Two-slit experiment with particles http://en.wikipedia.org/wiki/Double-slit_experiment

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 (s2= d2/dx2 + d2/dy2 d2/dz2). We will concentrate (for the most part) upon the version that does not involve time and restrict ourselves to one dimension. The book goes through the details of the “separation of variables” technique, and the result is (for one dimension): This is the “time-independent Schrodinger equation” (the version we will deal with exclusively, in one dimension).

Schrodinger’s Equation Conditions on the function y(x) y(x) must be finite or zero “everywhere” (except possibly for a set of measure zero). y(x) must be single valued For finite potentials both y(x) and dy(x)/dx must be continuous. The integral of |y(x)|2 must be finite (and is normally chosen to be equal to one for a “properly normalized” wavefunction) Behaviour at boundaries must be correct in order for all of these above conditions to be met, e.g. for a particle in a box, y(x) must be equal to zero at the box’s boundaries. There is a nice “live” graph showing the wavefunctions for the particle in a box (1-D infinite square well) problem at: http://bouman.chem.georgetown.edu/S02/lect13/lect13.htm Interesting “shooting algorithm example of ODE solving at: http://www.cse.illinois.edu/iem/ode/shoot/

P301 Lecture 19 “JITT question” Describe, as simply as you can, the “boundary conditions” that were imposed on the open and closed ends of organ pipes in P221 (or whatever first semester intro Physics course you took). The resonances of an open pipe are at full wavelengths while the resonance of closed pipes are at half wavelengths.. (3 People concentrated on the relationship between pipe length and wavelength.) The closed end of the pipe must be the location of a node. The open end of the pipe must be the location of an antinode. (6 people concentrated on this: This is exactly what “BOUNDARY CONDITIONS” refers to) At the closed end of the pipe, the wave has nowhere to go, so the velocity of the wave is zero and the pressure is large. At the open end of the pipe, the wave has no restraint imposed on its movement, so the pressure on the wave is small and the velocity is larger. (6 people concentrated on velocity and displacement conditions rather than just nodes and antinodes. 16 didn’t answer.

Particle-in-a-box (infinite square well) Particle cannot be anywhere out here. Particle cannot be anywhere out here. Particle must be in here somewhere DeBroglie and Heisenberg tell us that this picture is misleading; the particle may be in the box, but it is incorrect to draw it in there like the little blue dot I have used. What does Schroedinger’s equation tell us about a situation like this?

Schrodinger’s Equation Infinite Square Well In here Y must be continuous, and may be non-zero. Particle must be in here somewhere Particle cannot be anywhere out here. Out here, Y must be zero! Particle cannot be anywhere out here. Out here, Y must be zero! DeBroglie and Heisenberg tell us that this picture is misleading; the particle may be in the box, but it is incorrect to draw it in there like the little blue dot I have used. What does Schroedinger’s equation tell us about a situation like this?

Schrodinger’s Equation Solutions Infinite Square Well http://integrals.wolfram.com/index.jsp Useful web site for integrating

Schrodinger’s Equation Solutions Finite Square Well Unlike the infinite case, in this case, there will only be a finite number of bound states (i.e. states for which there are “classically forbidden” regions, where V>E).

Lecture 20 JITT question What is the most significant difference you see between the wave functions for a particle in an “infinite square well” and those for a particle in a “finite square well” of the same width? Which of well would you expect to have higher energy for its ground state (again, assuming both have the same width)? Differences: Particle can exist outside the well for the finite case (14 answers) Larger l/(more spread out) for the finite case (3 answers) Other 5 No answer (23 answers) Higher Energy in which well? Infinite case (14 answers) Finite case (5 answers) ? (1 answers)

Schrodinger’s Equation Solutions Finite Square Well 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=k tan(kL/2) ; Odd solutions k= -k cot(kL/2) 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: http://physics.indiana.edu/~courses/p301/F10/Lectures/ The fig. shows the marginal case for two even solutions (3 overall) a=p2

OPEN HOUSE 23 Oct. 2010 Volunteer via email to: anfoley@indiana.edu (Anne Foley) Or go to the front office.

Schrodinger’s Equation For the Simple Harmonic Oscillator, we have : V(x)=0.5k(x-xo)2 NOTE: here k is the spring constant, not a wave vector, and the angular frequency of the oscillator is w=(k/m)1/2 , just as in classical physics; but unlike classical physics now the energy is given by: En=(n+1/2)h w

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 = hw(n + ½) n = 0, 1, 2, …

Lecture 21 JITT question How would you expect the light absorption spectrum for an electron trapped in a harmonic potential to differ from that for an electron trapped in an infinite square well potential? Very few people seemed to understand what the question was asking.