7
votes
2answers
357 views

Quantization of a particle on a spherical surface

Suppose we have a particle of mass $m$ confined to the surface of a sphere of radius $R$. The classical Lagrangian of the system is $$L = \frac{1}{2}mR^2 \dot{\theta}^2 + \frac{1}{2}m R^2 \sin^2 ...
0
votes
0answers
54 views

quantization with a pure exponential potential

Given the Hamiltonian $$ H=p^{2}-ge^{-x}, $$ are the energies negative? If I impose the boundary condition $y(0)=0$ and $y(\infty)=0$, I get the condition for the energies $$ J_{2i \sqrt{E(n)}}(g) ...
3
votes
0answers
98 views

Geometric quantization AND nuclear physics

Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. Geometric quantization is one formalization of the notion ...
0
votes
4answers
247 views

When is energy discrete/quantized for a potential well?

Specifically, my question is: Should one expect energy quantization for a particle in the following potential well? More generally, how can one tell whether or not energy should be ...
1
vote
0answers
94 views

How functions become operators in quantum mechanics? [duplicate]

What used to be functions in the context of classical mechanics like position, linear momentum, angular momentum, etc in quantum mechanics are operators (these operators act on the state to get ...
8
votes
6answers
610 views

What is Quantization?

In classical mechanics you construct an action (involving a Lagrangian in arbitrary generalized coordinates, a Hamiltonian in canonical coordinates [to make your EOM more "convenient & ...
4
votes
1answer
257 views

What fundamental reasons imply quantization?

In classical wave mechanics, quantization can occur simply from a finite potential well. In quantum mechanics, the quantization is obtained from the Schrödinger equation, which is, to my knowledge, a ...
5
votes
3answers
304 views

Fundamentals of Quantum Electrodynamics

In quantum electrodynamics, the classical Hamiltonian is obtained from the classical electromagnetic Lagrangian. Then the classical electric and magnetic fields are promoted to operators, as is the ...
2
votes
0answers
194 views

The correspondence between Poisson bracket and Commutators in Quantum Mechanics

I don't understand canonical quantization. In passing from classical to quantum, one replaces the Poisson brackets with the commutators. I don't really understand this. How can we generally show that ...
6
votes
0answers
132 views

Quantization as a functor

Can anyone give an mathematical elaboration of the following statement: Quantization is a functor carrying the category of Hilbert space and linear maps to that of Symplectic manifolds satisfying ...
5
votes
1answer
451 views

Bohr-Sommerfeld quantization from the WKB approximation

How can one prove the Bohr-Sommerfeld quantization formula $$ \oint p~dq ~=~2\pi n \hbar $$ from the WKB ansatz solution $$\Psi(x)~=~e^{iS(x)/ \hbar}$$ for the Schroedinger equation? With $S$ the ...
5
votes
2answers
230 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.
6
votes
2answers
1k 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 ...
2
votes
4answers
375 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 ...
9
votes
1answer
546 views

exponential potential $ \exp(|x|) $

For $a$ being positive what are the quantisation 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
votes
2answers
275 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 ...
7
votes
1answer
359 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.
3
votes
1answer
281 views

Quantization of Nambu–Goto action in multiples of Planck's constant?

Isn't it possible? Quantization of Nambu–Goto action $$\mathcal{S} ~=~ -\frac{1}{2\pi\alpha'} \int \mathrm{d}^2 \Sigma \sqrt{{\dot{X}} ^2 - {X'}^2}~=~nh\qquad n \in\mathbb{Z}.$$
15
votes
4answers
886 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 ...
4
votes
2answers
1k 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 ...
2
votes
1answer
328 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
0answers
75 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 ...
9
votes
4answers
428 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 ...
5
votes
3answers
378 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 ...
7
votes
1answer
122 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 ...
10
votes
3answers
691 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 ...
15
votes
2answers
221 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 ...
2
votes
2answers
370 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 ...
16
votes
7answers
2k views

Does Quantum Mechanics assume space and time are continuous?

I was confused when I was listening to a Quantum Mechanics lecture online. Are space and time assumed to be continuous or discrete in Quantum Mechanics? I can see the question is vague, but this is ...