what is the difference between classical mechanics and quantum mechanics??
other than that classical mechanics dealing with macroscpic particles while quantum mechanics with microscopic particles
The Lie algebra of the Galilean group is the Lie algebra of symmetry of both CM & QM. Therefore, one can say that CM & QM are two different realizations of a single mathematical formalism. To see what I mean, consider the Lie bracket;
\left( a , b \right) = - \left( b , a \right)
which satisfies the conditions
\left( ab , c \right) = a \left( b , c \right) + \left( a , c \right) b
<br />
\left( a , \left( b , c \right) \right) + \left( c , \left( a , b \right) \right) + \left( b , \left( c , a \right) \right) = 0<br />
Now assume that
1) under the time translation t \rightarrow t + \delta t, the physical state, P(t), changes according to;
\delta P(t) = - \left( H , P(t) \right) \delta t
where H = p^{2} + V(q) is the Hamiltonian. In particular
\delta H = ( H , H ) \delta t = 0
gives the Lie algebra of the 1-parameter group of time translation. We also have
\dot{q}_{i} = ( q_{i} , H )
\dot{p}_{i} = ( p_{i} , H )
2) space translations; q_{i} \rightarrow q_{i} + \delta q_{i}, induce the following change;
\delta P(t) = - \left( p_{i} , P(t) \right) \delta q_{i}
Hence
\delta p_{j} = - ( p_{i} , p_{j} ) \delta q_{i}
assuming that \delta p_{i} and \delta q_{i} are independent, we find the algebra
( p_{i} , p_{j} ) = 0 \ \ \ (1)
Letting P(t) = q_{j}, we find the bracket
( q_{i} , p_{j} ) = \delta_{ij} \ \ \ (2)
3) for the "translations" p_{i} \rightarrow p_{i} + \delta p_{i}, we take
\delta P(t) = - \left( q_{i} , P(t) \right) \delta p_{i}
which gives
( q_{i} , q_{j} ) = 0 \ \ \ (3)
No more assumptions needed to complete Galilean invariance, since the fundamental brackets (1), (2) and (3) together with the definition of the angular momentum, J_{i} = \epsilon_{ijk}q_{j}p_{k}, are sufficient to derive the Lie algebra of the rotation group so(3);
( J_{i} , J_{j} ) = \epsilon_{ijk} J_{k}
Therefore, rotational invariance of the system is guaranteed once the dynamical variabes satisfy the fundamental brackets.
Now, if the Lie bracket is given by (Poisson's):
<br />
( a , b ) = \frac{\partial a}{\partial q_{i}} \frac{\partial b}{\partial p^{i}} - \frac{\partial a}{\partial p^{i}} \frac{\partial b}{\partial q_{i}} \equiv \{ a , b \}<br />
the above formalism gives you CM. But, when Lie bracket is realized by commutator;
( a , b ) = i(ba - ab) \equiv - i [ a , b ]
then, you are talking QM: since pq \ne qp, the dynamical variables cann't be ordinary numbers. Therefore, in QM, these variables are represented by operators and matrices. This means that the space of all possible physical states in QM is different from that in CM. The dynamical equations in both theories are linear. This leads to the fact that (in both theories) superposition of physically possible states is also physically possible. However, in QM ( again because of [q,p]=i ), the principle of superposition has a classically-weird meaning; While it is impossible for a classical (macro) object to be in two places at the same time, a quamtum (micro) object seems to be able to accomplish such classically-impossible task.
regards
sam
