QuantumPhysics.dvi

6 Quantum Mechanics Systems

Quantum systems associated with systems of classical mechanics are fundamental. In fact, it

is for these systems that we write the Schr ̈odinger equation; they will also admit a formulation

in terms of functional integrations over all possible paths, to be discussed in 221B. It is useful

to begin with a brief review of Lagrangian and Hamiltonian mechanics.

6.1 Lagrangian mechanics

At a most basic level, we describe systems by the time-evolution of the individual particles

that make up the system. Particlenis characterized in classical mechanics by its position

rn(t) at any given timet. It is a fact of Nature that its basic laws involve equations that are

first or second order in time derivatives, but not higher. Newton’s force lawF=Ma, for

example, is second order in time derivatives. As a result, the initial conditions of a classical

system are the positionsrnand velocitiesr ̇nof each of its constituent particles. The laws of

physics then yield the positions at later times.

The starting point of Lagrangian mechanics is a set of generalized positionsqi with

i= 1,···,N, describing all the degrees of freedom of classical particles. Fornparticles

in 3-dimensional space, for example, we useN = 3ngeneralized position variablesqi. The

associated generalized velocities are denoted by ̇qi=dqi/dt. Under the assumption of at most

second order time derivative evolution equations, Lagrangian mechanics will be completely

specified by a single function,

L(q,q ̇;t) =L(q 1 ,···,qN,q ̇ 1 ,···,q ̇N;t) (6.1)

referred to asthe Lagrangian. The associated Euler-Lagrange equations

d

dt

(

∂L

∂q ̇i

)

−

∂L

∂qi

= 0 i= 1,···,N (6.2)

arise as the solution to a variational principle of the action functional

S[q] =S[q 1 ,···,qN] =

∫t 2

t 1

dtL(q 1 ,···,qN,q ̇ 1 ,···,q ̇N;t) (6.3)

The notationS[q 1 ,···,qN] indicated that S is a functional of the path (q 1 (t),···,qN(t))

spanned fort∈[t 1 ,t 2 ]. To derive the Euler-Lagrange equations from the action, we perform

a variationδqi(t) on the pathqi(t), keeping the end points fixed,

δqi(t 1 ) =δqi(t 2 ) = 0 (6.4)