1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.13
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Complete set of commutating observables:
Compatible observables: Two observables are compatible if
corresponding operators commute
If operator do not commute then we call them non-compatible.
If two operators do not commute then the corresponding
observables cannot be measured simultaneously. The order
in which they are measured matters.
However if they commute then order of measurement does
not matter.

3. If two observables are compatible then their corresponding operators possesses a set of Common or simultaneous eigenstates (both for degenerate and non-degenerate case).
Proof: Let is non-degenerate state
is joint eigenstate of operators and

4. Let simulatenous eigenstates are denoted by
Theorem can be generalized to many set of mutually
compatible observables. And these posses joint eigenstates
Completeness and orthonormality condition are,

5. If operator has degenerate eigenvalues then the
specification of eigenvalue does not uniquely determine the
state of system.
For example: Free particle in one dimension, eigenvalues
of hamiltonian are double degenerate
Eigen values
Eigen functions: and
For free particle the momentum eigen value is and
corresponding eigen function is
Merely specifying the energy of the particle does not uniquely
specify the state of system. However specifying the
momentum help in determining the state of system
completely.

6. Among the degenerate eigenstates of only a subset will
be eigenstates of B.
Thus the set of states that are joint Eigenstates of both
operators is not complete.
To resolve degeneracy we introduce a third operator C which
commute with A and B. We can then construct a set of
Joint eigenstates of A, B and C which is complete.
If degeneracy still eixit we introduce a new operator.

7. Continuing in this way, we will finally obtain complete set
of commutating operators (CSCO).
A set of hermitian operators is called CSCO, if the operators
mutually commute and if the set of their common eigenstates
is complete and not degenerate (i.e. Unique)

8. Heisenberg’s Uncertainty principle:
In classical mechanics the particles position and momentum can be
determined precisely if we know the initial conditions.
However in wave mechanics we describe the particles by the wave
packet. The particle can be located anywhere with this wave packet.
The position of the particle becomes more and more definite as the size
Of the wave packet becomes smaller and smaller (see fig)
However in such situation the
average value of the wavelength
becomes uncertain
because in small wave packet
we have small number of
wavelengths present.
Since wavelength is related to the momentum and thus
momentum also becomes uncertain.

9. However if we consider the opposite case, i.e. the size of the wave
packet is larger then the wavelength can be determined precisely as
we have more no of waves inside the wave packet
and hence the momentum of the particle can also be determined
precisely.
However In this situation the position of the particle becomes
uncertain.
Thus we conclude that,
It is impossible to measure precisely and simultaneously both the
position and momentum of the particle to any desired degree of
accuracy.
If Δx is the uncertainty in measurement of the position and Δp is the
uncertainty in measurement of momentum of the particle then
(1)

10. A wave with well defined wavelength but ill
defined position
A wave with well defined position but ill defined wavelength
Uncertainties Δx and Δp are known as standard
Deviations.
Standard Deviation

11. Consider two hermitian operators Expectation values ------(1)
Derivation of general uncertainty relation:
Consider two hermitian operators
Expectation values
------(1)
Now consider operators
-----(2)
We have
(3)

12. We write the expectation values
(4)
Where,
(5)
The uncertainties are defined by
(6)

13. We operate the operators and on state
(7)
Using the Schwartz inequality we get
(8)
(9)

14. Using (8) and (9), we get
----(10)
We can write
(11)
Where, we used
antihermitian
Hermitian
Expectation values of Hermitian operator are real and
That the anti-hermitian imaginary

15. Expectation value is sum of real
and imaginary part Hence,
(12)
Last quantity is real positive, so we can write
-----(13)
Using (10) and (13), we get
-----(14)

16. Eq. (14) further written as
----(15)
Which is uncertainty relation.
For x- component of postion and momentum operator
We can write,