The quantization tag has no wiki summary.
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Bohr-sommerfeld quatnization from the WKB approximation
how can one prove the Bohr Sommerfeld quantization formula
$$ \oint p.dq =2\pi n $$
from the WKB ansatz solution for the Schroedinger equation ?? $ \Psi(x)=e^{iS(x)/ \hbar} $
with $ S $ the action ...
6
votes
1answer
80 views
Critical dimension in quantization of p-branes
So I have what might be a fairly basic question, but my understanding that in the quantization of the the string, or the 1-brane, there are conditions on the number of spacetime dimensions to ensure ...
4
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2answers
145 views
Integer physics
Are there interesting (aspects of) problems in modern physics that can be expressed solely in terms of integer numbers? Bonus points for quantum mechanics.
5
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0answers
78 views
Do semiclassical GR and charge quantisation imply magnetic monopoles?
Assuming charge quantisation and semiclassical gravity, would the absence of magnetically charged black holes lead to a violation of locality, or some other inconsistency? If so, how?
(I am not ...
4
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1answer
87 views
What is the action for an electromagnetic field if including magnetic charge
Recently, I try to write an action of an electromagnetic field with magnetic charge and quantize it. But it seems not as easy as it seems to be. Does anyone know anything or think of anything like ...
5
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2answers
170 views
Weyl Ordering Rule
While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian ...
3
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1answer
83 views
State space of QFT, CCR and quantization, and the spectrum of a field operator?
In the canonical quantization of fields, CCR is postulated as (for scalar boson field ):
$$[\phi(x),\pi(y)]=i\delta(x-y)\qquad\qquad(1)$$
in analogy with the ordinary QM commutation relation:
...
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0answers
17 views
Allowed Quantum States- Filkelstein and Rubinstein constraints
So basically i'm doing a report on Finkelstein and Rubinstein constraints. I have a system where the allowed quantum states satisfy ...
10
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1answer
387 views
Why one-dimensional strings, but not higher-dimensional shells/membranes?
One way that I've seen to sort-of motivate string theory is to 'generalize' the relativistic point particle action, resulting in the Nambu-Goto action. However, once you see how to make this ...
5
votes
2answers
120 views
Quantization of strings on a curved backgrond
usually when people want to quantize the string on flat background, they will try to find the the OPE of embeddings (by solving a green function in a 2D space) and use them to find the energy-momentum ...
7
votes
1answer
193 views
Canonical quantization in supersymmetric quantum mechanics
Suppose you have a theory of maps
$\phi: {\cal T} \to M$
with $M$ some Riemannian manifold,
Lagrangian
$$L~=~ \frac12 g_{ij}\dot\phi^i\dot\phi^j + \frac{i}{2}g_{ij}(\overline{\psi}^i ...
14
votes
7answers
593 views
Is the quantization of gravity necessary for a quantum theory of gravity?
The other day in my string theory class, I asked the professor why we wanted to quantize gravity, in the sense that we want to treat the metric on space-time as a quantum field, as opposed to, for ...
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0answers
63 views
Geometric quantization of a hydrogen atom
I want to know how to quantize a hydrogen atom as an example of geometric quantization. Apparently there is a derivation in the book "Geometric Quantization in Action: Applications of Harmonic ...
2
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4answers
183 views
What entities in Quantum Mechanics are known to be “not quantized”?
Since all the traditional "continuous" quantities like time, energy, momentum, etc. are taken to be quantized implying that derived quantities will also be quantized, I was wondering if Quantum ...
6
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0answers
223 views
exponential potential $ \exp(|x|) $
For $a$ being positive what are the quantization conditions for an exponential potential?
$$ - \frac{d^{2}}{dx^{2}}y(x)+ ae^{|x|}y(x)=E_{n}y(x) $$
with boundary conditions $$ y(0)=0=y(\infty) $$
I ...
3
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2answers
165 views
Definition of “Quantizing”
Could anyone explain to me what "quantize" means in the following context?
Quantize the 1-D harmonic oscillator for which
$$H~=~{p^2\over 2m}+{1\over 2} m\omega^2 x^2.$$
I understand that the ...
6
votes
1answer
169 views
Operator Ordering Ambiguities
I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity.
What does that mean?
I tried googling but to no avail.
5
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0answers
136 views
When can photon field amplitudes be written as field operators?
Suppose I have some classical field equation for two photon fields with amplitudes $A_1(z),A_2(z)$ (plane waves) given as
${A}_1=\alpha f(A_1,A_2) \\
{{A}_2}=\beta g(A_1,A_2) $
Under what ...
1
vote
1answer
195 views
Quantization of Nambu–Goto action in multiples of Planck's constant?
Isn't it possible? Quantization of Nambu–Goto action
$$\mathcal{S} ~=~ -\frac{1}{2\pi\alpha'} \int \mathrm{d}^2 \Sigma \sqrt{{\dot{X}} ^2 - {X'}^2}~=~nh\qquad n \in\mathbb{Z}.$$
3
votes
1answer
107 views
Poincaré group on quantum Klein-Gordon field (C*-algebraic scenario)
on the same topic as this question, I have been trying to fool around with the free real K-G field in flat spacetime on the C*-algebraic scenario (Haag-Kastler axioms, Weyl quantization, etc).
Since ...
4
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2answers
312 views
Path integral and geometric quantization
I was wondering how one obtains geometric quantization from a path integral. It's often assumed that something like this is possible, for example, when working with Chern-Simons theory, but rarely ...
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4answers
572 views
Reason for the discreteness arising in quantum mechanics?
What is the most essential reason that actually leads to the quantization. I am reading the book on quantum mechanics by Griffiths. The quanta in the infinite potential well for e.g. arise due to the ...
3
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2answers
154 views
Ordering Ambiguity in Quantum Hamiltonian
While dealing with General Sigma models (See e.g. Ref. 1)
$$\tag{10.67} S ~=~ \frac{1}{2}\int \! dt ~g_{ij}(X) \dot{X^i} \dot{X^j}, $$
where the Riemann metric can be expanded as,
$$\tag{10.68} ...
2
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1answer
145 views
Equivalence of classical and quantized equation of motion for a free field
Suppose a classical free field $\phi$ has a dynamic given in Poisson bracket form by $\partial_o\phi=\{H, \phi\}$. If we promote this field to an operator field, the dynamic after canonical ...
2
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0answers
100 views
Quantization and natural boundary conditions
The Euler-Lagrange equations follow from minimizing the action. Usually this is done with fixed (e.g. vanishing) boundary conditions such that we do not have to worry about any boundary terms. ...
3
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1answer
196 views
When can a classical field theory be quantized?
Given a classical field theory can it be always quantized? Put in another way, Does there necessarily need to exist a particle excitation given a generic classical field theory? By generic I mean all ...
8
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2answers
299 views
Dirac equation as canonical quantization?
First of all, I'm not a physicist, I'm mathematics phd student, but I have one elementary physical question and was not able to find answer in standard textbooks.
Motivation is quite simple: let me ...
0
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0answers
135 views
Bohr sommerfeld quantiztion rule and Gutzwiller trace
assuming we can evaluate the eigenvalue staircase $ N(E) $ in both manners with the Bohr-Sommerfeld quantization rule
$ N(E)2\pi \hbar = \oint _{C}p.dq $
and using the Gutzwiller trace $ N(E)= ...
3
votes
2answers
840 views
Bohr Model of the Hydrogen Atom - Energy Levels of the Hydrogen Atom
Why the allowed (stationary) orbits correspond to those for which the orbital angular momentum of the electron is an integer multiple of $\hbar=\frac {h}{2\pi}$?
$$L=n\hbar$$
Bohr Quantization rule of ...
3
votes
2answers
133 views
quantization of this hamiltonian?
let be the Hamiltonian $ H=f(xp) $ if we consider canonical quantization so
$ f( -ix \frac{d}{dx}-\frac{i}{2})\phi(x)= E_{n} \phi(x)$
here 'f' is a real valued function so i believe that $ f(xp) $ ...
3
votes
1answer
220 views
Dirac's quantization rule
I first recall the Dirac's quantization rule, derived under the hypothesis that there would exit somewhere a magnetic charge: $\frac{gq}{4\pi} = \frac{n\hbar}{2} $ with $n$ natural.
I am wondering ...
7
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2answers
683 views
Why do we use Planck's constant?
I have been trying to reason why energy packets (i.e. photons) are assumed to be quantized. I know this originated from Max Planck, but may someone explain why energy couldn't be emitted continuously ...
4
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3answers
348 views
Generalizing Heisenberg Uncertainty Priniciple
Writing the relationship between canonical momenta $\pi _i$ and canonical coordinates $x_i$
$$\pi _i =\text{ }\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial x_i}{\partial t}\right)}$$
...
2
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1answer
179 views
Computing a density of states of Hamiltonian $ H=xp$
How could I compute the integral
$$ N(E)~=~ \int dx \int dp~ H(E-xp) $$
the 'Area' inside the Phase space is taken for $ x \ge 0 $ and $ p\ge 0 $? The result should be
$$ N(E)~=~ ...
1
vote
1answer
83 views
understanding the oscillating part of the Gutzwiller trace
given the density of states according to Gutzwiller's trace formula
$ g(E)= g_{smooth}(E)+ g_{osc}(E) $
i know that the 'smooth' part comes from $ g_{smooth}(E)= \iint dxdp \delta(E-p^{2}-V(x)) $ ...
1
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0answers
69 views
shouldn't we add the oscillating terms into Bohr-Sommerfeld quantization formula
shouldn't be the quantization formula (in one dimension) equal to
$ N_{smooth}(E)+N_{osc}(E) = \oint_{C}p.dq $ ??
where the Oscillating term is just the correction from Gutzwiller trace formula or a ...
7
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3answers
294 views
Is the quantization of the harmonic oscillator unique?
To put it a little better:
Is there more than one quantum system, which ends up in the classical harmonic oscillator in the classial limit?
I'm specifically, but not only, interested in an ...
7
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0answers
132 views
Magnetic monopole and electromagnetic field quantization procedure
From the Maxwell's equations point of view, existence of magnetic monopole leads to unsuitability of the introduction of vector potential as $\vec B = \operatorname{rot}\vec A$. As a result, it was ...
5
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3answers
313 views
Some questions on observables in QM
1-In QM every observable is described mathematically by a linear Hermitian operator. Does that mean every Hermitian linear operator can represent an observable?
2-What are the criteria to say whether ...
6
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0answers
46 views
Pohlmeyer reduction of string theory for flat and AdS spaces
The definition of Pohlmeyer invariants in flat-space (as per eq-2.16 in Urs Schreiber's DDF and Pohlmeyer invariants of (super)string) is the following:
$ Z^{\mu_1...\mu_N} (\mathcal{P}) = ...
3
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0answers
71 views
Pohlmeyer reduction of string theory for flat- and AdS- spaces
The definition of Pohlmeyer invariants in flat-space (as per eq-2.16 in Urs Schreiber's DDF and Pohlmeyer invariants of (super)string) is the following:
$ Z^{\mu_1...\mu_N} (\mathcal{P}) = ...
8
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1answer
80 views
Quantum gravity at D = 3
Quantization of gravity (general relativity) seems to be impossible for spacetime dimension D >= 4. Instead, quantum gravity is described by string theory which is something more than quantization ...
9
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2answers
168 views
Virasoro constraints in quantization of the Polyakov action
The generators of the Virasoro algebra (actually two copies thereof) appear as constraints in the classical theory of the Polyakov action (after gauge fixing). However, when quantizing only "half" of ...
7
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1answer
66 views
How does one geometrically quantize the Bloch equations?
I've just now rated David Bar Moshe's post (below) as an "answer", for which appreciation and thanks are given.
Nonetheless there's more to be said, and in hopes of stimulating further posts, I've ...
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3answers
368 views
Rigorous proof of Bohr-Sommerfeld quantization
Bohr-Sommerfeld quantization provides an approximate recipe for recovering the spectrum of a quantum integrable system. Is there a mathematically rigorous explanation why this recipe works? In ...
14
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2answers
124 views
Geometric quantization of identical particles
Background:
It is well known that the quantum mechanics of $n$ identical particles living on $\mathbb{R}^3$ can be obtained from the geometric quantization of the cotangent bundle of the manifold ...
3
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1answer
300 views
Canonical quantization of quantum field
The canonical quantization of a quantum field prescribes that given a lagrangian, one can quantize the theory by imposing the commutation relations between the field operators and their conjugated ...
7
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1answer
105 views
What makes background gauge field quantization work?
[Again I am unsure as to whether this is appropriate for this site since this is again from standard graduate text-books and not research level. Please do not answer the question if you think that ...
3
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1answer
155 views
Question about the parity of the ghost number operator in BRST quantization
Given a Lie algebra $[K_i,K_j]=f_{ij}^k K_k$, and ghost fields satisfying the anticommutation relations $\{c^i,b_j\}=\delta_j^i$, the ghost number operator is then $U=c^ib_i$ (duplicate indices are ...
2
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2answers
250 views
Trouble with constrained quantization (Dirac bracket)
Consider the following peculiar Lagrangian with two degrees of freedom $q_1$ and $q_2$
$$ L = \dot q_1 q_2 + q_1\dot q_2 -\frac12(q_1^2 + q_2^2) $$
and the goal is to properly quantize it, following ...
