Energy levels for a different plancks constant

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Discussion Overview

The discussion revolves around the hypothetical implications of a significantly larger Planck's constant on energy levels in quantum systems. Participants explore how this change might affect various physical constants, atomic energy levels, and the underlying equations of quantum mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant inquires about the effects of increasing Planck's constant, specifically regarding the discretization of energy levels in bound systems.
  • Another participant questions the magnitude of the increase, suggesting a factor of 10^20.
  • Some participants discuss the implications of manipulating Planck's constant on fundamental equations, such as λν = c, and how this might alter the speed of light and quantum conclusions.
  • A reference is made to George Gamov's book, which presents a fictional scenario where Planck's constant is significantly increased, prompting a discussion on its effects on atomic energy levels and the Rydberg constant.
  • One participant posits that if Planck's constant increases, the energy levels would become closer together, seeking clarification on this point.
  • Another participant provides a mathematical perspective, stating that the Rydberg energy is proportional to Planck's constant squared, suggesting that energy levels would be reduced by the square of the increase in Planck's constant.
  • There is a discussion on the implications for the wave function and probability density if Planck's constant were to change, with questions about the use of the Time-Independent Schrödinger Equation in this context.
  • One participant notes that the atomic radial wave function scales with the Bohr radius, indicating that an increase in Planck's constant would lead to significant changes in the wave function and macroscopic density.

Areas of Agreement / Disagreement

Participants express a variety of hypotheses and models regarding the implications of a larger Planck's constant, with no consensus reached on the specific outcomes or effects. Multiple competing views remain regarding the nature of these changes and their implications for quantum mechanics.

Contextual Notes

Some discussions reference specific equations and constants, but the implications of these changes remain speculative. The discussion includes assumptions about other constants remaining unchanged, which may not hold in all scenarios.

bhsmith
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I am writing an essay about what would happen if Planck's constant changed to a much larger number.
What I am having a hard time with is the discretization of energy levels for bound system... how would that change if Plancks constant was larger?
Thanks for all the help in advance.
 
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How much larger are you talking about? Seeing how Planck's constant is very small in the first place, there are many numbers bigger than it.
 
say, 10^20 times larger.
 
Well here's how I thought of it... Think about the equation: λν = c, where c is the speed of light, lambda is Planck's constant, and v is frequency (you probably already knew that).

So, if you manipulated Plank's constant, then that equation would obviously increase the speed of light and thus, change everything that quantum physicists have concluded throughout history.

Just what I thought of. Idk if that would help.
 
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In George Gamov's book, The New World of Mr. Tompkins, Mr. Tompkins and a professor friend go on a quantum safari. The professor has increased the value of Planck's constant by a very large amount, "to about one." Here in the attached thumbnail is an excerpt from that chapter:

You should also look at the effect of Planck's constant on the atomic energy levels in atoms, assuming the electron mass is unchanged. In particular, look at the value of the Rydberg constant, and how it depends on Planck's constant.

Bob S
 

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Bob S said:
In George Gamov's book, The New World of Mr. Tompkins, Mr. Tompkins and a professor friend go on a quantum safari. The professor has increased the value of Planck's constant by a very large amount, "to about one." Here in the attached thumbnail is an excerpt from that chapter:

You should also look at the effect of Planck's constant on the atomic energy levels in atoms, assuming the electron mass is unchanged. In particular, look at the value of the Rydberg constant, and how it depends on Planck's constant.

Bob S

that is interesting
 
I have actually skimmed over that book, it's really interesting and I think it explains this stuff well. It helped me to understand the uncertainty principle and the diffraction,
but i still don't get the energy change.

Is this right...
since E=hf (f is frequency), then the Energy would increase if Planck's constant increases... but the frequency wouldn't change, correct?
 
I also read on another post that if Planck's constant were to increase, then the available values of E would be closer to each other.. is this correct? and if so, why?
 
  • #10
mv²/r = e²/4πεr² (Bohr_model#Electron_energy_levels)
mv²/r = mr(v/r)² = Centrifugal force
e²/4πεr² = Electrostatic force between 2 equal charges at a distance of r
mvr = mαcr = ℏ = h/2π = e²/4πεαc = angular momentum of particle in ground state of Bohr atom​

v² = α²c² = e²/4πεrm = e²v/4πεℏ
r = ℏ/mv = 4πεℏ²/e²m
v = αc = e²/4πεℏ = velocity of particle in ground state of Bohr model (of atom with nucleus of infinite mass and charge = 1)

v is independent of the mass of the particle​
 
  • #11
Let's start by assuming c (speed of light) is unchanged, e (charge of electron) is unchanged, and m (mass of electron) is unchanged.

We can write the Rydberg energy ERy (≈ 13.6 eV), which is the basis for all atomic energy levels, as

ERy = α2 mc2/2,

where α is the unitless fine structure constant, α = 2πe2/hc

So ERy = 2π2e4 mc2/h2c2

Because all the atomic energy level energies are proportional to the Rydberg energy, under the above constraints, all the atomic energy levels are reduced by the square of the increase in Planck's constant.

Bob S
 
  • #12
bhsmith said:
I am writing an essay about what would happen if Planck's constant changed to a much larger number.
What I am having a hard time with is the discretization of energy levels for bound system... how would that change if Plancks constant was larger?
Thanks for all the help in advance.

from Nonlinear Quantum Mechanics, NLQM (unlike of Standard Quantum Mechanics, SQM).



...In particular, Planck's constant is no longer a constant, but a function of the number of degrees in the system. Can the above experiment be used to infer the value of Planck's `constant', by treating the micro-mirror as a mesoscopic object? The departure from the linear theory is expected to become more and more pronounced as the mass of the mirror becomes closer to Planck mass...

.... What about High energy Compton scattering using nanostructured crystal of silver or gold Where to see the effect of change in value of Planck's constant, compton scattered electron or the scattered photon?...
 
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  • #13
bhsmith said:
I have actually skimmed over that book, it's really interesting and I think it explains this stuff well. It helped me to understand the uncertainty principle and the diffraction,
but i still don't get the energy change.

Is this right...
since E=hf (f is frequency), then the Energy would increase if Planck's constant increases... but the frequency wouldn't change, correct?

I would assume that is correct.
 
  • #14
Great, the part about the Rydberg constant was really helpful!
Now i have one more question...
If Planck's constant were to change by 10^20, how would that effect the square of the amplitude of the wave function?
That would be the probability density, so if i used to Time Independent Schroedinger Equation with the different Plancks Constant it would have an effect on it, but is that correct?
Would I use the Time- Independent equation?

And if the probability density changed then what effect would that have on objects?
 
  • #15
The atomic radial wave function R1s(r) scales as the Bohr radius a0 (≈0.53 Angstroms) as a03/2. a0 is proportional to h2. So if h is increased by a factor of 1020, a0 scales by a factor 1040 and R1s(r) scales up by a factor of 1060.

<R1s(r)* | R1s(r)> scales as 10120

Why do you need to use the time-dependent Schroedinger equation?

Because macroscopic density scales inversely to a03, the density decreases by a factor of 10120. Strange world.

You should review how Planck's constant affects all the other fundamental constants. See

http://pdg.lbl.gov/2002/consonepagerpp.pdf

Bob S
 

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