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Schrodinger’s Equation The above is taken from Wikipedia, and here the “Laplacian” operator in the first term on the right hand side is simply a short-hand for (  2 = d 2 /dx 2 + d 2 /dy 2 + d 2 /dz 2 ). We will concentrate (for the most part) upon the version that does not involve time and restrict ourselves to one dimension.

Schrodinger’s Equation Solutions Finite Square Well Recall from class that we can recast the transcendental equation into a dimensionless form, where the controlling parameter is the ratio of the potential well depth to the “confinement energy”. You can get the spreadsheet for the even solutions from the website: 301/F10/Lectures/ 301/F10/Lectures/ The fig. shows the marginal case for two even solutions (3 overall)  =  2 For this case, you must solve a transcendental equation to find the solutions that obey the boundary conditions (in particular, continuity of the function and its derivative at the well boundary). For the geometry we considered in class, this takes on one of two forms: Even solutions  =k tan(kL/2) ; Odd solutions  = -k cot(kL/2)

Schrodinger’s Equation Solutions Harmonic Oscillator There are an infinite number of possible states, since the potential is defined to keep going up (of course this is an idealization). Interestingly, the energy levels are evenly spaced in this case: E = h  (n + ½) n = 0, 1, 2, …

Schrodinger’s Equation Solutions Harmonic Oscillator n=10 n=0 Notice how the quantum probability distribution (blue curve) gets closer to the classical one (dashed line above) as n gets bigger.

Lecture 22 Potential Barrier: Classical <V o E >V o Transmitted Reflected

Lecture 22 Potential Barrier: Quantum Transmitted !! Reflected and E <V o You can watch a demonstration of the optical analog of this on YouTube at:

Identify two examples of phenomena discussed in the text where quantum mechanical tunneling is involved, and cast a vote for which one you would like to see discussed in greater detail in class. Alpha-decay (16 identified this; 12 voted for it) Tunnel diodes (13 identified this; 8 voted for it; we’ll cover later) STM (4 identified this, 1 voted for it) Internal reflection of light (4 identified it; 1 voted for it) Potential wells and barriers (6 identified these; 1 voted for it) Lecture 22 JITT question

A p-n junction is made when you join n-type (with impurity states near the top of the gap occupied with electrons) with a p-type (with empty impurity states near the bottom of the gap). The electrons in the (n- type) impurity states near the boundary can “fall down” and fill the impurity states in the p-type; this leaves what is called a “depletion zone” around the boundary where there are no carriers. Lecture 22 Semiconductor Physics n-type p-type

A p-n junction is made when you join n-type (with impurity states near the top of the gap occupied with electrons) with a p-type (with empty impurity states near the bottom of the gap). The electrons in the (n- type) impurity states near the boundary can “fall down” and fill the impurity states in the p-type; this leaves what is called a “depletion zone” around the boundary where there are no carriers. This also adds in a coulomb potential that shifts the energy bands near the interface Lecture 22 Semiconductor Physics n-type p-type

A p-n junction is made between two degenerately doped semiconductors. This makes this “depletion region” thin enough to tunnel through, and as a result the device shows negative differential resistance as the bias moves the occupied levels in the conduction band beyond (in the n material) above the valence band of the p material) Lecture 22 Tunnel Diode From S. M. Sze, “Physics of Semiconductor devices”

A simple apparatus for demonstrating Frustrated total internal reflection

Lecture 22 Intro to Radioactivity This chart shows the decay modes of the known nuclides, see the link for an interactive version with lots more information

Lecture 23 Potential Barrier: Alpha Decay The narrow “squarish” well is provided by the STRONG force, but once the  particle gets far enough away from the other nucleons, it feels the Coulomb repulsion. The  particle ‘leaks’ through the triangle-like barrier. The energy level (above 0) E , determines the half-life of the nucleus (i.e. how rapidly the particle tunnels out of the nuclear barrier). NOTE: for nuclei, their radius is given approximately by: R=1.2fm(A) 1/3 Where A is the atomic weight (and 1fm = m is one femto- meter, or 1 fermi) Alpha decay in radioactivity Nuclear potential

Lecture 23 Potential Barrier: Alpha decay The deeper the “bound” state is below the top of the barrier, the lower will be the kinetic energy of the alpha particle once it gets out, and the slower will be the rate of tunneling (and hence the longer the half-life). Figures from Rohlf “Modern Physics from  to Z o ”.

Quantum Tunneling in Chemistry McMohan Science 299, 833 (2003)

Lecture 23 Scanning Tunneling Microscope Basic operation of the STM. Piezo-electric actuators move the tip in three directions, typically keeping the current constant and the “surface” is mapped out by monitoring the “z” feedback voltage as the x-y directions are scanned. (the exponential dependence of current on separation is crucial to this).

Lecture 23 Scanning Tunneling Microscope “High-Resolution” STM measurement of the a surface of Si (7x7 reconstruction of the 111 surface, if you wan to be technical), along with a computer calculation of the electron density predicted for that surface, and a “stick” model of the atomic positions. Taken from the Omictron site:

The STM was the first of what is now a wide assortment of scanning probe microscopies for studying surfaces (Atomic Force, Magnetic Force, Scanning capacitance, Near-field Scanning Optical …). How would you expect the current in an STM to depend on the separation between the tip and the surface? “decrease as the distance increases” (14 responses) 1/x or 1/x 2 (Coulomb) (3 responses) Exponential decay with distance (2 responses) No response (22) What defines “the surface” in this technique? “the Surface” or “the sample” (3 responses) The “surface” atoms (7 responses) The “upper electrons” of the surface atoms (3 responses) other (4 responses) Lecture 23 JITT question

Lecture 23 Scanning Tunneling Microscope Fe atoms arranged by “hand” into a circle on a Cu surface (111 surface to be precise) The ripples inside the “corral” represent the states available to the electrons in the copper in the presence of these extra surface atoms. The STM “sees” the electrons, not the atoms. See also:

Lecture 23 Example questions From last year’s second exam: (6 Points) The Weak nuclear force is believed to be “mediated” by massive particles: the W +, W - and Z o. These particles have masses of approximately 100 GeV/c 2, and the interaction can be thought of as a phenomenon in which virtual particles of this type “pop” into existence briefly as they travel between the interacting particles. To keep the situation simple, assume that these virtual particles travel at a velocity of 0.1 c as they are exchanged. Use this picture to estimate the range of the weak force. Variation on the above: What is the “1/e distance” for the tunneling probability involved in this exchange of particles? T&R 6-44: A 1-eV electron has a probability of tunneling through a 2.5eV potential barrier. a. What is the probability of a 1-eV proton tunneling through the same barrier? b. (DVB) Assuming the barrier is flat, how wide is it?

Lecture 24 The Hydrogen Atom revisited Major differences between the “QM” hydrogen atom and Bohr’s model (my list): The electrons do not travel in orbits, but in well defined states (orbitals) that have particular shapes (probability distributions for the electrons, or linear combinations thereof) [8 responses, although expressed in about 8 different ways] New quantum numbers introduced ( l and m l ) [4 responses] The Energy levels are NOT tied directly to the angular momentum. [DVB] There are several different states with the same energy in the QM atom [DVB] Other 6 responses NOTE: the energy levels are (nominally) the same, until we account for subtle effects that lift degeneracy.

Lecture 24 Spherical Polar Coordinates r defines the sphere  defines the cone  defines the plane and the intersection of the three is the point of interest

Lecture 24 Laplacian But note that some references (e.g. Wikipedia), use an alternative notation for the Laplacian:  f AND, the even sometimes reverse the traditional roles of  and  There is a natural separation of variables for most cases in physics (where the potential energy depends only on r, not  or  ). Cartesian coords. Spherical coords.; from A. Goswami: Quantum Mechanics

Lecture 24 Spherical Harmonics

Lecture 24 Spherical Harmonics