Physics Lecture 11 3/2/ Andrew Brandt Monday March 2, 2009 Dr. Andrew Brandt 1.Quantum Mechanics 2.Schrodinger’s Equation 3.Wave Function Properties

3/2/ Andrew Brandt Bohr Energy Levels How much energy is required to raise an electron from the ground state of a Hydrogen atom to the n=3 state? Rydberg atom has r=0.01 mm what is n? E? Bohr atom explains energy levels Correspondence principle: For large n Quantum Mechanics  Classical Mechanics Defer lasers for now

Quantum Mechanics Bohr theory: -Successful for Hydrogen and one electron atoms -Does not explain some subtle aspects of spectral lines -Does not predict how atoms interact New theory of Quantum Mechanics (QM) needed QM makes predictions (often surprising) of remarkable accuracy Modern physics is based on QM QM deals with probabilities: the Bohr radius is most probable value of electron distance from nucleus, not exact value Classical mechanics, given initial conditions, momenta, and forces is deterministic, but due to uncertainty principle, QM is not is wave function of an object while is the probability of finding an object at a given place and time QM can be used to determine the wave function under the influence of external forces, and the wave function can then be used to obtain the momenum, angular momentum, and energy 3/2/ Andrew Brandt3

QM Wave Equation Recall general traveling wave solution: In QM, wave function  may be complex (unlike y), so generalize to where With and we obtain This represents a particle with energy E and momentum p moving freely in the positive x direction Interested in motion of particle when acted upon by forces This requires a fundamental differential equation (Schrodinger’s Equation) which cannot be rigorously derived Let’s do years of work in a couple of bullet points! 3/2/ Andrew Brandt4

Aside on Partial Derivatives How do I take the derivative of when both x and y are variables? Partial derivative holds one variable as fixed while take derivative with respect to the other. For the example above if y is fixed, so So correspondingly Note if f has no x dependence 3/2/ Andrew Brandt5

Deriving Schrodinger Equation Consider wave equation: ; can take partial derivative with respect to x and obtain which we can rewrite as: This gives an association between quantity p and the operator, implying that multiplying  by the momentum is equivalent to operating on it with a differential operator Similarly so Consider energy of a particle where U is some form of potential energy due to position in a field (EM field for example) Operating on the wave function with this energy equation gives the time dependent Schrodinger Equation in one dimension: in 3 dimensions 3/2/ Andrew Brandt6

Probability Density In classical physics, solutions to wave equations are real (not imaginary) A complex wave function has no physical significance so need for probability Probability Density:, where if then For a particle, the integral of the probability density over all space must be one (one particle): (this can be used to obtain the normalization of the wave function) 3/2/ Andrew Brandt7

Properties of   must be “well-behaved” (good psi) -continuous and single-valued -derivatives finite and single valued -normalizable (  0 as x  ) Schrodinger Equation is linear so if  1 and  2 are solutions then is a solution (where a 1 and a 2 are constants) Wave functions add, not probabilities 3/2/ Andrew Brandt8 can’t ignore interference terms [board]