Energy levels for a different plancks constant

[bhsmith]

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.

 

[chug.-.chug]

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.

 

[bhsmith]

say, 10^20 times larger.

 

[chug.-.chug]

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.

 

[chug.-.chug]

I got curious and found this:
http://en.wikipedia.org/wiki/Matter_wave

Those equations would obviously be manipulated as well...

 

[Bob S]

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

 

[chug.-.chug]

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

 

[bhsmith]

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?

 

[bhsmith]

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?

 

[granpa]

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​

 

[Bob S]

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 E_Ry (≈ 13.6 eV), which is the basis for all atomic energy levels, as

E_Ry = α² mc²/2,

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

So E_Ry = 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

 

[yoda jedi]

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?...

 

[chug.-.chug]

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.

 

[bhsmith]

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?

 

[Bob S]

The atomic radial wave function R_1s(r) scales as the Bohr radius a₀ (≈0.53 Angstroms) as a₀^3/2. a₀ is proportional to h². So if h is increased by a factor of 10²⁰, a₀ scales by a factor 10⁴⁰ and R_1s(r) scales up by a factor of 10⁶⁰.

<R_1s(r)^* | R_1s(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