1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.3
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Dirac Notations: The physical state of the system is
represented by the state vectors. These state vectors
are independent of the basis.
e.g. for ordinary vectors, the components of vector may take
different values in different co-ordinate system but the
meaning of the vector does not change from one co-ordinate
system to other.
The Dirac introduced a new formalism to make the state
vectors independent of particular representation.

3. The state of dynamical system in quantum mechanics under Dirac notations are represented by ket vectors.
We use the symbol to represent ket. For example state
vector will be represented by just putting it in middle of
above symbol i.e.
Ket vectors belongs to the Hilbert space.
Ket vectors can be multiplied by complex numbers and may be added together to give other ket vectors

4. Each state of a dynamical system is represented by a
ket vector and if the state result from the superposition
of other states, then the ket of that state can be
expressed as linear superposition of other ket vectors.
By superposing a state with itself we cannot form
a new state but we get original state again

5. We now introduce the concept of Bra vectors. These are
element of the dual space corresponding to the vector space.
To understand, we consider a number α which is a function
of a ket vector
i.e. Corresponding to each ket vector there corresponds
one number α. We also suppose that the function is a linear
One.
The number may be looked upon as scalar product of that
with some new vector. This new vectors are called
Bra vectors.

6. The bra-ket is used to represent the scalar product
(also known as inner product) of two functions
Bra vector help in extracting the amplitude of
something to happen when the system is in state

7. Note that
In wave mechanics one deal with wave function
But in general formalism of quantum mechanics we deal
with the abstract ket vectors
We can express state vectors in terms of some particular
representation e.g. Position representation
Scalar product in co-ordinate representation is

8. Properties of Ket and Bra:
Every Ket has corresponding bra

9. Properties of Scalar product: The scalar product
is not same as This is because the scalar product is
a complex number and hence order matters a lot.
We write
We can write
If both state vectors are real then

10. More properties of scalar product:

11. Norm is real and positive:
Schwarz inequality:
This relation is analogous to relation in real Euclidean space

12. Triangle inequality:
Orthogonality:

13. Orthonormal states:
Forbidden states: The products like or
are forbidden if belongs to same Hilbert space.
However two vectors belongs different spaces e.g. one belong
to orbital space and other to spin space then we can define
These are called tensor product.

14. If a ket is not normalized, then we can form a normalized Ket as|
If a ket is not normalized, then we can form a normalized
Ket as
With property,

16. Example: Ket are represented by column matrices
And Bras are represented by row matrices.
Consider the following two kets
Find bra corresponding to
Ans: Bra is the complex conjugate of transpose of ket.

17. Find the scalar product
Products does not make sense
Because columns are not multiplied with columns and
rows are not multiplied by rows.