1. Peter Atkins • Julio de Paula Atkins’ Physical Chemistry
Eighth Edition
Chapter 8
Quantum Theory:
Introduction and Principles
Copyright © 2006 by Peter Atkins and Julio de Paula

2. Fig 8.29
Integral of the function f(x) = (sin x)(sin 2x)
sin x
sin 2x

3. Superpositions and Expectation Values
Recall that for a particle in 1-D when A = B,
Let’s operate on Ψ with momentum operator,
Not an eigenfunction! However, Ψ is a linear
combination (sum) of exp(ikx) and exp(-ikx)
exp(ikx) and exp(-ikx) correspond to the two momenta
Superposition principle: When a system is in a range
of states, its Ψ is a linear combination of the Ψi’s:
Ψ = c1Ψ1 + c2Ψ

4. Ψ = c1Ψ1 + c2Ψ = ΣckΨk
Ψk’s are a complete set
Quantum mechanically:
A single measurement gives the one eigenvalue
of that Ψk (that instantaneous state of the system).
The probability of a certain eigenvalue being observed
is proportional to |ck|2.
The average of a large number of observations is given
by the expectation value 〈Ω〉 of the operator .
    
    
    
〈Ω〉 is the weighted average of a
large number of observations.

5. Heisenberg uncertainty
Fig The wavefunction for a 1-D particle
at a well-defined location.
In this case, Ψ(x) is a wave packet
formed by superposition of
an infinite number of Ψ(x).
What is the particle’s momentum?
Quantitatively:
In this example, Δq = 0, so Δp = ∞
Heisenberg uncertainty
principle

6. Fig 8.31 The wavefunction for a 1-D particle
with an ill-defined location.
Examples of complementarity:
Position and momentum
Wave and particle nature
Lifetime and bandwidth

7. More generally, the uncertainty principle applies to complementary observables in terms of their operators:
e.g.,
If the two observables
Ω1 and Ω2 are said to be complementary.
When the effect of the operators depends on their
order of application (as above), the operators do not
commute.

8. Different outcomes of operator order of application expressed in terms of the commutator:
Now, if two operators do not commute,
they are complementary, and cannot be determined simultaneously.
If two operators commute,
they are not complementary, and can be determined with the same degree of precision.

9. Young’s double-slit experiment
Single electrons are emitted from the source at rate of 10/s.
Single electron events build up to from an interference pattern in this double-slit experiment.