2. Chapter Overview State Functions QM Operators & CM Variables
Observables & Eigenvalues
Commutators
Hermitian Functions
Hermitian Operators & Orthogonality
Commuting 2 or more Eigenfunctions
Fourier Coefficients
Time-Dependent Schrödinger Equation

3. State of System &  The Setup Only Normalized ‘s Allowed Postulate 1
Dynamic Variables in Classical Mechanics:
E.g. position (x), momentum (p), energy (E)
These variables are also known as observables
Total probability of finding a particle somewhere:
For 3D:
Classical Mechanics states:
Any state is specified by (x,y,z) with momentum (px,py,pz) at any time t
Newton’s laws of motion rule in classical systems:
In General:
State of System & 
Normalization Factor:
Postulate 1
The state of a quantum mechanical system is completely specified by a function Ψ(r) that depends upon the coordinate of the particle and on time. This function, called the wave function or the state function, has the important property that Ψ*(r,t)Ψ(r,t)dx is the probability that the particle lies in the volume element dxdydz located at r at time t.
No Naughty Wavefunctions
For well-behaved wavefunctions:
 must be continuous
’ must be continuous
* must be single-valued
must be finite – otherwise  cannot be normalized

5. The Angular Momentum Operator QM Operators & CM Variables
Postulate 2
For every observable in classical mechanics there is a corresponding linear operator in quantum mechanics.
- I.E. x in CM corresponds to in QM - See Table 4.1 on p. 148 𝑥
The Angular Momentum Operator
QM Operators & CM Variables
- Classically:
- Quantum Mechanically:

7. Observables & Eigenvalues
Suppose …
POSTULATE 3
Observables & Eigenvalues
POSTULATE 4

9. Commutators & Manipulating Operators
To be an Commutator
Applying more than one Operator
Commutators & Manipulating Operators
Or not to be a Commutator aka a Noncommutator
Notation, Notation, Notation

10. Commutators & Manipulating Operators
Back to Heisenberg
Commutators & Manipulating Operators
When 2 operators do not commute they cannot be measured at the same time
If 2 operators commute they can be measured at the same time
What does it all mean?

12. Hermitian Operators What they are & how they work
Operators which when applied to one of their eigenfunctions produce a real eigenvalue regardless of whether the operator and/or the eigenfunction is complex
Example using Momentum
Hermitian Operators
Order of operation is the key
Your book also defines Hermitian as:
If you reverse the order you must take the conjugate and reverse to whom the operator is applied
Application of Dirac
POSTULATE 2 (Revisited)
To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics

14. Hermitian Operatros & Orthonormality
Formal Statement of Orthonormality
Eigenfunctions of Quantum Mechanical Operators are Orthogonal to each other
Hermitian Operatros & Orthonormality

15. Hermitian Operatros & Orthonormality
Are the two Eigenfunction Orthogonal?
Hermitian Operatros & Orthonormality
Yes they are perpendicular/orthogonal to each other

17. Commuting Operators share Eigenfunctions
See proof in the text (p )
Commuting Operators share Eigenfunctions
Commutators share the same set of eigenfunctions which can be measured simultaneously

19. Fourier Coefficients (FCs)
From where they come
Fourier
Coefficient
FC’s & Probability
Fourier Coefficients (FCs)
We use a similar equation to find the average energy

21. Time Dependent Schrödinger Equation
POSTULATE 5
We start with separation of variables
Time Dependent Schrödinger Equation

23. SKIP!!
Two-Slit Experiment