1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.15
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Harmonic oscillator:
Classic harmonic oscillator is a mass m attached to
Spring of force constant k.
Hooke’s Law,
----(1)
Solution:
------(2)
Where,
Angular frequency,
-----(3)

3. Potential energy:
Parabola
-----(4)
The sum of the kinetic and 
potential energies in a
simple harmonic oscillator 
is a constant, i.e.,
KE+PE=constant.
The energy oscillates back and forth
between kinetic and potential,
going completely from one
to the other as the system oscillates.

4. Any potential is approximately parabolic in the
neighbourhood of a local minimum.
Expanding V(x) in a Taylor series about minimum
(5)

5. Subtracting from potential given in Eq. (5)
will not effect as adding constant does not effect
force.
As x0 is minimum, so
Also neglecting higher order terms as long as
is small.
So we get from (5),
-----(6)
Above Eq describe simple harmonic oscillation about
x0 with spring constant
Any oscillatory motion is approximately simple
Harmonic as long as amplitude is small.

6. Quantum Harmonic oscillator:
Hamiltonian operator for a particle of mass m which oscillate
with an angular frequency ω under the one dimensional
harmonic potential is
(1)
Two methods to solve:
Power series method in which one solve Schrodinger Eq
Ladder or Algebraic method where we use operator
algebra (Matrix method)

7. Algebraic method: Consider two dimensionless
Hermitian operators
------(2)
We write Hamiltonian of Eq (1) now using above
operators,
------(3)
Introducing two dimensionless, non-hermitian
Operators,
------(4)

8. Note that,
(5)
Where,
----(6)
Hence
(7)
(8)

9. Using (8) in (2),
(9)
Number operator or occupation number operator. It
is Hermitian.
Also,
i (10) So
(11)

10. Energy Eigenvalues:
Hamiltonian is linear in number operator
and thus both commute and thus have simultaneous
Eigenstates.
We write (12)
and
(13)
is energy eigenstates.
Using (9) and (13), we get
(14)
n is positive integer. We will prove it!

11. Physical meaning of :
Note that , Using (9) and (11)
(15)

12. Using (13) and (15), we cna write
(16)
(17)
Note from above,
and are eigenstates of
with eigenvalues
respectively. Action of is to generate
new energy states that lower and higher energy by
one unit

13. = Lowering operator or anihilation operator
= Raising operator or creation operator
Also know as ladder operators.
Now we prove the following:

14. Using the identity
and , we can write
------(18)
Hence,
Combining with ,we get
--(19)
---(20)

15. Thus we can write
(21)
Cn is a constant is normalize for all values of n
(22)
Also
(23)

16. From (22) and (23), we get ------(24)
Note that n cannot be negative
Using (24) in (21), we get
(25)

17. Show that
----(26)
The energy spectrum of harmonc oscillator is
Discrete.
(27)

18. Energy Eigenstates:
Using (26), we can write the eigenvectors as,
(28)

19. Orthogonality and completeness condition:
form complete and orthonormal basis.3
(29)

20. Energy eigenstates in position space:
In position representation, we write the operator
as,
(30)
Where, Dimension of length.
Annihilation and creation operators are written as
(31)
(32)

21. Using (31), in position space is written as,
(33)
------(34)
Where
Solution of (34) is,
(35)

22. Normalizing we get the constant A
----(36)
-----(37)
Ground state wave function is
(38)
Which is a Gaussian function.

23. For 1st excited state

24. For 2nd and 3rd excited state:

25. For nth state, we write in general,
(39)

26. The probability of finding the particle outside the classical
allowed range is non-zero.
Also in odd states the probability of Finding the particle is zero
at the centre.

27. When n is very large (say 100) then the quantum picture
resemble the classical case.

28. Matrix representation of various operators:

32. Expectation values of operators and Heisenberg Principle:

33. From above two equations we conclude that the
expectation value of potential energy is equal to
the expectation value of K.E. And both are equal
to the half of total energy.

34. Heisenberg Uncertainty relation: