02 dirac notation.pdf
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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
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
DIRAC
bras
kets
Each state is denoted by a ket |>. Individual kets are distinguished by the labels placed
inside the ket symbol |A>, |B>, etc
kets
1
we can add two
R
many, many
R
or even have
if x varies
Additio
contin
n
uously
2
i
i
A B
L
Q x X dx
C C
C
C
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
Multiplication
i A AC
Complex number
Cont.
1 2 1 2
addition of two identical kets
A A A A C C C C
CM: addition of 2 identical statesnew 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
a vector that yields a complex number by doing the scalar multiplication with a ket is a:
BRA
have the same properties as ,
and are defined by their scalar product with every
'
completel
'
y
number number
B A A B A B A
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
as it happens with vectors, the scalar product of
numberbra ket bra ket
,
i j
i j
B A b a
Cont.
*
for every ket A there is a bra A which is the complex conjugate of A
orA A A
** *associated for the ket , the bra is A A A A C C C C
the scaler product is
................complexnumberB A A B
?A
? 0 unless A 0A A A A
if A 0 and B are orthogonal
if 0 is to
B A
x y x y
* and A B A B A B A B C C C C
the scalar product is
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 1A A
' lenght ' ' ? lengthA A A A
phase does not change neither the length or directionof state!
Even when using a fixed-length bracket ( 1) there is a phase factor
which is not defined '
'
ii
length
A e e AA A
and ' have the SAME DIRECTIONA A
Operators An operator is a rule that transforms a ket (or bra) in another ket (or bra)
Every observable is associated with an operator
ˆˆ GF A B
ALL Quantum Mechanical operators are LI (not all operators are linNEAR ear)
1 2 1 2
Properties of linear operators:
ˆ ˆ ˆ ˆ ˆ and A A A A A A C C
Properties of operators:
ˆ ˆˆ ˆSummation is distributive
ˆ ˆˆ ˆProduct is associative
A A A
A A
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