import numpy as np
pi = np.pi
# half Angstrom radius, taken as the reference radius of the orbit.
radius = 2*0.5292*(10**(-10)) ## metres
# supposed average kinetic energy of electron (-1) in hydrogen molecule
velocity = 2.186*(10**6) ## metres per second.
# electron mass.
mass = 9.109*(10**(-31)) ## kilograms. (?)
print("Planck's constant given electron with:\n radius r, velocity v, mass m is approximately: ", pi * radius*velocity*mass, " Joule Seconds per Cycle. ")
Planck's constant given electron with: radius r, velocity v, mass m is approximately: 6.620953893160361e-34 Joule Seconds per Cycle.
What is Planck's constant $h$?
Planck's famous constant $h$ from [Plank, M. (1901) On the Law of Distribution of Energy in the Normal Spectrum. Annalen der Physik, 4, 553.] is obviously a famous constant in physics, but I don't think it's well understood. Recently in the course of our studying Newton-Weber equations we have cause to review Planck's definition. Ralph Sansbury's treatment of Planck's constant here is also useful.
The constant $h$ is defined by Planck as representing the action of an electron charge $(-1)$ at radius $r=1 A$ one Angstrom unit around a hydrogen nucleus of charge $+1$. More specifically $h$ is the action of the kinetic energy of the electron $e^{-}$ around an imaginary circular orbit of radius $1$ Angstrom units, i.e. Planck's constant is defined $$h:= T . (\text{travel time}) = \frac{1}{2}~\mu v^2( 2\pi r / v ) =\pi \mu v r ~~~\text{Joule seconds per cycle}.$$
What is strange in Planck's definition? It appears to contain arbitrary choices. For example Planck simply asserts (posits) that the hydrogen atom has an (average) radius of 1 Angstrom, and travelling at a given average velocity $v=2.186 \times 10^6$ metres per second.
To quote from Planck's Nobel Prize speech here:
- "The explanation of the second universal constant [h] of the radiation law was not so easy. Because it represents the product of energy and time (according to the first calculation it was 6.55 x 10-27 erg sec), I described it as the elementary quantum of action. Whilst it was completely indispensable for obtaining the correct expression for entropy [...] it proved elusive and resistant to all efforts to fit it into the framework of classical theory. As long as it was looked upon as infinitely small, that is, for large energies or long periods of time, everything went well; but in the general case, however, a gap yawned open in some place or other, which was the more striking, the weaker and faster the vibrations that were considered."
[And what is Planck's solution to this problem?]
- "The foundering of all efforts to bridge the chasm soon left little doubt. Either the quantum of action was a fictional quantity, then the whole deduction of the radiation law was in the main illusory and represented nothing more than an empty non-significant play on formulae, or the derivation of the radiation law was based on a sound physical conception. In this case the quantum of action must play a fundamental role in physics, and here was something entirely new, never before heard of, which seemed called upon to basically revise all our physical thinking, built as this was, since the establishment of the infinitesimal calculus by Leibniz and Newton, upon the acceptance of the continuity of all causative connections."
Remarks
Planck's belief in universal constants is a direct consequence of his accepting Kirchoff's erroneous law of emission, which is no law at all. The failure of Kirchoff's theory on emission has been extensively documented by Prof. Dr. P.M. Robitaille, being in fact error and violating the 0th law of thermodynamics (definition of Temperature as intensive property). "Kirchhoff's Law of Thermal Emission: Blackbody and Cavity Radiation Reconsidered".
It's not immediate from reading Planck's 1905 paper that $h$ is related to the orbit of an electron in hydrogen. This was apparently Bohr's insight in viewing the oscillators in the cavity walls as electron in hydrogen orbit seen "from the side".
The very definition of $h$ appears to assume that the electron has a certain circular orbital structure around the hydrogen nucleus. As Planck describes, his methods were not successful with weaker and faster oscillations, i.e. deeper into atomic structure.
Planck makes direct reference to Boltzmann, a probabilistic model for resonators, the need to define two absolute constants $h/k$ to obtain the normalization constants for probabilities. Can we more directly explain what motivated Planck?
It seems that perhaps Planck found in the cavity oscillators an inexplicable stability. It must appear somewhat contradictory that a Hertz type oscillator need not radiate? This is the same tension within the Bohr planetary atomic model, but which tension disappears in the Weber planetary atomic model. Planck's definition of $h$ in terms of the hydrogen atom can be somewhat generalized to the Weber model of the atom, where we have precession.
A Weberian type definition of Planck's constant would need examine the action of an electron $e^-$, namely the integral of the kinetic energy of $e^-$ multiplied by time $ \int T \cdot t $. For Planck defines $h$ in units of Joules/Herz, which means Joule seconds per cycle. But cycle of what? Evidently of the electron $e^-$ in orbit, but then we need to modify the meaning of cycle given the precession of the orbit. Strictly speaking, the orbit is not periodic, but we can still consider the angular variable $\theta$. This will be discussed below.
It seems that fundamentally the Coulombian and Maxwellian model of the hydrogen atom, imagined as an electron in orbit around the positive hydrogen nucleus $H^+$, was always in tension with the development of the atom. "How could the electron in orbit not be radiating energy under acceleration as predicted by Maxwell equations?" "How does the atom not blow apart or collapse?" The history of 20th century physics is that this question was never answered! It was evaded by the assumption of some reference unit of action, namely the kinetic energy of an electron in a supposedly circular orbit around the hydrogen nucleus.
Sansbury-Weber Planetary Models:
There is another option to define the Planck constant given it's variational definition.
As we view it, the electron orbit in a hydrogen atom is impossible for Coulomb-Maxwellian's to model. However Sansbury's amazing idea is that a system only radiates if there is a net loss of energy to the environment. It's possible for the electron to be communicating energy with the hydrogen nucleus yet there being zero net radiation to any observer. As Sansbury describes, it's possible by superposition for the net radiation to be zero. This is Sansbury's amazing insight: there is radiation, but here is no net radiation!
Sansbury also suggests a variational definition of Planck's constant, namely by evaluating the Planck constant as a functional on orbits! For Planck's definition readily suggests the more variational action interpretation:
For electron orbits in stable isolated two-body hydrogen system, evaluate the action integral $\int_\gamma T dt$ along trajectories. In Weber systems the trajectories are not strictly periodic since there is precession around the centre of mass.
The angular momentum $L=r\mu r \dot$ is conserved in our isolated system, and this yields $$\frac{d\theta}{dt} = \frac{L}{\mu ~r^2}.$$
The equation for $\dot{\theta}$ can be formally integrated to obtain $$\theta(t)-\theta(t_0)=\frac{L}{\mu} \int_{t_0}^t \frac{ds}{r(s)^2}.$$ Here $s$ is "dummy variable" used for integration. This expression is only formal at this point because we still need to find the radial function $r(t)$ of the particle as a function of time.
Comparing to Planck's constant, we are looking for times $t^*$ of complete revolution, and this means finding $t^*$ such that $\theta(t^*)-\theta(0)=2\pi$. This leads to the equation: $$2\pi \mu ~/ L = \int_{0}^{t^*} \frac{ds}{r(s)^2}.$$ When the particle is in circular orbit, then the integrand is constant and we obtain the usual formula for $t^*$ as equal to "circumference divided by speed". (Question: is this integral a useful or useless identity for $t^*?$)
How to Compute this Planck Action?
Our strategy is simply to plot some two-body orbits, then directly integrate $1/r^2$ and solve for the period $t^*$. There is a tension here because of precession, and it's really not clear that the integral properly represents an action.
"""
To upload the Weber notebooks we need run %run <path to file>.
Except this file and the path needs to be memory accessible /readable
to the session RAM of googel colab. This is why we need to upload the
Weber notebooks to the session storage.
- Find direct line of code which imports Weber notebook from url as read-only.
- Or have the file uploaded locally everytime the command is run.
- the %run command works better with .py files, so we need method to export .ipynb --> .py
"""
%run /content/Weber.py
#import Weber
"""
Now we also need compute the angular momentum of the system and rescale to find the exact time value.
Another idea: we are learning the lesson of selecting the centre of rotation to compute the angular momentum.
In other words, the r involved in our definition of L is *not* the same rho radial distance.
The correct procedure is to redefine r to be the distance from the centre of mass. Does this improve the answer?
"""
mu=0.5
# Angular momentum L = r m r theta' #
# i don't think this is correct ... the r term is not the relative distance but the distance from the centre of mass,
# which is the centre of rotation. (This doesn't much improve our answers...)
state0=[[[ 0.6 , 0. , 0. , 0. , 0.033, 0. ],
[-0.6 , 0. , 0. , 0. , -0.033, 0. ]], [[1.0, -1.0], [1.0, -1.0]]] ;
y = IW(state0, epsilon=0.1, dt=0.001, tEnd=10)
fig = plt.figure(figsize=(10, 10))
ax = fig.gca(projection='3d')
plt.plot(y[0,0,:], y[0,1,:], y[0,2,:], "g", label='position of particle x0(t) with -1 charge')
plt.plot(y[1,0,:], y[1,1,:], y[1,2,:], "b", label='position of particle x1(t) with +1 charge')
#plt.plot(y[2,0,:], y[2,1,:], y[2,2,:], "r", label='position of particle x2(t) with +1 charge')
plt.legend(loc='best')
plt.show()
def L(x, mu=0.5): #two body
x1,x2 = x
# mu = (k[0][0] * k[1][0])/(k[0][0]+k[1][0])
dx = np.subtract(x2[0:3], x1[0:3])
# rho = norm(dx)
dv = np.subtract(x2[3:6], x1[3:6])
# rhat = np.dot(1/rho, dx)
# rprime = np.dot(rhat, np.dot(rhat, dv))
# test = np.subtract(dv, np.dot(rhat, np.dot(rhat, dv)))
return np.cross(dx, dv)*mu #dx, dv are the not the distances from the origin of rotation.
print(L(y[:,:,0]))
ell = norm(L(y[:, :, 1]))
print(ell)
index = len(y[0, 0, :])
#print(length)
# note the sign on U, this is only for q1.q2 < 0 case.
# for electrons q1.q2 > 0.
energy_list= []
for j in range(index):
energy_list = np.append(energy_list, U(y[:, :, j]) + T(y[:, :, j], state0[1]) )
print("Energy list: ", energy_list) #no conservation of energy! Integrator is not accurate!!
#angular_list= []
#for j in range(index):
# angular_list = np.append(angular_list, 2*np.pi*mu/norm(L(y[:, :, j])) )
print("The 2*pi*mu/L values : ", 2*np.pi*mu/norm(L(y[:, :, 0])))
[0. 0. 0.0396] 0.039599999999999996 Energy list: [0.83333333 0.83333334 0.83333335 ... 0.77958989 0.77958734 0.7795848 ] The 2*pi*mu/L values : 79.33314781792406
index = len(y[0, 0, :])
timestep=0.001
#time_list = np.linspace()
print(index)
tList = np.linspace(0, timestep*index, index)
print(tList)
print("Our two lists have equal length?: ", len(tList)==index)
invSquareRhoList = [ rho(y[0, :, j], y[1, :, j] )**(-2) for j in range(index)]
thetaList = [ np.arctan(y[0, 1, j]/y[0,0,j]) for j in range(index)]
print("Theta angles of particle x1 :", thetaList)
plt.plot(tList, invSquareRhoList, 'r-', label='r(t)**-2')
#plt.plot(t_points[0:N1], sol.y[3], 'b:', label='velocity v0(t)')
#plt.ylabel('relative distance')
plt.xlabel('time')
plt.legend(loc='best')
plt.show()
plt.plot(tList, thetaList, 'r-', label='theta of x1')
#plt.plot(t_points[0:N1], sol.y[3], 'b:', label='velocity v0(t)')
#plt.ylabel('relative distance')
plt.xlabel('time')
plt.legend(loc='best')
plt.show()
#Plot of 1/rr is periodic... that period is approximately 4000*dt.
#However this periodicity is *not* the time period for the angle.
#This is because the angular return distance is *not* equal to the radial
#period. So we *do* need to integrate the plot!
# Defining the angle theta directly via theta = arctan(y/x) gives us closer agreement.
# I.e. the periodicity is more easily viewed as the time difference between poles of arctan !
# If we define L = mu * r x v, then can we alternatively derive the angle via:
# |L| = mu |r| |v| sin(theta)
#
#
#
10001 [0.0000000e+00 1.0001000e-03 2.0002000e-03 ... 9.9989998e+00 9.9999999e+00 1.0001000e+01] Our two lists have equal length?: True Theta angles of particle x1 : [0.0, 5.5000047564910346e-05, 0.00011000028527903334, 0.0001650009032937346, 0.00022000209176453248, 0.00027500404085307236, 0.0003300069407291012, 0.00038501098157244244, 0.0004400163535749705, 0.0004950232469425866, 0.0005500318518971942, 0.0006050423586786748, 0.0006600549575468656, 0.0007150698387835351, 0.0007700871926943624, 0.0008251072096109142, 0.0008801300798926252, 0.0009351559939287771, 0.0009901851421404795, 0.001045217714982651, 0.001100253902946003, 0.001155293896559021, 0.0012103378863899512, 0.0012653860630487844, 0.0013204386171892437, 0.0013754957395107716, 0.00143055762076052, 0.0014856244517353393, 0.0015406964232837711, 0.0015957737263080417, 0.0016508565517660552, 0.001705945090673392, 0.0017610395341053026, 0.0018161400731987118, 0.0018712468991542164, 0.001926360203238088, 0.00198148017678428, 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import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import interp1d
from scipy.interpolate import interp2d
from scipy.integrate import quad
# from scipy.integrate import solve_ivp
mu=0.5
# y_f = interp1d(tList, invSquareRhoList, 'linear')
y_f = interp1d(tList, invSquareRhoList, 'cubic')
def integral(t):
return quad(lambda x: y_f(x), 0, t)[0]
x = np.linspace(0,10,500)
y = y_f(x) # note advantage of pure function definition!
plt.plot(x,y, label="1/r squared")
plt.plot(x, [integral(t) for t in x], label="integral over [0,t] of 1/r squared")
plt.plot(x,[2*np.pi*mu/ell for t in x] , label="nominal value of 2*pi*mu/L")
plt.legend()
plt.show()
print(2*np.pi*mu/ell)
91.53824748222007