1. 1 Reading: QM Course packet – Ch 5 BASICS OF QUANTUM MECHANICS

2. 2 We will state two things without proof, and you'll see why they are reasonable, later. 1. In the "position representation" or "position basis", the position operator is represented by the variable x: 1. In the "position representation" or "position basis", the momentum operator is represented by the derivative with respect to x: 1. This follows if you accept (2). The energy operator is: Now let's think about eigenfunctions of these operators (worksheet)

3. 3 If the momentum operator operates on a wave function and IF AND ONLY IF the result of that operation is a constant multiplied by the wave function, then that wave function is an eigenfunction or eigenstate of the momentum operator, and its eigenvalue is the momentum of the particle. operator eigenvalue not all states are eigenstates – and if the are not, they can be usually be written as superpositions of eigenstates if a state is an eigenstate of one operator, (e.g. momentum), that state is not necessarily an eigenstate of another operator, though it may be.

4. 4 Look more closely at the momentum eigenfunction (in the position representation) or momentum eigenstate: 1. Why did we change C to p? And why the subscript? 2. What is the probability distribution for this state? 3. Is it normalized? Normalizable? 4. It is degenerate (new word, maybe?) 5. What sort of particle would be represented by this function? 6. Where is this particle "located"? Could we "improve" the description of a particle by localizing it? Position eigenstates: This is a useful (but a bit pathological) representation of a position eigenstate: 1. Normalizable? 2. Otherwise reasonable?

5. 5 Review language of PH425 Kets and wave functions Probability density Operators – position, momentum, energy Eigenfunctions Mathematical representations of the above BASICS OF QUANTUM MECHANICS REVIEW