52
votes
Accepted
How can Planck's constant take different values?
In a purely classical (Newtonian) universe, quantum effects would be absent, and the way to pretend this is true mathematically is to allow Planck's constant to approach zero, and see what the ...
- 97.8k
44
votes
Accepted
Does spin really have no classical analogue?
You're probably overthinking this. "Spin has no classical analogue" is usually a statement uttered in introductory QM, where we discuss how a quantum state differs from the classical idea of ...
- 129k
38
votes
Why did the Bohr Model Successfully calculate some of the energy levels in hydrogen?
What bugs my is that Bohr derives the energy from very few assumptions and sets up the solution through a natural force balance. Why is it that a faulty model can deduce the energy levels?
Bohr's ...
- 105k
31
votes
Accepted
On Groenewold's Theorem and Classical and Quantum Hamiltonians
You probably need to internalize Ivan Todorov's accessible Quantization is a mystery. Your best bet for addressing your questions is Geometric quantization, not phase space quantization that you ...
- 66.3k
24
votes
Accepted
Why does the expectation value in quantum mechanics correspond to the classically measured value?
In general, there is no such thing as a "classically measured position" for a generic quantum system/state. Some situations are simply not well-modeled by classical physics, and Ehrenfest's ...
- 129k
21
votes
Accepted
Why does electron orbital circumference have to be in multiples of de Broglie wavelengths?
I will assume you are familiar with the properties of waves such as interference and diffraction.
Consider an electron orbiting the nucleus. By de Broglie's hypothesis, we would consider it to be a ...
- 3,207
20
votes
Accepted
Is there a second-order non-linear addition to Maxwell's equations?
In Quantum Electrodynamics by A. I. Akhiezer; V. B. Berestetskii, photon-photon scattering is considered in paragraph 54, and the effective classical Lagrangian for nonlinear vacuum electrodynamics ...
- 1,123
18
votes
Why does electron orbital circumference have to be in multiples of de Broglie wavelengths?
Nothing actually. It was quite a wild guess by Bohr and supplied him with the spectrum of hydrogen. Pretty good guess indeed.
- 47.9k
17
votes
Accepted
Do Maxwell's equation describe a single photon or an infinite number of photons?
Because photons do not interact, to very good approximation for frequencies lower than $m_e c^2 / h$ ($m_e$ = electron mass), the theory for one photon corresponds pretty well to the theory for an ...
- 22.8k
17
votes
Accepted
Contradiction in my understanding of wavefunction in finite potential well
Even classically, particles with a fixed total energy spend more time near the turning points since this is where the motion is the slowest. The probably of finding the particle in a small region ...
- 47.9k
16
votes
Why does electron orbital circumference have to be in multiples of de Broglie wavelengths?
De Broglie suggested the existence of matter waves and gave a relation between their wavelength and momentum.
$\lambda=\frac{h}{p}$ ,
He said that this relation is completely general. It can be ...
- 4,927
16
votes
Does spin really have no classical analogue?
It's seemingly unappreciated by many people that there are different classical limits of quantum mechanics. At least there are two, a particle limit where you take $\hbar\to 0$ and $ω\to\infty$ while ...
- 28.7k
14
votes
Do Maxwell's equation describe a single photon or an infinite number of photons?
Do Maxwell's equation describe a single photon or an infinite number of photons?
Both.
(i) The single photon wave function is a solution of the free Maxwell equations in vacuum, and any nonzero ...
- 45.7k
14
votes
Accepted
Period of the propagator of quantum harmonic oscillators
Good observation. OP's eq. (2) should be amended with a metaplectic correction/Maslov index: There is a caustic at every half-period, which leads to a phase factor $\exp\left(-\frac{i\pi}{2}\right)$ ...
- 213k
14
votes
Is there a second-order non-linear addition to Maxwell's equations?
In addition to what the other answer already mentions, the non-linear extension required to include QED effects classically can also be found in [Jackson].
But the claim the question makes, that &...
- 5,791
12
votes
Transition from quantum to classical mechanics
The heuristic that compares the action $S$ to Planck's constant is vaguely useful as an initial criterion, but the limit from quantum to classical mechanics is rather more subtle, in ways that make ...
- 135k
12
votes
Accepted
Is the Moyal-Liouville equation $\frac{\partial \rho}{\partial t}= \frac{1}{i\hbar} [H\stackrel{\star}{,}\rho]$ used in applications?
"Used in anger" or "killer ap"? To my knowledge, no problem has been solved in the phase-space quantization language that was not solvable in the other two formulations/pictures (Hilbert space or path ...
- 66.3k
12
votes
Why did the Bohr Model Successfully calculate some of the energy levels in hydrogen?
Bohr's model actually is part of the "old quantum theory" prior to the advent of wave and matrix mechanics that is based on the concept of classical mechanics that the motion of particles (electrons) ...
- 10.1k
12
votes
Contradiction in my understanding of wavefunction in finite potential well
Imagine a perfectly elastic ball dropped vertically onto a flat surface. The ball heads for the point of lowest potential, ie the ground, but because of conservation of energy it bounces back to its ...
- 28.4k
11
votes
Does spin really have no classical analogue?
An essential difference is that there is no representation of spin in ordinary $3D$ space$^\dagger$. Unlike the spherical harmonics $r^\ell Y_{\ell m}(\theta,\varphi)$ which can be expressed in terms ...
- 47.9k
10
votes
Why does the expectation value in quantum mechanics correspond to the classically measured value?
Expectation values of quantum mechanics correspond to classically measured values by construction. Indeed, if the classical theory were not a limiting case of the quantum one, this latter would be ...
- 65.1k
10
votes
What happens to branching in the Many-Worlds Interpretation of quantum mechanics in the limit when Planck's constant goes to 0?
We learn from quantum mechanics courses that one recovers classical mechanics in the limit when Planck's constant goes to zero. This can be seen in the path integral formulation. This is why ...
- 10.2k
9
votes
Accepted
Why are we allowed to let $\hbar \ \rightarrow \ 0$ in the semi-classical regime?
We never "let $\hbar \to 0$". As you said, that doesn't make sense because $\hbar$ is dimensionful, and also fixed in our universe.
What we mean by $\hbar \to 0$ is that we're considering only ...
- 105k
8
votes
Accepted
Difference between QFT In curved spacetime, semiclassical, and quantum gravity?
QFT in curved spacetime: The spacetime metric is fixed. It doesn't react to the state of the quantum fields, which are described by operators on a Hilbert space. The fixed metric field defines what &...
- 55k
8
votes
Accepted
How does the linearity of the Schroedinger equation reflect the interactions?
There are different equivalent formulations of quantum mechanics, and how an interaction is represented in each formulation is different.
In the Schrodinger picture, operators are time independent ...
- 55.5k
8
votes
How can Planck's constant take different values?
Theories that are said to have real parameters really have a multidimensional parameter space, and the real parameters are coordinates in that space.
Often, multiple points in the parameter space ...
- 28.7k
8
votes
Change of variable in Schrödinger's first paper
Schrodinger begins with
$$H(q, \frac{\partial S}{\partial q})=E$$
When the symbol $S$ stands for the action the quantity
$$
\frac{\partial S}{\partial q} = p\;,
$$
is the momentum. (More precisely, ...
- 23.3k
7
votes
Accepted
What is the quantum analog of classical infinitesimal displacement?
The object $\mathrm ds^2$ is not really an infinitesimal displacement, but a metric. In non-relativistic QM, the metric is the standard, Euclidian one
$$
\mathrm ds^2=\mathrm d\boldsymbol r^2
$$
so ...
7
votes
Landau & Lifshitz's Approach (contour method) on the WKB connection formulas
I) L&L are referring to a linearized model where the TISE
$$\begin{align} \hbar^2\psi^{\prime\prime}(x) +P^2(x) \psi(x)~=~&0,\cr P(x)~:=~ \sqrt{2m (E-V(x))}~=~& |P(x)| e^{i\theta(x)}, \...
- 213k
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