Quantum Mechanics for Mathematicians Flashcards
Discuss state of system in classical mechanics. Basic priniple
State of system at instance in time is specified by value of coordinates q in M=R^n and its generalized velocities v in R^n
The basic principle of CM is that, given the state of the system at an initial time t, its motion is determined at all times (past and future)
Motion described by the classical trajectory q(t) = (q1(t), … , qn(t))
Define: Lagrangian system, Lagrangian function
How does this determine the motion of a system?
variation, infinitesmal variation, action functional, principal of least action
This is a more powerful replacement for Newton’s laws.
A Lagrangian system is defined by a real-valued smooth function L:R^n x R^n X R –> R
(q,v,t) –> L(q, v, t) called the Lagrangian function
Consider all paths in R^n connecting points p0 and p1.
A variation Q with fixed ends is just a homotopy (family of paths)
Derivative w.r.t. epsilon is called infinitesmal variation
The action functional S: {Paths} –> R of a Lagrangian system is
S(lambda) = integral_t0^t1 L(q(t), q’(t), t) dt
The motion of the Lagrangian system is determined by the principle of least action: a path q(t) describes the motion of the system <=> it is an extremal of the action functional i.e. dS(q_eps)/deps |eps = 0 = 0
where q_eps is an arbitrary variation of q(t) with fixed end points.
pg 1-3
Discuss/prove how principle of least action leads to the equations of motion.
Thm. (Euler Lagrange EOM) A path q(t) is an extremal of the action functional iff it satisfies the equations of motion EOM:
pg 4-5
Discuss example of N interacting particles in R^3
pg 4 - book
pg 6-7
Define: integral of motion
Example?
aka conserved quantity - is a smooth function I: R^n x R^n –> R s.t. d/dt I(q(t), q’(t)) = 0.
Energy : If Lagrangian does not depend on time, then energy is conserved.
pg 7
book 4-5
Define: symmetry of L
Discuss action of diffeo on L
pg 8
book 5
What is Noether’s Thm? Proof?
Thm. Assume L invariant under 1-parameter group of diffeomorphisms, g_s, s in R. Then the following is an integral of motion:
I(q, q’) = sum_i=1^n dL/dq’_i(vector field)
Pf. Compute
pg 9-10
