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 ketLength and sign are irrelevant

DIRAC

bras

kets

Each state is denoted by a ket |>. Individual kets are distinguished by the labels placedinside 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

ii

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

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 ji j

B A b a

*

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

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* and A B A B A B A B C C C C

Length and phase

LENGTH

vectors A A

bras and kets A A

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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

OperatorsAn 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

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