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Dirac Notation
States can be added to yield a new state  Superposition
To describe STATES, we use vectors.
VECTORS represent STATES
Each vector can have finite or infinite number of elements
bras
DIRAC 
kets
Each state is denoted by a ket |>. Individual kets are distinguished by the labels placed
inside the ket symbol |A>, |B>, etc
A vector has direction and length, and so do the kets
A state of a dynamical system = direction of ket
Length and sign are irrelevant
kets
Multiplication
Addition
Ci A  A
we can add two
R  C1 A  C2 B
Complex number
many, many
R   Ci L
i
or even have
Q   C  x  X dx if x varies continuously
If a state is the superposition of 2 states, then the corresponding ket is the linear
combination of 2 other kets
are independent if no one can be expressed as a linear combination of the others
addition of two identical kets
C1 A  C2 A   C1  C2  A  A
CM: addition of 2 identical statesnew state
QM: addition of 2 identical states same state
CM: state can have 0 amplitude (no motion)
QM: |ket> CANNOT have 0 amplitude,
STATE  direction of vector , and if there is a vector, there is a length.
bras
To each ket |A>, there corresponds a dual or adjoint quantity called by Dirac a bra; it is
not a ket-- rather it exists in a totally different space
a vector that yields a complex number by doing the scalar multiplication with a ket is a:
BRA
as it happens with vectors, the scalar product of
bra ket  bra ket  number
B A   bi  a j
i, j
have the same properties as
,
and are completely defined by their scalar product with every
B
A
 A'   B A  B A'
number
number
for every ket A there is a bra A which is the complex conjugate of A
A  A or A
*

A ?
for the ket C A , the associated bra is C A  C A  C* A
*
*
the scaler product is
B A  ................  A B complex
number
A A ?
A A  0 unless A  0
CA B  C* A B
A CB  C A B
and
if A B  0  A and B are orthogonal
if x  y  0
 x is  to y

Length and phase
LENGTH
vectors 
A  A
bras and kets 
A A
The direction of the vector defines the dynamical state, and the length is not important
 We can always use normalized vectors
A A 1
Even when using a fixed-length bracket (length  1) there is a phase factor
which is not defined
A '  ei A
 A '  ei A
A and A ' have the SAME DIRECTION
A ' lenght 
A' A'  ?
 A length
phase does not change neither the length or directionof state!
Operators
An operator is a rule that transforms a ket (or bra) in another ket (or bra)
Every observable is associated with an operator
Notation: I use ^ (as most other authors do), Fayer’s book uses underline
F  ˆ A
Properties of operators:
G  B ˆ
ˆ  ˆ  A  ˆ A  ˆ
ˆ  ˆ  A  ˆ  ˆ A 
Summation is distributive
Product is associative
A
ALL Quantum Mechanical operators are LINEAR (not all operators are linear)
Properties of linear operators:
ˆ  A1  A2
  ˆ
A1  ˆ A2
and
ˆ C A  Cˆ A