1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.16
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. Symmetries and Conservation Laws
Symmetries leads to conservation law
Time- translation : Energy conservation
Space translation: Linear momentum
Conservation
Rotational symmetry: Angular momentum
conservation

3. Infinitesimal unitary transformations
State vectors and operators transform under
Infinitesimal unitary transformation As
----(1)

4. Where we used fact that is First order Taylor expansion of
----(1)
------(2)
Where we used fact that is
First order Taylor expansion of

5. -----(1)
Note and Taylor expansion of
is ,
We can write
---(2)

6. Using (10), we can write the transformation eq
for Position operator as
-----(3)
Which shows that is generator of
Infinitesimal transformations.

7. Finite Unitary transformations
Finite unitary transformations are generated
From successive infinitesimal transformations.
For example, applying time translation N times
In steps we get
----(1)
In above Hamiltonian is the Generator of
time translation.

8. Applying (1) on , we get
--(4)
Applicaion of will yield
----(5)

9. Transformed position vector can be calculated Using ----(6)
And we have
(7)
Where, we used
Linear momentum in is generator of spatial
translation as can be seen from (5) and (7).

10. Symmetries and Conservation Laws
Hamiltonian of a system under transform as
----(8)
If [ , ] =0 , then [ , ] = 0.
This show that Hamiltonian is invariant under
i.e.,
---(9)

11. Also, if does not depend upon time explicitly,
then
--(10)
is conserved.

12. Conservation of energy and linear momentum
Time translations are generated by time
evolution operator
As discussed earleier, generator of time
Translation is Hamiltonian H.
Thus Hamiltonian H will commute with the
H (generator) and thus H is conserved i.e.,
Energy is conserved.

13. Linear momentum and postion is invariant
Under :
For isolated system ,
Thus, P is cosnerved: