The Old Quntum Theory

©Fernando Caracena 2013

Models of the Atom

John Dalton (1766-1844) is famous for his atomic hypothesis, which made plain the laws of chemistry:

1. (Chemical) Elements are made of extremely small particles called atoms.
2. Atoms of a given element are identical in size, mass, and other properties; atoms of different elements differ in size, mass, and other properties.
3. Atoms cannot be subdivided, created, or destroyed.
4. Atoms of different elements combine in simple whole-number ratios to form chemical compounds.
5. In chemical reactions, atoms are combined, separated, or rearranged.

It turned out that the atoms of Dalton were composite structures, which could be further broken down into positively and negatively charged particles. The stability of atoms came from their quantum mechanical organization that operates in a restricted energy range, when hit by particles with enough energy they break apart into charged components.

The experiments of J. J. Thompson (1856-1940)with cathode rays (beams of electrons) suggested that the atoms consisted of positive and negatively charged particles that are bound together by their mutual, electrostatic attraction.

As the cathode rays carry a charge of negative electricity, are deflected by an electrostatic force as if they were negatively electrified, and are acted on by a magnetic force in just the way in which this force would act on a negatively electrified body moving along the path of these rays, I can see no escape from the conclusion that they are charges of negative electricity carried by particles of matter.

—J. J. Thomson^[12]

As to the source of these particles, Thomson believed they emerged from the molecules of gas in the vicinity of the cathode.

If, in the very intense electric field in the neighbourhood of the cathode, the molecules of the gas are dissociated and are split up, not into the ordinary chemical atoms, but into these primordial atoms, which we shall for brevity call corpuscles; and if these corpuscles are charged with electricity and projected from the cathode by the electric field, they would behave exactly like the cathode rays.

—J. J. Thomson^[12]

Thompson developed an erroneous model of the atom based on his experiments. He pictured atoms as composite structures that consisted of electrons moving in a positively charged, background matrix to form a neutral atom. This model was called the "plum pudding model" of the atom.

Ernest Rutherford (1871-1937) conducted a series of experiments, in which alpha particles shot through gold foils were scattered in such a way that suggested that the atom was very unlike the Thompson model. Some alpha particles were scattered at very wide angles; indeed, they were almost back scattered. The effect was like firing a pistol into a pillow and observing that some of the bullets ricocheted all over the place. The logical conclusion would have been that the pillow contained some hard, heavy objects concealed among the feathers. The electrons were known to be about 2,000 times lighter than the particles that make up the nucleus of an atom: protons and neutrons. Electrons, in effect, would be the feathers in the pillow. The remaining parts of the atom must have been heavy, small and tight. On this basis, Rutherford proposed that atoms consisted of small, heavy nuclei, around which orbited the much lighter electrons.

Instability of any classical model of the atom

Electrons held in tight orbits about an atomic nucleus are constantly being accelerated. Maxwell's equations predict that any accelerated charge will radiate electromagnetic waves. Since the electric potential is infinitely deep as the orbit of a particle approaches zero radius, there is nothing in classical theory to stop a Rutherford-type atom form collapsing to nothing in a burst of radiation. Quantum theory was able to account for the stability of the atom as well as the nature of the spectral line components of its radiation.

The Bohr Atom

Niels Bohr (1885-1962) spent some time in England in 1911 where he met with J.J Thompson and Ernest Rutherford. After returning to Denmark, Bohr applied the emerging ideas of quantum mechanics to the Rutherford model of the atom, from which he was able to account to some extent for the character of atomic spectra and the stability of the atom.

Rydberg formula for hydrogen

The spectra of the various elements were very useful in chemical analysis, because the elemental components in a chemical compound could be identified from their emission spectra. Spectroscopists put together empirical formulas that described the various line spectral components of the element. The simplest series corresponded to the hydrogen emission spectra, in which the wavelengths of the various spectral lines could be written in terms of pairs of integers (n and m)

1/λ_mn = R (1/n²-1/m²)                                                                   (1a)

where

R ≈ 1.097 x 10⁷ m^-1                                                                         (1b)

is called the Rydberg constant. Bohr attempted and succeeded in predicting the Rydberg formula for hydrogen (1) based on ideas of quantum mechanics.

Bohr's reasoning ran somewhat as follows:

Consider a circular orbit of an electron about a proton, which forms a hydrogen atom. The only orbits allowed by the wave nature of the electron are those in which the wavelength of the electron wraps around the circumference of the orbit an integral number of times (n)

2 π r /λ = n                                                                                   (2a)

2 π r h/ λ = n h.                                                                            (2b)

Using the relation between the momentum and and wave length of an electron,

m_e v = h/ λ,                                                                                     (3)

and (2a), we can write

(2 π r ) m_e v = n h,                                                                         (2c)

or

m_e v r = n h/(2 π )                                                                          (2d)

L = n ħ,                                                                                          (2e)

where L is the orbital angular momentum of the electron,

L = m_e v r                                                                                         (2f)

and ħ is a modified version of Planck's constant,

ħ= h/(2 π ).                                                                                    (2g)

Let us write (2e) directly in terms of the mass and velocity of the electron

m_e v r= n ħ.                                                                                     (2h)

Note that here we have used m_e for the mass of the electron, which should not be confused with the integer m that describes an energy level.

Apply Newton's law of force and acceleration to the electron orbiting a proton under an electrostatic attraction

F = m_e dv/dt                                                                                    (4a)

- k e²/r² = - m_e v²/r,                                                                         (4b)

where k= 1/(4πε₀)                                                                         (4c)

or

m_e v² r = k e².                                                                                  (4d)

Additionally, we can solve (4b) for the kinetic energy of the electron in terms of the magnitude of its potential energy

½ m_e v² = ½ k e²/r                                                                           (4e)

which means that the magnitude of the kinetic energy is equal to half that of the potential energy .

The total energy of the electron is given by the sum of the kinetic energy (positive) and the potential energy (negative)

E = ½ m_e v²  - k e²/r                                                                          (5a)

or

E = ½ k e²/r  - k e²/r                                                                         (5b)

or

E = -½ k e²/r.                                                                                    (5c)

The negative sign of the energy indicates that the electron is in a bound state. The value of the energy necessary to bring the total energy of the electron to zero, which represents a free electron at rest, is called the binding energy. Above this threshold (E>0) the electron is in a free state and in motion.

Using (4d) and (2h) form the following equation

(m_e v² r)/(m_e v r)² = k e²/(n² ħ²)                                                     (6a)

, from which we solve for 1/r

1/(m_e r) = k e²/(n² ħ²)                                                                     (6b)

1/ r = k e² m_e/(n² ħ²).                                                                      (6c)

The value of the reciprocal of r substituted in (5c) gives the values of the bound states corresponding to the integer n,

E_n = -½ (k e²)²m_e/ħ² 1/n² .                                                               (6d)

Note that as n increases in (6d), the energy of that state of motion approaches zero from the negative side, so that

E_m - E_n > 0,  if m > n .

When an electron jumps from the orbit corresponding to m (a higher energy state) to that corresponding to n (a lower energy state), the resulting energy difference is just the amount of energy radiated away as a photon, because of the conservation of energy

E_m= E_n + h ν,                                                                                    (7a)

or

h ν_mn = E_m - E_n                                                                                   (7b)

h c/λ_mn = E_m - E_n                                                                                (7c)

1/λ_mn = (E_m - E_n)/(2π ħ c)                                                                (7d)

Use (5d) to rewrite this equation as

1/λ_mn = (k e²)²m_e/(4π c ħ³)(1/m² -1/n² ).                                           (8a)

R=k²e⁴ m_e /(4π c ħ³).                                                                          (8b)

A short calculation using python code (see listing below) gives a value of

R= 10973731.   m^-1,

which agrees well with the value given by Google's calculator that is obtained in searching for Rydberg Constant:

10 973 731.6 m^-1.

The transition to the Modern Quantum Theory

Although the Bohr model of the atom is somewhat successful, it looks contrived. Specifically, it pictures the electron in a well defined circular orbit, which is not allowed by its quantum nature as specified by the Heisenberg Uncertainty relations (see below). Further, it fails to account for the relative brightness of the various spectral lines, which manifests transition probabilities, the strongest representing common events and the weakest, rare events.

A simple criticism of the Bohr is that it is two dimensional, whereas atoms are three dimensional. Matter is constructed of entities that are fully three dimensional. Bohr atoms maybe could form a three dimensional stack like coins, but that would result in very odd bulk properties fro hydrogen. What is needed is a fully three dimensional quantum theory, which will be developed in the next blog.

Heisenberg Uncertainty Principle

The idea that the subatomic particles behaved like waves (in fact, that all matter has wave properties) implied that the precision, with which certain pairs of quantities could be measured  was inherently limited. Werner Heisenberg  proposed that the precision with which the energy of an event (ΔE) could be measured and the time, over which it was so determined (Δt), were subject to the following relation:

ΔE Δt ≥ h,                                                                              (9a)

where h is Palnck's constant, which in current measurements is given by

h=6.62606957(29)10^-34 m² kg s^-1.

Heisenberg's corresponding uncertainty relation for the precision in paired, simultaneous measurements of momentum and position is

Δp Δx ≥ h.                                                                              (9b)

The Heisenberg Uncertainties are related to the bandwidth theorem of communications theory

Δk Δx ≥ 1,                                                                               (10a)

where k is the magnitude of the wave vector, which is also called the wave number

k=2π/λ.                                                                                 (10b)

The corresponding expression in the frequency domain is

Δω Δt ≥ 1,                                                                               (10c)

where

ω = 2π ν.                                                                                (10d)

In quantum theory, the wave vector is related to the momentum of a particle as follows

p = ħ k.                                                                                  (10e)

Correspondingly

E = ħ ω.                                                                                  (10f)

The proof of (10e) and (10f) is left as an exercise for the reader.

Multiplying both sides of (10a) and (10c) by  ħ, we obtain the Heisenberg uncertainties to a greater tolerance

ΔE Δt ≥ ħ                                                                                  (11a)

and

Δp Δx ≥ ħ.                                                                                 (11b)

Actually, in the full quantum theory, the uncertainties on the RHS of equations (11a) and (11b) can be further narrowed to ħ/2 as a result of the pairs of variables x and p, and t and E having special operator properties, which are discussed in the next section.

Old text books on quantum mechanics demonstrate the Heisenberg Uncertainty principle by proposing that here is an unpredictable disturbance of position by measurements of a particle's momentum, and visa versa. Recent experiments cast doubt on this line of reasoning, although they do not do away with the  Heisenberg Uncertainty principle. What appears to be the case is that the Heisenberg Uncertainty principle is an intrinsic property of the quantum state relating to the limits of information, which are possible to know about it.

They do not commute

As physicists struggled with the concepts of quantum mechanics, Dirac cut to the core of the quantum problem. He noticed that the dynamical variables in reality were operators that operated on states that are complex vectors of indefinite dimensions, of both infinite and fineit dimensions. As operators, the dynamical variables do not act algebraically as numbers, which can be multiplied in any order; rather the order of their product is important. For example, consider the product of the x and p_x as operators, which satisfy a set of commutation relations

[x, p_x] = i ħ,                                                                                   (12a)a

where the commutator is defined as

[x, p_x] ≡ x p_x -p_x x.                                                                           (13)

The operators act on any state vector to the right of them and none to the left.  The product of more than one operator with a vector means that there is a succession of operations to be performed.

Three commutation relations exist for each pair of position coordinates and corresponding momenta, and for that of energy and time:

[y, p_y] = i ħ,                                                                                     (12b)

[z, p_z] = i ħ,                                                                                     (12c)

[H, t] = -i ħ.                                                                                     (12c)

The last equation (12c) represents the commutation rule between the energy operator (H) and time. This the energy operator, which is called the Hamiltonian of the system wahen the energy is written as a function of all the dynamical variables and corresponding momenta.

The Heisenberg uncertainty relations can be derived from the commutation relations (12) and applying rules for computing expectation values of the dynamical variables, which will not be done here.

Incidentally, P. A. M. Dirac was a very intelligent physicist with a powerful mind, a very reserved behavior and a man of very few words. " His colleagues in Cambridge jokingly defined a unit of a dirac, which was one word per hour." The young Dirac resembled the character of Sheldon in TV program, the Big Bang Theory,except that Sheldon is perhaps too verbose.

Graham Farmelo has written a book worth reading about Dirac,which is called: The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom.

Python Code

ipython --pylab
from pylab import *
eps = 8.85418782e-12   # m-3 kg-1 s4 A2e
qe= 1.60217646e-19

me=9.10938215e-31 #(45) kg
c= 299792458. # m/s
h=6.626068e-34 #m^2 kg / s
k = 1/(4*pi*eps)
hbar=h/(2 *pi)
R=(k**2) * (qe**4) * me/(4*pi *c *hbar**3)

print R

# R= 10973731                m^-1