Ch 5: Section 8 Expectation values Uncertainties and Operators Expectation Value O In contrast to classical mechanics, quantum O O O O mechanics tells us that we cannot precisely know all quantities such as position and momentum. In QM, we model matter as a wave. Find a wave function Ψ, and from that a probability density |Ψ|2. We can also use the wave function to find an expectation value. An expectation value is an average of some quantity. You could measure the expectation value of some quantity through repeated experiments and averaging Expectation Value for x O |ψ(x)|2 is the probability per unit length. O So the probability of being in a length of dx around x is |ψ(x)|2 dx. O To find the average of some quantity, you multiply each value by its probability and then add; here adding means integrating: ∞ ̅ = � () 2 −∞ O The overhead bar means average or expectation value Expectation Value for Q O Finding the expectation value of other observables such as momentum, energy (or angular momentum) requires somewhat different approach: ∞ � = � Ψ ∗ (, )�Ψ(, ) −∞ where � is an operator for the quantity O Operators are found in Uncertainty in a Quantity Δ = Square first, then take average. 2 − �2 Average first, then square. Homework O Ch 5: 59, 60, 61 and 62 O Due Thursday 12NOV15 (same as Test 3) O Hint follow Example 5.5

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# Ch 5: Section 8 Expectation values Uncertainties and Operators