Quantum Mechanics and the Microscopic World: Understanding Atoms and Quanta, Lecture notes of Classical Physics

Download Quantum Mechanics and the Microscopic World: Understanding Atoms and Quanta and more Lecture notes Classical Physics in PDF only on Docsity! PHYSICS –PHY101 VU © Copyright Virtual University of Pakistan 147 The Microscopic World •ATOMS--10-10 m •NUCLEI--10-14 m •NUCLEONS--10-15 m •QUARKS--??? Summary of Lecture 42 – QUANTUM MECHANICS 1. The word "quantum" means packet or bundle. We have already encountered the quantum of light - called photon - in an earlier lecture. Quanta (plural of quantum) are discrete steps. Walking up a flight of stairs, you can increase your height one step at a time and not, for example, by 0.371 steps. In other words your height above ground (and potential energy can take discrete values only). 2. Quantum Mechanics is the true physics of the microscopic world. To get an idea of the sizes in that world, let us start from the atom which is normally considered to be a very small object. But, as you can see, the atomic nucleus is 100,000 times smaller than the atom. The neutron and proton are yet another 10 times smaller than the nucleus. We know that nuclei are made of quarks, but as yet we do know if the quarks have a size or if they are just point-like particles. 3. It is impossible to cover quantum mechanics in a few lectures, much less in this single lecture. But here are some main ideas: a) Classical (Newtonian) Mechanics is extremely good for dealing with large objects (a grain of salt is to be considered large). But on the atomic level, it fails.The reason for failure is the uncertainty principle - the position and momenta of a particle cannot be determined simultaneously (this is just one example; the uncertainty principle is actually more general). Quantum Mechanics properly describes the microscopic - as well as macroscopic - world and has always been found to hold if applied correctly. b) Atoms or molecules can only exist in certain energy states. These are also called "allowed levels" or quantum states. Each state is described by certain "quantum numbers" that give information about that state's energy, momentum, etc. c) Atoms or molecules emit or absorb energy when they change their energy state. The amount of energy released or absorbed equals the difference of energies between the two quantum states. d) Quantum Mechanics always deals with probabilities. So, for example, in considering the outcome of two particles colliding with each other, we calculate probabilities to scatter in a certain direction, etc. docsity.com PHYSICS –PHY101 VU © Copyright Virtual University of Pakistan 148 ( )u ν ν 4. What brought about the Quantum Revolution? By the end of the 19th century a number of serious discrepancies had been found between experimental results and classical theory. The most serious ones were: A) The blackbody radiation law B) The photo-electric effect C) The stability of the atom, and puzzles of atomic spectra In the following, we shall briefly consider these discrepancies and the manner in which quantum mechanics resolved them. A)Classical physics gives the wrong behaviour for radiation emitted from a hot body. Although in this lecture it is not possible to do the classical calculation, it is not difficult to sh 3 ow that the electromagnetic energy ( ) radiated at frequency increases as (see graph). So the energy radiated over all frequencies is infinite. This is clearly wrong. The correct calcu u ν ν ν lation was done by Max Planck. Planck's result is shown above, and it leads to the sensible result that u( ) goes to zero at large . He assumed that radiation of a given frequency could only be emitted and absorbed in ν ν ν quanta of energy . If the electromagnetic field is thought of as harmonic oscillators, Planck assumed that the total energy of this large number of oscillators is made of finite energy elem hε ν= ents . With this assumption, he came up with a formula that fitted well with the data. But he called his theory "an act of desperation" because he did not understand the deeper reasons. B hν ) I have already discussed the photoelectric effect in the previous lecture. Briefly, Einstein (1905) postulated a quantum of light called photon, which had particle properties like energy and momentum. The photon is responsible for knocking electrons out of the metal - but only if it has enough energy. C) Classical physics cannot explain the fact that atoms are stable. An accelerating charge always radiates energy if classical electromagnetism is correct. So why does the hydrogen atom not collapse? In 1921 Niels Bohr, a great Danish physicist, made the following hypothesis: if an electron moves around a nucleus so that its angular momentum is ,2 ,3 , then it will no⋅ ⋅ ⋅ t radiate energy. In the next lecture we explore the consequences of this hypothesis. Bohr's hypothesis called for the quantization docsity.com PHYSICS –PHY101 VU © Copyright Virtual University of Pakistan 151 N S ↑ ↓ 11. The simplest system for discussing quantum mechanics is one that has only two states. Let us call these two states "up" and "down" states, and . They could denote an electron with s ↑ ↓ pin up/down, or a switch which is up/down, or an atom which can be only in one of two energy states, etc. Things like were called kets by their inventor, Paul Dirac. For definiteness ↑ let us take the electron example. If the state of the electron is known to 2 1 be , then the probability of finding the electron with spin up is 3 3 2 P( ) , and with spin down is P( ) 3 Ψ = ↑ + ↓ ↑ = ↓ = 1 2 2 2 1 2 1 . More generally: c denotes 3 an electron with P( ) c and P( ) . This means that if we look at look at a large number of electrons all of which are in state , then the nu c c N Ψ = ↑ + ↓ ↑ = ↓ = Ψ 2 2 1 2 1 2 1 2 mber with spin up is and with spin down is . We sometimes call c a , and c and . N c N c c quantum state c quantum amplitudes Ψ = ↑ + ↓ 12. The Stern-Gerlach experiment illustrated here shows an electron beam entering a magnetic field. The electrons can be pointing either up or down relative to any chosen axis. The field forces them to choose one of the two states. That the beam splits into only two parts shows that the electron has only two states. Other particle beams might split into 3,4, ⋅ ⋅ ⋅ 1 2 1 2 1 2 13. If and are the amplitudes of the two possibilities for a particular event to occur, then the amplitude for the total event is . Here and are complex numbers in general. a a A a a a a= + 2 2 1 2 1 2 2 * * * 1 2 1 1 2 2 1 2 But the probability for the event to occur is given by . In daily experience we add probabilities, but in quantum mechanics we add amplitudes: P A a a P P P P a a a a a a a a = = + = + = + = + + * * *2 1 1 2 1 2 2 1 * * 1 2 2 1 . The cross terms are called interference terms. They are familiar to us from the lecture on light where we add amplitudes first, and then square the sum t a a P P a a a a a a a a + = + + + + 2 2 1 2 o find the intensity. Of course, if we add all possible outcomes then we will get 1. So, for example, in the electron case ( ) ( ) 1. Note that amplitudes can be complex but probP P c c↑ + ↓ = + = abilities are always real. docsity.com PHYSICS –PHY101 VU © Copyright Virtual University of Pakistan 152 14. Let us return to the double slit experiment discussed earlier. Here the amplitude for an electron wave coming from one slit interferes with the amplitude for an electron wave coming from the other slit. This is what causes a pattern to emerge in which electron are completely absent in certain places (destructive interference) and are present in large numbers where there is constructive interference. So what we must deal with are matter waves. But how to treat this mathematically? 15. The above brings us to the concept of a "wave function". In 1926 Schrödinger proposed a quantity that would describe electron waves (or, more generally, matter waves). The wavefunctio• 2 n ( , ) of a particle is the amplitude to be at position at time . The probability of finding the particle at position between and (at time ) is ( , ) . Since the parti x t x t x x x dx t x t dx Ψ • + Ψ + 2 - cle has to be somewhere, if we add up all possibilities then we must get one, i.e. ( , ) 1. ( , ) is determined by solving the "Schrodinger equation" which, unfortunately, I x t dx x t ∞ ∞ Ψ = • Ψ ∫ shall not be able to discuss here. This is one of the most important equations of physics. If it is solved for the atom then it tells you all that is possible to know: energies, the probability of finding an electron here or there, the momenta with which they move, etc. Of course, one usually cannot solve this equation in complicated situations (like a large molecule, for example) and this is what makes the subject both difficult and interesting. 16. For an electron moving around a nucleus, one can easily solve the Schrodinger equation an 2 d thus find the wavefunction ( , ). From this we compute ( , ) , which is large where the electron is more likely to be found. In this picture, the probability of finding the el x t x t Ψ Ψ ectron inside the first circle is 32%, between the second and first is 44%, etc. docsity.com