Intermediate Quantum Mechanics: Lecture 1 - Classical Physics vs. Quantum Mechanics, Lecture notes of Quantum Physics

Download Intermediate Quantum Mechanics: Lecture 1 - Classical Physics vs. Quantum Mechanics and more Lecture notes Quantum Physics in PDF only on Docsity! Intermediate Quantum Mechanics Notes for Lecture 1 (1/21/15) Structure of Physics, Classical Physics and Quantum Mechanics vs. Classical Physics The Structure of Physics • The two pillars of physics are the Special Theory of Relativity (SR) and Quantum Mechanics (QM). All the rest of physics is supported by these two pillars. Any experimental disagreement with with either SR or QM would be the collapse of all of physics. This might happen but is extremely unlikely. Any experiment showing disagreement with either theory should be considered highly suspect. These theories are very well established and no one is doing serious research to test them. It would almost certainly be a waste of time. The active areas of research in physics are in those areas supported by these two pillars: atomic physics, nuclear physics, solid state physics, astrophysics, elementary particle physics, cosmology, etc. In these areas we don’t understand everything and the forefront of knowledge is advancing. • Classical physics (essentially Newtonian Mechanics) is a very good approxima- tion at the scale of our everyday lives. It is very intuitive to us and satisfies our “common sense”. That’s because a working knowledge of classical physics has great survival value and so our brains evolved to “understand” classical physics. Quantum Mechanics and Special Relativity don’t make common sense. The regimes where their effects are manifest are not encountered in our daily lives and therefore there is no survival value to having a working knowledge of them. (At least not until modern times, of course). The only things that matter in determining whether a theory is correct are: − Do it agree with all experiments (measurements)? − Is it mathematically consistent? Whether or not it makes common sense to us is irrelevant. In fact, relativity and quantum mechanics almost always don’t. A serious mistake that students new to these theories make is to think that their calculation is wrong because it doesn’t “make sense”. • Special Relativity was put forward by Albert Einstein in a set of two papers in 1905. It is important in the regime where velocities are close to the speed of light. v ∼ c = 3.0× 108 m/s At that speed you could go around the world about 7.5 times in one second. • Quantum Mechanics was developed slowly in fits and starts over a period of about thirty years from 1900 to 1929. It was the work of many great physicists: Einstein, Planck, Bohr, Heisenberg, Schrodinger, Pauli, Born, Dirac, etc. It is important in the regime where distance is comparable to Planck’s constant over the momentum. x ∼ h p Planck’s constant, h, is the fundamental constant of QM. We more often use h̄ = h/2π. As we will see, the energy of particle is related to its frequency. We can write this as either E = hν or E = h̄ω. They are of course equivalent. We will often have expressions with oscillatory functions. It’s much more convenient to write, for example, cos(ωt) than cos(2πνt). That’s really the only reason for choosing h̄ over h. • The value of h̄ is 1.05×10−34 J·s. It’s good to see this number to appreciate how small it is when using macroscopic (SI) units. But, when we deal with quantum mechanics we are studying particle with very small masses and energies. It make no sense to calculate in term of Joules. Almost everybody can remember the value of c. The way to remember the value of h̄ is by the product of h̄ and c. h̄c = 200 eV·nm = 200 MeV·fm [The more precise number is 197 but 200 is good enough. An electron volt (eV) is the kinetic energy gained by an electron when it falls through a voltage difference of 1 V (1 eV = 1.6 × 10−19J). One Fermi (fm) = 10−15 m is about the size of the proton.] • The program of physics is very simple to state and is the same for both classical physics and QM. It involves the following. − Select the closed system that you are studying. This could be a single particle or several particles with various types of interactions. − Determine what are the possible states of the system and how they are specified. − Determine how the system evolves (moves from state to state) in time. This is the dynamics. Classical Physics • In classical physics, the dynamics is given by Newton’s Law: −dU(x) dx = F (x) = ma If we know the position of the particle, then we know the force acting on it and therefore the rate of change of the velocity. dv = a dt = F m dt This is not enough to know how the particle will evolve. We have to be given the velocity as well as the position at some particular time. The state of the system, in this case a single particle in one-dimension, is given by specifying its position and velocity. Then we have: dx = v dt dv = F m dt