PH 401 Dr. Cecilia Vogel Lecture 6

Review Outline  Representations  Momentum by operator  Eigenstates and eigenvalues  Free Particle time dependence  Fourier Synthesis and Analysis

Representation  The wavefunction contains all the info available about the state of the particle.  momentum info can be found by Fourier analysis.  The momentum amplitude ALSO contains all the info available about the state of the particle.  position info can be found by Fourier synthesis.  Each is just a different representation of the particle’s state.

x Operator  Consider   To find expectation value of position, we can  multiply  by x  then multiply by  * and integrate  For this reason  position is said to be represented by the “multiply by x” operator

p Operator  Consider   For each partial wave  bring down the momentum value

p Operator  Consider  To find expectation value of momentum, we can  take deriv of  with respect to x, and multiply by -i   then multiply by  * and integrate  For this reason  momentum is said to be represented by the “-i  ∂/∂x” operator

x Operator  Position is said to be represented by the “multiply by x” operator   Momentum is said to be represented by the “-i  ∂/∂x” operator

K Operator  Also to find expectation value of a function of momentum,  For example, K=p 2 /2m  KE is said to be represented by the operator

Energy Operator  Consider   For a wave with definite frequency  the time dependence is e -i  t.  So, we get   =E  For this reason  energy is said to be represented by the operator

Schroedinger Eqn  Identify the operators in the Schroedinger eqn 

Representation  The wavefunction contains all the info available about the state of the particle.  The momentum amplitude ALSO contains all the info available about the state of the particle.  Each is just a different representation of the particle’s state.  The state is fundamental.  state = “the way it is” state

Eigenstate of momentum  A state with definite momentum in the position representation is   (x) =Ae ip 1 x/   the momentum operator p op = -i  d/dx acting on this state yields  p op  (x) = p 1  (x)  just a # times the original state  Math: when an operator acting on function= # times original function  then that state is an eigenfunction of that operator

Eigenstates and eigenvalues  Thus   (x) =Ae ip 1 x/   is an eigenfunction of momentum  the constant, p 1 is called the eigenvalue  Generally  If Q  =q a ,  where Q=operator, q a =number  then  is an eigenstate of Q  with eigenvalue q a

Eigenvalues and measurement  Also   (x) =Ae ip 1 x/   is an eigenfunction of momentum  and if you measure momentum, you will definitely find value p1  Generally  If Q  =q a ,  where Q=observable operator, q a =number  then when you measure observable Q  you will definitely find value q a

PAL  Consider 1.Show that  (x,0) is continuous 2.Find 3.Find 4.Find  p (Don’t worry about units.) Do all derivatives and set up all integrals for most of the credit. Solve for full credit.