1. 5. Quantum Theory 5.0. Wave Mechanics
The Hilbert Space of State Vectors
Operators and Observable Quantities
Spacetime Translations and Properties of Operators
Quantization of a Classical System
An Example: the One-Dimensional Harmonic Oscillator

2. 5.0. Wave Mechanics Cornerstones of quantum theory:
Particle-wave duality
Principle of uncertainty
Planck’s constant
History:
Planck: Empirical fix for black body radiation.
Einstein: Photo-electric effect → particle-like aspect of “waves”.
de Broglie: Particle-wave duality.
Thomson & Davisson: Diffraction of electrons by a crystal lattice.
Schrodinger: Wave mechanics.

3. Wave mechanics
State of a “particle” is represented by a (complex) wave function Ψ( x, t ).
Probability of finding the particle in d 3x about x at t =
P = probability density if
Ψ( x, t ) is called normalized if c = 1.
P d 3x = relative probability if Ψ cannot be normalized.
E.g., free particle:
(1st) quantization: →
r-representation:

4. Hamiltonian : →
Time-dependent Schrodinger equation :
c.f. Hamilton-Jacobi equation
Time-independent Schrodinger equation :

5. 5.1. The Hilbert Space of State Vectors
Specification of a physical state: Maximal set of observables M = { A, B, C, …}.
→ Pure quantum state specified by values { a, b, c, … } assumed by S.
Assumption:
Every possible instantaneous state of a system can be represented by a ray (direction) in a Hilbert space. (see Appendix A.3)
Hilbert spaces are complex linear vector spaces with possibly infinite dimensions.
φ | = 1-form dual to | φ  = | φ †
Inner product is sesquilinear :  α, β  C, →
Norm / length / magnitude of | φ  is

6. | φ  is normalized if
Maximal set M = { A, B, C, …} → pure states are given by | a, b, c, … .
Probability of measuring values { a, b, c, … } from a state | Ψ  is → 
orthonormality → →
Completeness 
else

7. If a takes on continuous values,
Example: 1-particle system with x as maximal set.
orthonormality
completeness
Ψ (x) = wave function

8. 5.2. Operators and Observable Quantities
E.g., identity operator:
Linear operators:
 α, β  C
Observables are represented by linear Hermitian operators.
Let the maximal set of observables be M = { A, B, C, …}.
If we choose | a,b,c,…  as basis vectors, then the operators
are defined as
Eigen-equations eigenvalues eigenvectors …

9. Expectation value of A:
Adjoint A† of A is defined as: →
A is self-adjoint / hermitian if

10. Consider 2 eigenstates, | a1  and | a2 , of A.→
If A is hermitian,
Hence, →
Eigenvalues of a hermitian operator are all real. →
Eigenstates belonging to different eigenvalues of a hermitian operator are orthogonal.

11. Algebraic Operations between Operators
Addition:
Product:
Commutator:
Analytic functions of A are defined by Taylor series.
Ex. 5.7
Caution:
unless
If B is hermitian, then →
( A is unitary )

12. 5.3. Spacetime Translations and Properties of Operators
Time Evolution: Schrodinger Picture
Each vector in Hilbert space represents an instantaneous state of system.
U is the time evolution operator
Normalization remains unchanged : →
U is unitary
Setting
we can write
where
If H is time-independent, →
H = Hamiltonian
c.f. Liouville eq.

13. Heisenberg Picture
| Ψ (t)  is not observable.
Observable:
for time independent H →
 A is conserved if it commutes with H.
H is conserved.
Classical mechanics:
Possible rule:
( Not always correct )
Example: Canonical commutation relations

14. Alternative derivation:
Classical translational generator: →
All components of x or p should be simultaneously measurable → →
Consider → →
See Ex.5.3 for higher order terms → 
H independent of x (translational invariant)

15. Classical mechanics:
Conserved quantity ~ L invariant under corresponding symmetry transformation
Quantum mechanics:
Conserved operator = generator of symmetry transformation
Example: Angular Momentum

16. 5.4. Quantization of a Classical System
Canonical quantization scheme:
Classical Quantum
Schrodinger
Heisenberg
Difficulties:
Generalized coordinates may not work.
Remedy: Stick with Cartesian coordinates.
Ambiguity.
E.g., AB when
Possible remedy: use
Constrainted or EM systems with
Remedy: use pi .
Reminder: velocity is ill-defined in QM.

17. Wave functions:
x-representation
(Taylor series) →
Many bodies:

18. 5.5. An Example: the One-Dimensional Harmonic Oscillator
Some common choices of maximal sets of observables:
{ x } : coordinate (x-) representation
{ p } : momentum (p-) representation
{ E } : energy (E-) representation
{ n } : number (n-) representation
n-representation
Basis vectors are eigenstates of number operator:
Lowering (annihilation) operator:
Raising (creation) operator:
Canonical quantization:

19. → → → … → →
Setting cn and bn real gives

20. Exercise: Show that if n is restricted to the values 0 and 1, then the commutator relation
must be replaced by the anti-commutator relation

21. For the harmonic oscillator, if we set
then →
Hence, basis { | n  } is also basis of the E-representation.
The nth excited state contains n vibrons, each of energy  ω. → →

22. x-representation: →
where → 
so that
with
Using
one can show
where
and

23. The x- and p- representations are Fourier transforms of each other.
where
so that
Similarly
where
In practice, ψ (x) is easier to obtain by solving the Schrodinger eq.
with appropriate B.C.s

24. Let V(x) → 0 as |x| → .
For E < 0, ψn(x) is a bound state with discrete eigen-energies εn.
For E > 0, ψk(x) is a scattering state with continuous energy spectrum ε(k).
However, scattering problems are better described in terms of the S matrix, scattering cross sections, or phase-shifts.