short seminar on application of linear algebra

1. MATHEMETICAL
FORMULATION OF
QUANTUM MECHANICS

2. Contents
• Quantum Mechanics
• Basic Principles of Quantum Mechanics
• Postulates of Quantum Mechanics : Analogy
with Linear Algebra
• Schrödinger picture of quantum mechanics
• Summary

3. QUANTUM MECHANICS
 Branch of physics dealing with physical
phenomena at microscopic scales, where the
action is on the order of the Planck constant.
• The dual - particle-like and wave-like
behaviour and interactions of energy and
matter.

4. BASIC PRINCIPLES OF QUANTUM
MECHANICS
1. Physical States
2. Physical Quantities
3. Composition
4. Dynamics

5. BASIC PRINCIPLES OF QUANTUM
MECHANICS
• Physical States
Every physical system is associated with a Hilbert
Space H.
Every unit vector in the space corresponds to a
possible pure state of the system.
The vector is represented by a function known as
the wave-function, or ψ-function.

6. BASIC PRINCIPLES OF QUANTUM
MECHANICS
• Physical Quantities
Hermitian operators in the Hilbert space
associated with a system.
Their eigenvalues represent the possible
results of measurements of these quantities.

7. BASIC PRINCIPLES OF QUANTUM
MECHANICS
• Composition
The Hilbert space associated with a complex
system is the tensor product of those associated
with the simple system.
H1⊗H2

8. BASIC PRINCIPLES OF QUANTUM
MECHANICS
• Dynamics
Contexts of type 1:
‘Schrödinger's equation’ : gives the state at any other
time U|vt> → |vt′>
 U is deterministic
 U is linear
If U takes a state |A> onto the state |A′>, and it takes the
state |B> onto the state |B′>,
then it takes any state of the form α|A> + β|B> onto the
state α|A′> + β|B′>.

9. BASIC PRINCIPLES OF QUANTUM
MECHANICS
• Dynamics
Contexts of type 2 ("Measurement Contexts"):
Collapse Postulate.
The eigenstate getting collapsed is a matter of
probability, given by a rule known as
Born's Rule: prob(bi) = |<A|B=bi>|2.

10. POSTULATES OF QUANTUM MECHANICS :
ANALOGY WITH LINEAR ALGEBRA
• Each physical system is associated with a
(topologically) separable complex Hilbert
space H with inner product .
< ᵩᵩ
| >
• Rays (one-dimensional subspaces) in H are
associated with states of the system.

11. POSTULATES OF QUANTUM MECHANICS :
ANALOGY WITH LINEAR ALGEBRA
• Physical observables are represented by
Hermitian matrices on H.
The expected value (in the sense of probability
theory) of the observable A for the system in
state represented by the unit vector
|ᵩ Є H
> is < ᵩ |ᵩ
|A >
• A has only discrete spectrum, the possible
outcomes of measuring A are its eigenvalues.

12. • More generally, a state can be represented by
a so-called density operator, ᵩwhich is a trace
class, nonnegative self-adjoint operator
normalized to be of trace 1.
• The expected value of A in the state is tr(A ᵩ)
• If is the orthogonal projector onto the one-
dimensional subspace of H spanned by , then
• tr(A ᵩ = < ᵩ ᵩ
) |A| >

13. Schrödinger picture of quantum mechanics
the dynamics is given as follows:
• The time evolution of the state is given by a differentiable
function from the real numbers R, representing instants of
time, to the Hilbert space of system states. This map is
characterized by a differential equation as follows:
If | ᵩ > denotes the state of the system at any one
(t)
time t,
where H is a densely-defined self-adjoint operator, called the
system Hamiltonian , i is the imaginary unit and h is
the reduced Planck constant. As an observable, H corresponds
to the total Energy of the system.

14. Summary
Quantum system --- Mathematical Formulation
Possible states --- Unit Vectors
State Space --- Hilbert Space
Observable --- Self- adjoint Linear Operator
Each eigenstate of an observable corresponds to an
eigenvector of the operator, and the associated
eigenvalue corresponds to the value of the observable
in that eigenstate.

15. THANKS
The orbitals of an electron in a hydrogen atom are
eigenfunctions of the energy.