Principles of Quantum Mechanics What is Quantum Mechanics?

1. Principles of Quantum Mechanics What is Quantum Mechanics? QM is the theory of the behavior of very small objects (e.g. molecules, atoms, nuclei, elementary particles, quantum fields, etc.) Why Quantum Mechanics? One of the essential differences between classical and quantum mechanics is that physical variables that can take on continuous values in classical mechanics (e.g. energy, angular momentum) can only take on discrete (or quantized) values in quantum mechanics (e.g. the energy levels of electrons in atoms, or the spins of elementary particles, etc).

2. QM is weird! QM deals with concepts that are counter intuitive, and impossible to visualize in classical mechanical terms. Some concepts defy common sense, e.g. - a) superposition (of states, quantum systems can be in more than one discrete state at a time) b) non-locality (spooky action at a distance) c) non determinism (QM is essentially stochastic) d) non reality (some “interpreters of QM” claim that QM implies that there is no independent reality) e) uncertainty (one cannot have full simultaneous knowledge of certain physical quantities, e.g. the position and momentum of an elementary particle) f) wave-particle duality (waves behave like particles, & vice versa!)

3. Bullets hits Machine Gun Double Slit Screen The next few slides deal with experimental phenomena which historically motivated the introduction of basic quantum concepts. The most famous one is arguably the “double slit experiment” which illustrates several of the weirdo phenomena mentioned earlier. Here it is. Absorbing Screen

4. Waves intensity Wave Generator Double Slit Screen So what kind of pattern would you expect if you shot electrons through the double slit? Would it be bullet-like or wave-like? Its wave-like! But how? Electrons are particles. They leave a flash on the absorbing screen at localized points. How do they interfere? Which slit do they go through? Mysterious!

5. What happens if the electron emission rate is so low, that only one electron goes thru the slit at once? Will there be a wave-like pattern? Yes. Somehow the single electrons are interfering with themselves! What! How? Does a single electron go through both slits (to cause the interference)? How can an electron that causes a localized flash on the absorbing screen go through both slits? Can an electron be in two places at once? Counter common sense. As Feynman said “No one understands quantum mechanics!” Lets try to determine which slit the single electron goes thru, by shining light on it and detecting the deflected photon. The light has to have a very high frequency, to have a very small wavelength. The smaller the wavelength, the bigger the kick of the photon on the electron. This kick causes the wave-like pattern to disappear and become a bullet-like pattern! We know which slit the electron went thru but we lost the interference (the wave-like) pattern!

6. So we cant have both the knowledge of which slit the electron went thru AND have the interference pattern, so how can we ever know which slit it went thru when there IS a wave-like pattern? Maybe we can NEVER know? So, QM postulates that there two wavelike phenomena (one from each slit) that interfere at the absorbing screen and that these two “waves” influence where the electron will hit the screen. These “waves” were thought at first to be real physical waves (de Broglie, Schrodinger, (later)Bohm, etc), but most QMechanics interpret these waves as “probability waves” or “probability amplitudes”, whose squares gives the probability distributions (a la statistics) of finding the electron in a given region of space, etc. The mathematical formulae of QM is generally accepted (they work!), NOT the interpretation of what the mathematical formulae mean. 100 years of controversy - still!

7. Further Quantum Phenomena e.g. a) Einstein’s Explanation of the Photo-Electric Effect b) de Broglie’s idea that particles have wave properties c) Heisenberg’s Uncertainty Principle d) The Stern-Gerlach Experiment e) Schrodinger’s wave equation Newton (17th century) thought that light consisted of particles. Then Young’s double slit experiments with light (19th century, i.e. interference fringes) implied that light was a wave phenomenon! Then Einstein came along (20th century) and said that light consisted of “photons” i.e. localized discrete bundles of light energy (i.e. particles again!)

8. The Photo-Electric Effect. Experiments around the turn of the 19th/20th centuries showed some weird phenomena going on with the so-called “photo-electric effect”. This effect could NOT be explained with classical physics. It is a primary example of quantum physics whose explanation earned Einstein the Nobel Prize. The effect is that shining light on certain metal surfaces, caused electrons to be emitted. From classical physics one would argue that increasing the intensity of the light, would mean the energy of the emitted electrons would increase, but NO. The electron energy would only increase if the frequency of the light increased! Below a certain frequency, NO electrons were emitted, independently of the intensity of the light???!

9. To explain this puzzling phenomenon, Einstein introduced the Concept of the “photon” (= light particle). He postulated that A photon had an energy proportional to its frequency f. Hence E = hf (where h is a constant of proportionality called Planck’s constant). For an electron to be emitted, it has to be given a minimum energy by the photon to overcome the attraction of the metal. Call this minimum energy M. Hence the energy En of the emitted electron is En = hf - M If f is too small, En < 0, so no electrons are emitted! Hence light consists of photons, i.e. particles!

10. de Broglie’s idea that particles have wave properties In the 1920s the Frenchman Louis de Broglie was pondering on the particle nature of waves (e.g. photons) and hit on the idea that maybe the reverse was also true, that maybe there was a kind of symmetry in nature, i.e. waves behave like particles, so maybe particles behave like waves!!!??? When de Broglie was defending his PhD thesis with this weirdo idea, his examiners found it too radical, so they sent the thesis to Einstein, who loved it. de Broglie used ideas from Einstein’s relativity, to derive a formula for the wavelength of the wavelike phenomenon depending on the particle’s momentum. From relativity for a photon, p = E/c

11. For a photon, E = hf, but c = lf (where l is the wavelength) So p = E/c = hf/c = hf/lf = h/l p =h / l de Broglie postulated that the same formula would apply to a particle, i.e. =h / pwhere l is the wavelength of the particle’s associated wave. “Matter waves”. This formula was tested experimentally in the late 1920s by the son of J.J. Thompson (the discoverer of the electron). He obtained interference patterns from electrons reflecting off a metal surface. Hence J.J. discovered the particle nature of electrons, and his son discovered the wave nature of electrons - the ironies of history!! This wave-particle duality is an important part of quantum physics!

12. Heisenberg’s Uncertainty Principle This famous principle of quantum physics says basically that there are physical quantities whose values cannot be known accurately simultaneously, e.g. the momentum and position of an electron. Try to measure the momentum and position of an electron. This would not be difficult conceptually using classical physics thinking, Since light would be considered as having continuously varying Energies and hence minimal disturbance on the electron being Observed. But in quantum physics, light consists of photons which have discrete (quantized) energies, with a kick! Bouncing a photon off an electron will give the electron a kick and disturb its momentum.

13. To obtain an accurate measurement of the position of the electron the photon used must have a small wavelength (a large wavelength would not have enough resolution to locate the electron, hence there would be an uncertainty in the electron’s position Dx). But a small wavelength for a photon, means a large energy and hence momentum (p = E/c), which gives the electron a kick and Hence an uncertainty in the electron’s momentum measurement. The measurement (using a photon - what else is there to use?!) disturbs the electron. Measurement is NOT gentle as in classical physics. There is thus a trade off between knowing accurately (and simultaneously) the momentum of the electron and its position.

14. A more quantitative analysis (see the handout) shows that the product of the two uncertainties, Dx and Dp is a constant - DxDp ~ h where h is again Planck’s constant (Planck was the first person to discover (1900) the value of this constant, so it is named after him). It appears that the uncertainty principle is purely a measurement phenomenon. Quantum physicists think that undisturbed particles (wavicles) just don’t have exact positions and momenta! A consequence of the HUP is that particles do NOT have trajectories! So what happens to the notion of an external reality with QM????!!