Microscopic World & Heisenberg Uncertainty Principle

Lecture XI : Feb 26, 2018 Summary of Last Class: Microscopic world & Heisenberg Uncertainty Principle (I) Stability of an atom If we magically turned off the uncertainty principle, atoms would vanish in a flash. (II) Zero Point Energy • Empty space can never be completely empty. Particles can spontaneously pop into existence and then disappear. Imagine a proton and antiproton spontaneously created from the vacuum with kinetic energy 1.0 MeV each. How long can they live ??. ~ −25 Using ∆E∆t = 2 , with ∆E= 1Mev, we find ∆t = 1:8 × 10 sec. • Particle in a box can never be at rest as the uncertainty in its position is the size of the box and that implies that its momentum cannot be zero. • Motion of electrons in an atom is an example of zero point energy. • Tunneling: Unlike classical physics, particle can end up on the other side of the barrier even if it does not have enough energy. Tunneling can be understood from Heisenberg uncertainty principle, where the width of the barrier survey as the uncertainty in position, that leads to uncertainty in momentum and hence energy. Uncertainty principle & macroscopic world Consider, for instance, the implications of the uncertainty principle for a baseball. Conceivably, the baseball could fly off unpredictably due to its intrinsically uncertain momentum. The more precisely we can locate the baseball in space, the larger is its intrinsic momentum. So, let’s consider a pitcher who is so sensitive that he can tell if the baseball is out of position by, for instance, the thickness of a human hair, typically 0.1 mm . According to the uncertainty principle, the baseball’s intrinsic speed due to quantum effects is about 10−29 m/s. This is unbelievably slow. For instance, the time for the baseball to move quantum mechanically merely by the diameter of an atom would be roughly 20 times the age of the universe. Obviously, whatever might give a 1 pitcher a bad day, it will not be the uncertainty principle. SPIN of an Electron MUST READ: <http://www.lorentz.leidenuniv.nl/history/spin/ goudsmit.html> It was found that electron is like a little “magnet”. The strength of its magnetic property is ~ characterized by “spin”, which we denote as S. For electrons, S = 2 . The spin of the proton, neutron is also the same. Since it is multiplied by ~, it is purely a quantum mechanical quantity. That is, classically, there is no magnetic moment associated with electron, proton... HISTORY: Wolfgang Pauli in 1924 was the first to propose a doubling of electron states due to a two-valued non-classical “hidden rotation”. In 1925, George Uhlenbeck and Samuel Goudsmit at Leiden University suggested the simple physical interpretation of a particle spinning around its own axis, in the spirit of the old quantum theory of Bohr and Sommerfeld. Ralph Kronig anticipated the Uhlenbeck-Goudsmit model in discussion with Hendrik Kramers several months earlier in Copenhagen, but did not publish it because when they discussed with Pauli, he did not support. The ( unsatisfactory) mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it. In other words, relativistic theory implies are particles like electrons have to have spin, even though electrons may not be moving close to speed of light . BOSONS and FERMIONS All particles of nature have spin that is either: 1 3 5 (a) Half-integer multiples of ~ , that is, 2 ~; 2 ~; 2 ~... Such particles are called Fermions. 2 (b) or Integers, like 0; ~; 2~; 3~::::, such particles are called Bosons. Examples are photons ( spin 1), π-mesons (spin zero), gravitons ( spin -2 ). 1 1 (c) There are NO particles in nature that have spin 3 ~, 8 ~ etc... Pauli Exclusion Principle and Atoms with more than one electron Each quantum state ( specified by the quantum number such as n ) of an atom can accommodate only two electrons. Known as the Pauli exclusion principle, it explains the buildup of elements, the Periodic Table. What is ~ ??? • ~ determines the minimum uncertainty in classical observables... • Spin of all elementary particles is either an integral or half integral multiple of ~. I. IMPORTANT REMARKS • Pauli Exclusion Principle... Pauli’s New Force 3 Pauli’s exclusion principle that two electrons cannot be in the state state ultimately prohibits two electrons too close together. The “effective force” generated by Pauli exclusion principle is called the “exchange force”.Note it is not a new kind of force – it is not really a force – (there are only four types of fundamental forces in nature). It is an EFFECTIVE force that is, it can be viewed as a kind of force. • NOTE: Pauli exclusion principle applies only to Identical Fermions– • When we swap two electrons, there is NO way to detect the swap – experimentally or even in principle. • Pauli exclusion principle explains why we cannot walk through a solid wall, even though it is 99 percent empty. The electrons of our hand cannot penetrate the “wall” atoms because they are ruled by Pauli exclusion principle. • We understand why matter is made up of only fermions. If bosons were the building blocks of matter, it will be unstable. • Yes, quantum physics is the key to understand all physical phenomena ( almost all... ), only in principle. The devil is in the details.... Even though we know all the forces, we cannot always solve Schrodinger¨ equation.... Shocking things – How Can it Possibly Be so Weird ????? READING: Chapter 7 from Quantum Physics for Poets. Home Work Assignment II ( Due on March 19, 2018) Write about two pages a quantum phenomenon that you find most weird and puzzling. State clearly your reasons ... 4.

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