\begin{aligned}
t'&=t\\x'&=x-vt
\end{aligned}\right\} \quad \Longrightarrow\quad \dot{x}'=\dot{x}-v
$$
and therefore $p'=p-mv$.
If $p'=-i\hbar\nabla' $, then $\nabla'=\nabla-iv/\...
- 868
0
votes
1
answer
399
views
Galilean relativity in QM
Intro
I've been trying to show that the generator of boosts can be written in operator form as can be seen here, as:
$$ B = \sum_i m_i x_i(t) - t \sum_i p_i $$
As a reminder the transformation ...
- 705
1
vote
0
answers
153
views
Mass, Spin, Internal Energy and 1-Particle States in Galilean Quantum Mechanics
I have been reading an article discussing the unitary representation of Galilean group and non-relativistic quantum mechanics. The link to the article is given below.
http://arxiv.org/abs/1107.2442
...
- 1,110
3
votes
1
answer
2k
views
Naive interpretation of Galilean invariance of the TDSE
I was told today by someone smarter than myself that the time-dependent Schroedinger equation in one dimension was invariant under a Galilean transformation of $(x,t)$, namely under
$$\begin{cases}x'=...
- 576
3
votes
1
answer
308
views
Representations of Galilei group
Show that the operator $U(\alpha, \beta) = e^{i(\alpha \hat{x}^2 + \beta \hat{p}_{x}^2)}$ can represent the space reflection of the 1D Galilei group: $x \to -x; t \to t$.
I don't really know anything ...
- 177
1
vote
0
answers
136
views
Galilean Transform
I tried to solve a problem using two different ways and I had some trouble, the problem is:
We define a symmetry transform of the expected value of $\vec{P}$ like this:
$$\langle \psi|\vec{P}|\psi \...
- 103
2
votes
1
answer
2k
views
Generator of Velocity Transformations - Galilean Transformations
Under a Galilean transformation, the coordinates and momenta of any system transform as:
$$ t \rightarrow t',\\ \vec r\:' = \vec r + \vec vt,\\ \vec p\:' = \vec p + m\vec v $$
where $\vec v$ is ...
- 379
6
votes
2
answers
542
views
Does Galileo's Tower of Pisa argument contradict quantum mechanics?
(My questions are at the end, but they may not mean much without explanation below.)
Galileo argued that because the mass of a falling object can always be redistributed in ways that asymptotically ...
- 22.2k
28
votes
4
answers
23k
views
Galilean covariance of the Schrodinger equation
Is the Schrodinger equation covariant under Galilean transformations?
I am only asking this question so that I can write an answer myself with the content found here:
http://en.wikipedia.org/wiki/User:...
- 3,802