Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.16 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

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

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

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

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

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

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.

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

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).

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)

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

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.

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