THE ATOM Announcements: First midterm is 7:30pm on Sept. 26 Today overview of the atom; a survey of 3.1-3.6 and 3.10-3.12. You are responsible for the ideas and vocabulary but not all of the math in the development of the basic theory. Plan for next few lectures: Will begin Chapter 4 which covers the ideas and experiments that lead to quantum mechanics. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 1
Energy and momentum of massive & massless particles Remember our triangle relation: But photons are massless so For a massless particle like a photon, this reduces to so the energy and momentum are the same Keep in mind that momentum is a vector and energy is a scalar As you will find out soon, Einstein won the Nobel Prize for finding the energy of a photon is f = frequency and h is Planck s constant: doesn t work. What does? This can then be used to obtain the photon momentum http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 2
Clicker Question 1 Set frequency to AD The correct formula for Newton s 2 nd law in special relativity is: A. B. C. D. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 3
Clicker Question 1 Set frequency to AD The correct formula for Newton s 2 nd law in special relativity is: Forces still work in relativity, pretty much A. the same as in classical physics. B. C. D. Since we need to allow for massless particles, we need to use for Newton s 2 nd law. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 4
Four vectors Position and momentum are 3-vectors since in a 3D space, they measure three things In the 4 dimensions of spacetime we have 4-vectors which are a 3-vector plus a 4 th scalar. Whether we put the 4 th scalar at the end or beginning of the order list depends on the convention. The 4 th scalar associated with the position 3-vector is, of course, time or ct: (x,y,z,ct) or (ct,x,y,z) and the invariant spacetime interval is The 4 th scalar associated with the momentum 3-vector is energy or E/c to keep the units straight: (p x,p y,p z,e/c) or (E/c,p x,p y,p z ) and the invariant is http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 5
E = http://www.colorado.edu/physics/phys2170/ Tachyons if they exist Rela%vis%c Expressions for energy and momentum of a par%cle with rest mass m and velocity u are given by: mc 2 1 u 2 /c 2 p = mu 1 u 2 /c 2 Hence the mass of a par%cle must be imaginary, if it is traveling with a speed u > c for the measurable quan%%es E, p to remain real. SeCng m = iµ, we find: E = µc 2 u 2 /c 2 1 p = µu u 2 /c 2 1 0 < E < µc < p < At infinite velocity a tachyon carries no energy and has a finite momentum µc. We find as tachyon loses energy, it accelerates. u > c
Why is the atom so important? Special relativity has effects that are typically not observable except when objects move very fast (close to the speed of light). One exception is rest mass energy which exists no matter what the speed. Quantum mechanics has effects that are typically not observable except when objects are very small (like atomic sizes). Thus, the development and understanding of quantum mechanics is intimately tied to the discovery and understanding of the atom. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 7
A useful picture of atoms The center of the atom is composed of protons and neutrons bound together in a very small nucleus Nuclei radii range from about 1 to 8 fermi (1 fermi = 1 fm = 1x10-15 m) The rest of the atom is basically empty with electrons flying around. The electrons extend out to ~0.1 nm, aka 1 ångström (1 Å) Electrons have q= 1.6x10-19 C and protons have q=+1.6x10-19 C. Protons and neutrons have about the same mass (938.3 MeV/c 2 and 939.6 MeV/c 2 ); electrons are ~1800 times lighter (0.511 MeV/c 2 ). http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 8
A useful picture of atoms Atomic number Z gives # of protons (and # of electrons for a neutral atom). Atomic mass number A gives number of nucleons (protons + neutrons) The actual atomic mass is not just the sum of the masses of protons, neutrons, and electrons. Why? Binding energy (of protons and neutrons together) reduces the mass. When protons and neutrons fuse (nuclear fusion) energy is released (which is made up for by the loss of mass). This is what powers the sun (and the hydrogen bomb). Atomic mass unit u is used to measure atomic masses. It has a value of 931.5 MeV/c 2. http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 9
Clicker question 2 Set frequency to AD electron X B J.J. Thomson discovered the electron in 1897 V How will the electron be deflected in the B-field? A. Into the page B. Out of the page C. Upwards D. Downwards E. No deflection at all Hint: Remember http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 10
Clicker question 2 Set frequency to AD electron X B J.J. Thomson discovered the electron in 1897 V How will the electron be deflected in the B-field? A. Into the page B. Out of the page C. Upwards D. Downwards E. No deflection at all Hint: Remember Thomson observes the particle bending downward which means negative electric charge (from the right hand rule). http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 11
Measuring q/m for the electron (e/m) The force by a magnetic field is perpendicular to the velocity so it causes centripetal acceleration. If we knew the electron velocity we could find q/m. X B E Solution: Add an external downward E-field and adjust until the particle goes straight. V Nobel Prize 1906 http://www.colorado.edu/physics/phys2170/ Physics 2170 Fall 2013 12