Ch 5: Section 8
Expectation values
Uncertainties and
Expectation Value
O In contrast to classical mechanics, quantum
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
O Ch 5: 59, 60, 61 and
O Due Thursday
12NOV15 (same as
Test 3)
O Hint follow Example

Ch 5: Section 8 Expectation values Uncertainties and Operators