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The modern atom

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The modern atom

Chapter 23

Successes:

- combining successfully Rutherford’s “solar system” model, with the Planck hypothesis on the quantified energy states at atomic level + Einstein’s photons

- explaining the atomic emission and absorption spectra

- explaining the general features of the periodic table

- a first “working” model for the atom

Failures

- a mixture of classical and quantum ideas (electrons move classically on orbits, but their possible energy states are quantified)

- postulates that on the allowed orbits electrons do not radiate

(conflict with Maxwell’s theory)

- could not account for the maximal electron numbers on one shell

- could not explain splitting of the spectral lines in magnetic fields

- it is a non-relativistic theory although the speed of the electrons is close to c

- Electromagnetic waves (photons) have both particle and wave properties
- D’Broglie proposed that all particles in the Universe could present this dualism:
- each entity in the Universe exhibits both particle and wave properties

- Wavelength associated to the particles:
(h: Planck’s constant, m: mass, v: speed)

- electrons in atoms have also wave properties
- electrons form standing waves
in atoms --> explains Bohr’s

quantum hypothesis:

- wave nature of the electron proved experimentally (Davisson & Gremer)

- Experimental results if electrons were normal particles
- Experimental results if electrons
behave like waves

- Conclusions: Electrons
interfere like photons

wave nature of particles

important when mv is

small (otherwise is too

small to produce detectable effects)

- Waves associated to particles--> matter-waves
- matter-waves characterized by their amplitude, called wave-function:
- Intensity of the wave ~ 2
- What is oscillating? Intensity of what varies?
- Intensity of matter waves represents the likelihood or probability of finding an electron at that location and time
- new view of physics --> quantum mechanics
- new rules and equations which describe the dynamics of the matter waves
- fundamental equation: Schrodinger’s wave equation --> gives
- Schrodinger’s equation a very complicated second-order differential equation, can be solved analytically only in some simple case (Solving Schrodinger’s equation = finding
- knowledge of provides all possible information about atomic particles

- Behavior of a classical particle closed in a box
- Behavior of a quantum particle closed in a box
- I. One dimensional box
- properties found by solving Schrodinger’s equation

- possible : standing waves--> quantized wavelength

- quantized values of the possible energies

- one quantum number characterizing the states

- II. Three dimensional (real) box
- three dimensional standing waves for

- quantized energy values

- three quantum numbers distinguish the states

- Schrodinger’s equation is analytically solvable for the H atom
- Schrodinger’s equation cannot be solved analytically for atoms with more than one electron
- Very complicated standing waves, hard to visalize
- no classical electron orbits, replaced by probability clouds...
- discrete energy spectra, characterized by three quantum numbers
- n: associated with the electronsenergy (n=1,2,3,4…..)

- l : associated the angular momentum’s magnitude (l=0,1,2…n-1)

- ml: characterizing the direction of the angular momenta (it’s projection on a given direction) (ml=-l, -l+1, -l+2, ….l-1,l)

- an additional quantum number: ms not explained by the classical theory of Schrodinger
- ms characterizes the spin of the electron
- two possible orientations: up and down (dependent on the spinning direction)

- ms = +1/2 or -1/2

- four quantum number label the possible states of the electrons in atoms

- Exclusion principle: No two electrons can have the same state (no two electrons can have the same set of quantum numbers) (Wolfgang Pauli, 1924)
- atomic shells: a group of allowed states, where the energy is very close
- chemical properties determined by the number of electrons in the outermost incomplete shell
- first shell: possible states: n=1, l=0, ml=0, ms=+1/2 or -1/2 (total: 2)
- second shell: n=2, l=0, ml=0, ms=+1/2 or -1/2
n=2, l=1, ml=1,0,-1, ms=+1/2 or -1/2 (total: 8)

- third shell: n=3, l=0, ml=0, ms=+1/2 or -1/2
n=3, l=1, ml=1,0,-1, ms=+1/2 or -1/2 (total: 8)

- fourth shell: n=4, l=0, ml=0, ms=+1/2 or -1/2
n=3, l=2, ml=2,1,0,-1,-2, ms=+1/2 or -1/2

n=4, l=1, ml=1,0,-1, ms=+1/2 or -1/2 (total: 18)

- ……..
- filling of the states with electrons, after increasing energy of the states (electrons always occupy the most lower allowed energy state)

- Schrodinger equation --> the atomic particles motion are governed by probability laws (very hardly accepted by the physics community)
- Werner Heisenberg another description and interpretation of the quantum world (more mathematical description with matrix algebra)
- Both description leads to the UNCERTAINTY PRINCIPLE
There are some quantities which cannot be determined together exactly (independently how good and precise apparatus we use), better we know one of them, worst is our knowledge about the other one

Examples: I. momentum and position in the direction of one axis:

(better we know the momentum, of a particle less

we know about it’s energy)

II. Energy and time of the measurements

( longer the time for the measurement, smaller is the

uncertainty in the measured energy value)

- Complimentary principle: all entities in our physical word exhibit both wave and particle properties, these two aspects are complimentary. Better we know one aspect of the particle, worst is our knowledge about the other.
Example: If we know the position (particle aspect), we have no idea about the momentum which determines the wavelength (wave aspect)

- classical Newtonian determinism (mechanistic view) does not hold!
- strong argument against predetermined Universe

- In quantum mechanics the future becomes a statistical (or probabilistic) issue
- We have only probabilities for predicting each of the possible futures (nature will select among them, with probabilities revealed by the theory of quantum mechanics)

- Summary:
- Bohr’s model was not complete, since it combined classical and quantum concepts in a non-rigorous manner

- De’ Broglie’s revolutionary hypothesis: waves associated to all particles (wavelength inversely proportional to its momentum) matter-waves

- Atomic particles confined to finite spaces form standing-wave patterns that quantize their physical properties

- Quantum particles are described by the wave-function

- Square of the wave-function: probabilistic interpretation

- Wave-function determined by the Schrodinger equation

- There are four quantum numbers associated with the allowed states of the electrons in an atoms

- The electrons fill the available states, respecting Pauli’s exclusion principle and the minimal energy principle

- There is an indeterminacy of knowledge resulting from the wave-particle duality (Heisenberg’s uncertainty principle)

- New philosophical view on determinism: --> probabilistic determinism