1. School of Mathematical and Physical Sciences PHYS1220 22 August, 20021 PHYS1220 – Quantum Mechanics Lecture 3 August 22, 2002 Dr J. Quinton Office: PG 9 ph 49-21-7025 phjsq@alinga.newcastle.edu.au

2. School of Mathematical and Physical Sciences PHYS1220 22 August, 20022 Early Models of the Atom Between 450 - 410 BC, Leucippus of Miletus (now Turkey) and Democritus of Abdera (now Thrace, Greece) postulated that matter was made of fundamental, indivisible units called atoms This fact was widely accepted by 1900 The periodic table was well underway at the time Discovery of radioactivity in mid 1890s posed a problem in that if particles smaller than the atom existed, perhaps they were not indivisible at all J.J. Thomson proposed the ‘plum-pudding’ model of the atom comprised a uniform positively charged sphere, with negatively charged electrons After J.J. discovered the electron in 1897 (Nobel Prize 1906), he amended his model to include the idea that the electrons should be moving

3. School of Mathematical and Physical Sciences PHYS1220 22 August, 20023 Rutherford’s Experiments "All science is either physics or stamp collecting” Rutherford was the first to experiment with the radiation from radioactivity named a, b and g particles He found that a and b particles were charged Won the 1908 (Chemistry!) Nobel Prize Rutherford (~1911) experiments with a particles and gold foil The result Majority of the alpha particles went straight through Occasional scattering (even backward) ”as if you fired a 15 inch shell at a sheet of tissue paper and it came back and hit you.”

4. School of Mathematical and Physical Sciences PHYS1220 22 August, 20024 Rutherford’s “Planetary” Model of the Atom Since most alpha particles were undeflected, the atom must be mostly empty space The only way back scattering could occur is if there is a concentrated positive charge at the nucleus The nucleus must therefore contain more than 0.999 of the atom’s mass Theorized a new model of the atom A tiny but massive, positively charged nucleus that is surrounded by electrons that orbit some distance away. The electrons must be moving, otherwise they would fall into the nucleus due to Coulomb interaction Measured closest distance between alpha particle and nucleus from its KE which is converted to electric PE Nuclear radius must be smaller than this

5. School of Mathematical and Physical Sciences PHYS1220 22 August, 20025 Atomic Spectra Heated solids emit a continuous spectrum of EM radiation rarefied gases can also be excited to emit photons intense heating or dielectric breakdown (low pressure gases) emitted spectra are comprised of discrete lines emitted wavelengths are unique to each element Balmer (1885) studied the visible lines of hydrogen emission, showed that they are related by where R = 1.0974 x 10 7 m -1 (determined experimentally) is called the Rydberg constant and n=3,4,5,6 for visible lines at 656, 486, 434 and 410 nm respectively Later found to extend into UV region ending at 365nm (the lines get too close together to distinguish). This corresponds with the n →  case Balmer Series

6. School of Mathematical and Physical Sciences PHYS1220 22 August, 20026 Atomic Spectra II Later experiments with hydrogen by others showed similar emission line series existed in the UV and IR regions, with patterns similar to the Balmer series. The formula also fitted, but ½ 2 needed to be replaced by 1/1 2, 1/3 2 etc to suit the particular series Lyman Series (UV) Paschen Series (IR)

7. School of Mathematical and Physical Sciences PHYS1220 22 August, 20027 Failings of Rutherford’s Model Rutherford’s model of the atom was superior to Thomson’s plum pudding model, but it did have a few shortcomings Unable to explain why atoms emit discrete line spectra Orbiting electrons could have any energy based upon their orbital motion and distance from the nucleus Orbiting charges accelerate (centripetally). They should emit radiation like any other accelerating charge, lose energy and spiral into the nucleus As they spiral inward, their orbital frequency increases, thus the frequency of photons they emit should also increase. Therefore, they should lose even more energy and decay even more rapidly Basically, there are two major flaws with Rutherford’s atomic model Continuous emission spectra should be emitted It predicts that atoms are unstable  

8. School of Mathematical and Physical Sciences PHYS1220 22 August, 20028 Bohr Model of the Atom Neils Bohr thought that Rutherford’s model had merit, but needed to include some of the newly developing quantum theory to make it work (Bohr studied in Rutherford’s lab in 1912) Planck and Einstein had shown that the energy of oscillating charges must change in discrete amounts. Einstein argued that in changing energy states, a photon would be emitted with energy equal to that change Bohr (1913) argued that perhaps electrons in the atom may also behave in this way. Electrons don’t radiate with just any energy, but rather must do so in a quantised fashion. He developed the following theory of the atom from these ideas

9. School of Mathematical and Physical Sciences PHYS1220 22 August, 20029 Electrons are only allowed to exist in specific circular orbits (called stationary states) with definite energies and they do so without emitting radiation. Photon emission only occurs when an electron jumps from one stationary state to another of lower energy If the electron exists in stationary states with fixed orbits, then the angular momentum must be quantised where n is an integer and r n is the radius of the n th possible orbit It’s important to note that this postulate was made by Bohr to make the model work. He found that this condition was needed but he did not understand why at the time Bohr Model of the Atom II Bohr’s Quantum Condition

10. School of Mathematical and Physical Sciences PHYS1220 22 August, 200210 Bohr Model of the Atom II The allowed orbits are numbered 1,2,3,… etc according to the value of n, which is called the (principle) quantum number Consider an electron in an orbit of radius r n Coulombic force between electron and nucleus = centripetal force The smallest orbit (n=1) for hydrogen (Z=1) is called the Bohr Radius Radius of the n th orbit

11. School of Mathematical and Physical Sciences PHYS1220 22 August, 200211 The total energy is sum of kinetic and potential energies Substituting Bohr’s quantum condition for v and the expression for r n For hydrogen (Z=1) So energy is quantised Lowest energy level is E 1, called the ground state E 2 – first excited state E 3 – second excited state Bohr Model of the Atom III E (eV) 0 -13.6 -3.40 -1.51 n=1 n=2 n=3 n= 

12. School of Mathematical and Physical Sciences PHYS1220 22 August, 200212 Bohr Model of the Atom IV Although energies are negative, the orbit closest to the nucleus, corresponding with n=1, has the lowest energy For two charges, the electric PE is only zero as their separation approaches infinity, so an electron with zero KE must have n   to become free of the atom An atom’s Binding energy (or ionisation energy) is the energy required to remove a ground state electron measured for hydrogen to be 13.6eV. Removing an electron from the lowest state n=1 (E=-13.6eV) to E=0 (n   ) requires 13.6 eV of energy and thus corresponds completely with. Finally, for a relaxation transition from state n to state n’, the emitted photon will have a wavelength given by

13. School of Mathematical and Physical Sciences PHYS1220 22 August, 200213 Bohr’s model accurately predicts the emission spectrum of hydrogen n’=1 – Lyman series n’=2 - Balmer series n’=3 - Paschen series Bohr won the Physics Nobel Prize in 1922 Bohr Model of the Atom IV

14. School of Mathematical and Physical Sciences PHYS1220 22 August, 200214 The Correspondence Principle Any theory must match the well-known laws of classical physics if the conditions match the classical case. This is known as The Correspondence Principle Recall from special relativity that when v<<c, the theory must simplify to Newtonian physics eg relativistic kinetic energy becomes if you take the expansion and make v<<c In Quantum Mechanics, the same idea applies in going from microscopic to macroscopic situations When a quantum number approaches , the system being described should behave in a way that is consistent with classical physics Eg in the Bohr model, the discrete energy levels E n get closer and closer together and as n → , they essentially become ‘continuous’ The same applies for r n and L as n → . The exercise is left to the student

15. School of Mathematical and Physical Sciences PHYS1220 22 August, 200215 Applying de Broglie’s Hypothesis to Atoms Bohr did not explain why orbits were quantised in his model de Broglie applied his hypothesis by considering the wave nature of electrons and applied it to a circular orbit Standing wave modes sustained on a string have nodes at the ends de Broglie argued that the electron was a circular standing wave that must close on itself, otherwise it would destructively interfere with itself and die out very quickly

16. School of Mathematical and Physical Sciences PHYS1220 22 August, 200216 Wave nature applied to orbits In order for a standing wave to be sustained, there must be an integral number of wavelengths around the circle’s circumference which is Bohr’s Quantum Condition! The moral of the story? the wave nature of electrons is inescapable It is integral to the nature of electrons and electron states in the atom! This approach began what is now Quantum Mechanics