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:
∞
𝑥𝑥̅ = � 𝑥𝑥 𝜓𝜓(𝑥𝑥) 2 𝑑𝑑𝑑𝑑
−∞
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:
∞
𝑄𝑄� = � Ψ ∗ (𝑥𝑥, 𝑡𝑡)𝑄𝑄�Ψ(𝑥𝑥, 𝑡𝑡)𝑑𝑑𝑑𝑑
−∞
where 𝑄𝑄� is an operator for the quantity
O Operators are found in
Uncertainty in a Quantity
Δ𝑄𝑄 =
Square first, then
take average.
𝑄𝑄2 − 𝑄𝑄�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