1. Operators A function is something that turns numbers into numbers An operator is something that turns functions into functions Example: The derivative operator In quantum mechanics, x cannot be the position of a particle Particles don’t have a definite position Instead, think of x as something you multiply a wave function by to get a new wave function x is an operator, sometimes written as x op or X There are lots of other operators as well, like momentum

2. Expectation Values Suppose we know the wave function  (x) and we measure x. What answer will we get? We only know probability of getting different values Let’s find the average value you get Recall |  (x)| 2 tells you the probability density that it is at x We want an expectation value It is denoted by  x  For any operator, we can similarly get an average measurement

3. Sample Problem A particle is in the ground state of a harmonic oscillator. What is the expectation value of the operators x, x 2, and p? Note:  x  2   x 2  More on this later Note: Always use normalized wave functions for expectation values!

4. The Hamiltonian Operator In classical mechanics, the Hamiltonian is the formula for energy in terms of the position x and momentum p In quantum, the formula is the same, but x and p are reinterpreted as operators Schrodinger’s equations rewritten with the Hamiltonian: Advanced Physics: The Hamiltonian becomes much more complicated More dimensions, Multiple particles, Special Relativity But Schrodinger’s Equations in terms of H remain the same The expectation value of the Hamiltonian is the average value you would get if you measure the energy

5. Sample Problem A particle is trapped in a 1D infinite square well 0 < x < L with wave function given at right. If we measure the energy, what is the average value we would get? Compare to ground state: Often gives excellent approximations

6. Tricks for Finding Expectation Values We often want expectation values of x or x 2 or p or p 2 If our wave function is real, p is trivial To find p 2, we will use integration by parts

7. Recall:  x 2    x  2. Why? The difference between these is a measure of how spread out the wave function is Define the uncertainty in x: Uncertainty We can similarly define the uncertainty in any operator: Heisenberg Uncertainty Principle

8. Sample Problem A particle is in the ground state of a harmonic oscillator. Find the uncertainty in x and p, and check that it obeys uncertainty principle Much of the work was done five slides ago We even found  p , but since  is real, it is trivial anyway Now work out p 2 : Now get the uncertainties