1. PH 301 Dr. Cecilia Vogel Lecture 2

2. Review Outline  matter waves  probability, uncertainty  wavefunction requirements  Matter Waves  duality eqns  interpretation

3. Probabilty and Normalization  the probability of the particle being in a volume of space  the probability of the particle being in all of space, should be 1 (100%)  If the integral over all space =1, the wavefunction is “normalized”  Only normalized wavefunctions can be used to find absolute probability NORMALIZATION

4. Probabilty and Averages  The average value of x can be found by averaging the possible values of x  but some are more probable than others  so the average is weighted by the probability density EXPECTATION VALUE

5. Probabilty and Averages  The expectation value of any function of x can be found similarly: EXPECTATION VALUE

6. Uncertainty  The uncertainty in x is  a measure of the spread in possible values of x  It is not  measurement error  nor lack of knowledge  The wavefunction is really spread out over many x values  like a water wave that strikes many points on the shore

7. Uncertainty and Averages  The uncertainty in x can be found as the root mean square (rms) deviation UNCERTAINTY DEF UNCERTAINTY CALCULATION  The uncertainty can more easily be calculated using

8. Uncertainty Example  An electron in 1 st excited state of an infinite 1-D square well 1-nm long has a wavefunction that is zero outside the box and inside the box equal to (x in nm)  The uncertainty can be calculated using Mathcad Mathcad

9. Uncertainty Principle  The uncertainty in position is not restricted  Can be arbitrarily small  But uncertainty in position and momentum can’t both be arbitrarily small

10. Wavefunction Requirements  Mathematically, a wavefunction can be any function,  so long as it is normalized.  BUT to describe a real physical particle  the wavefunction must obey the laws of physics.  The law of physics that applies to wavefunctions  of non-relativistic particles  is the Time Dependent Schroedinger Eqn

11. TDSE  The Time Dependent Schroedinger Equation:  cannot be derived  agrees with empirical observation  describes the time evolution of a particle, given its environment  (like F=ma for classical particles).

12. TDSE  The Time Dependent Schroedinger Equation in 1-D:  The Time Dependent Schroedinger Equation in 3-D:

13. Four Requirements  The wavefunction of a physical particle 1.must obey TDSE 2.must be normalizable must be finite everywhere must approach zero as x, y, z approach ± ∞ 3.must be continuous  no physical quantity should change by finite amount for an infinitesimal change in position 4.must have cont. first spatial derivative.  anywhere V is finite  actually a consequence of TDSE