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 ...
niels nielsen's user avatar
42 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 ...
ACuriousMind's user avatar
  • 125k
37 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 ...
knzhou's user avatar
  • 102k
28 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 ...
Cosmas Zachos's user avatar
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 ...
ACuriousMind's user avatar
  • 125k
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 ...
Pritt Balagopal's user avatar
17 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.
ZeroTheHero's user avatar
  • 45.5k
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 ...
ZeroTheHero's user avatar
  • 45.5k
16 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 ...
Sean E. Lake's user avatar
  • 22.5k
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 ...
Mitchell's user avatar
  • 4,907
15 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 ...
benrg's user avatar
  • 26.2k
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 ...
Arnold Neumaier's user avatar
13 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 $\exp\left(-\frac{i\pi}{2}\right)$ ...
Qmechanic's user avatar
  • 203k
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 ...
Emilio Pisanty's user avatar
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 ...
Cosmas Zachos's user avatar
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) ...
Frederic Thomas's user avatar
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 ...
Marco Ocram's user avatar
  • 26.2k
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 ...
ZeroTheHero's user avatar
  • 45.5k
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 ...
Roger V.'s user avatar
  • 58.8k
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 ...
alanf's user avatar
  • 7,542
9 votes

Why is it OK to keep the quadratic term in the small $\hbar$ approximation?

You are right to be concerned by this procedure, since a Taylor expansion in powers of $\hbar$ itself lacks a clear-cut physical meaning. This is because $\hbar$ is a dimensionful constant: its value ...
Mark Mitchison's user avatar
9 votes

Lack of Maslov index in the path integral formalism

Apart from a full proof of the Gutzwiller's formula in the context of the Feynman path integral (FPI), then OP is essentially asking the following: Does the FPI know about the metaplectic ...
Qmechanic's user avatar
  • 203k
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 ...
knzhou's user avatar
  • 102k
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 &...
Chiral Anomaly's user avatar
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 ...
benrg's user avatar
  • 26.2k
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, ...
hft's user avatar
  • 20.1k
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)}, \...
Qmechanic's user avatar
  • 203k
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 ...
AccidentalFourierTransform's user avatar
7 votes

Why does electron orbital circumference have to be in multiples of de Broglie wavelengths?

This comes from the Bohr-Sommerfeld quantization formula, which can be derived from the semiclassical WKB approximation of quantum mechanics, cf. e.g. this Phys.SE post. The quantization condition ...
Qmechanic's user avatar
  • 203k

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