1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.11
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Representation in continuous bases:
Ket, bra and operators are represented in the continuous
basis by continuous matrices by noncountable infinite
matrices,
Orthonormality condition of base kets of contiuous basis
is expressed Dirac’s continuous delta function
Dirac delta function

3. Completeness relation for continuous basis is defined by
Integral
State vector is expanded as
where b(k) is projection.

4. Recall that the norm of discrete base ket was finite but
norm of continuous base ket is infinite

5. Properties of Dirac Delta function

6. Ket is represented in continuous basis by a single column
matrix which has continuous and infinite number of
Comopnets b(k),
Bra is represented by row matrix

7. Operators are represented by square continuous matrices
whose rows and columns have continuous and infinite
number of components,

8. Position representation:
Here basis consist of infinite set of vector
We have
Position operator, Eigenvalue
Orthonormality and completeness conditions are,

9. State vector can be expanded as,
Above eq. Give us the wave function for state vector.
give us the probability of finding
The particle in volume d3r.

10. The scalar product between two state vector is defined as
Operator is Hermitian,

11. Momentum Representation:
Basis of momentum representation is
We have
Momentum operator momentum vector
Orthonormality and completeness conditions in
Momentum representation are,

12. Momentum space wave function
Probability of finding system’s momentum in volume
element d3p is
Scalar product between two states is given in momentum
Space by

13. Connection between momentum and position
representation:
We try to establish connection between position and
momentum of state vector
We write
-----(1)

14. Similarly we write
------(2)
Eq (1) and (2) implies that are Fourier
transform of each other.
Fourier tranform of function f(r) is defined by
(3)

15. Thus we have, comparing (1) and (3)
Inverse transform is,

16. Position wave function is
Fourier transform is
Show that if is normalized then also

17. Find momentum operator in position representation:
Find position operator in momentum representation