1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.7
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Eigen values and Eigen vectors:
Given any dynamical variable A (represented by operator), how we find and characterize those particular states of the system in which A may have a definite value ?
If we measure A in large number of system which are identically prepared and each is represent by same state ψ, under what conditions all these systems will yield
same sharp value?
What kind of state of system corresponds to a probability
distribution of the value of A i.e. peaked at a and has no spread?

3. If we have some such state, then the standard deviation
of A in this state (also called determinate state) would be
zero. So
(1)
Note that we take <A> = a in 1st line because every
measurement of dynamical observable gives us the value a
and therefore the average will also be a.
Since is hermitian and thus is also hermitian
Therefore we moved over to 1st term in the inner product.

4. Only function whose inner product with itself vanishes is 0.
Thus from Eq. (1), we writ Or
(2)
The state vector is called the eigenvector and a is called
the eigenvalue.
Eq. (2) is known as eigenvalue equation or eigenvalue
problem.

5. Eigenvalue probelm for unit operator with eigenvalue 1
Note the following:
Thus,

6. Note: Eigenvalue is a number not an operator or function.
The collection of all Eigen values of an operator is called
the spectrum.
If two or more linearly independent Eigen functions share
same Eigen value then the spectrum is called degenerate.

7. Theorem 1: For a Hermitian operator all of its eigenvalues
are real and the eigenvector corresponding to different
Eigenvalues are orthogonal.
Proof: Note the following two Eqns
(1)
(2)

8. Subtract (2) from (1), using fact that is hermitian operator,
We get
We have now two cases

9. Eigenvectors of non-degenerate Hermitian operator are
unique and form orthonormal basis.
These eigenvectors are linearly independent.
Theorem: The Eigenvectors corresponding to equal
eigenvalues of a degenerate hermitian operator are not unique
and also not necessarily orthogonal.
However we can construct a orthogonal set of eigenvectors for
degenerate Hermitian operator using the Schmidt
orthonormalization procedure.

10. Theorem:If two hermitian operators and commute
and if has no degenerate eigenvalue, then each eigenvector
of is also an eigenvector of In addition, we
Can construct a common orthonormal basis that is made of
Joint eigenvectors of and
Proof: is hermitian operator with no degenerate
Eigenvalue.
For each eigenvalue of there corresponds one
eigenvector

11. Above eq shows that is an eigenvector of
with Eigenvalue an . But if is only linearly eigenvector
of with eigenvalue an, then must be differ from
by a constant. So we can write
So above Eq show that is also eigenvector of .

12. Let an is degenerate eigenvalue of
will also be eigenvector of operator A with
Eigenvalue an.
But is not necessarily eigenvector of .
Explain!

13. Theorem: The eigenvalues of a unitary operator are complex
numbers of moduli equal to one. The eigenvectors of a unitary
operator that has no degenrate eigenvalues are mutually
Orthogonal.
Proof:

15. Theorem:
The eigenvalues of an anti-Hermitian operator
are either purely imagninary or equal to zero.
The eigenstates of a Hermititian operator define
a complete set of mutually orthonormal basis states.
The operator is diagonal in this Eigenbasis with its diagonal
elements equal to the eigenvalues. The basis set is
unique if the operator has no degenerate eigenvalues and
not unique (in fact it is infinite) if there is any degeneracy.

16. Unitary Transformations:
How ket, bra and operators transform under unitary
Transformation?
Ket and bra
------(1)
Operator: Let
Transform to (2)
Using (1), =

17. Multiplying both sides by

18. Properties of unitary transformations:
If is hermitan then the transformed operator
The Eigenvalues of and are same i.e.
Proof:

19. 3. Commutators which are equal to complex numbers
remain unchanged under unitary transformations
4. Complex number remain unchanged
Under unitary transformations.

20. Let , then from last Eq, we can write the scalar
Products invariant under unitary transformations
Norm of state is conserved
Verify
Discuss finite and infinite unitary transformations.

21. Infinitesimal unitary transformation
Consider the transformation
Generator of
transformation
Infinitely small
Real parameter
will be unitary if is Hermitian and
is real:

22. Transformation of state vector under above
Where
For operator
Note:

23. Finite unitary transformation
Finite unitary transformation can be generated
by successive infinitesimal unitary
Transformation
N is integer and α is some finite parameter.
We write

24. Transformation of operator
where we used
Note: