1. PH 401 Dr. Cecilia Vogel Lecture 6

2. Review Outline  Eigenvalues and physical values  Energy Operator  Stationary States  Representations  Momentum by operator  Eigenstates and eigenvalues

3. Recall Operators  Operating on  with the x-operator means  multiply  by x  Operating on  with the p-operator means  take deriv of  with respect to x, and multiply by -i   Averages found by integrating:  for example

4. Eigenvalues  Consider the momentum operator  -i  ∂/∂x  Eigenstates of this operator look like  Ae ik 1 x =Ae ip 1 x/   Prove it  p  =p 1   Eigenvalue = p 1.

5. measured values  Also recognize  Ae ik 1 x =Ae ip 1 x/   is a state with definite momentum  If you were to measure the momentum, you would find p 1. No doubt.

6. Eigenvalues and measured values  Generally  Eigenstates of any observable operator  Are states with definite value of the observable  and that definite value is the eigenvalue  A measurement will produce a value equal to the eigenvalue, for sure.

7. KE 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

8. PE Operator  Since x-operator is “multiply by x”,  Any function of x operator is “multiply by function of x”  For example, potential energy operator, V, is represented by V(x)

9. 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

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

11. Energy Eigenstate= Stationary State  Energy Eigenstate is a state with  definite value of energy  zero uncertainty in energy  time dependence  (x,t)=  (x)e -iEt/   Probability density independent of time  hence “stationary” on average  prove it  all averages and uncertainties independent of time  obeys the TISE – why?

12. Time Independent Schrödinger Equation  Plug stationary state wavefunction:   (x,t)=  (x)e -iEt/   into the time DEpendent SE:

13. PAL 11/29/10  Consider 1.Find. Does it depend on time? 2.Find. Does it depend on time? (Don’t worry about units.) Do all derivatives and set up all integrals. DO NOT SOLVE any integrals, EXCEPT those that are zero because the integral of an odd function over even interval = 0.