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Minute No. 2, the Quantum Mechanics
Quantum systems Mes71 are described by the wave function
(complex function of coordinates and time ),
and by the Hamilton operator that can be obtained
from the classical Hamiltonian by a procedure called quantization.
We define operators that correspond to each classical object,
e.g., the classical coordinates and momenta are quantized as,

(7) 
Then, the Hamilton operator is, more or less, obtained by
inserting these operators into the classical Hamiltonian, i.e.,

(8) 
This is not an exact science, because the function of operators
cannot be uniquely defined for a given function of variables;
one has to also define the order in which the operators act.
Well, in fact the quantization provides us only with general rules
on how to start the quantum mechanics based on our knowledge of the
classical mechanics. One can also subscribe to the point of view
that we must axiomatically define the quantum system by specifying
its Hamilton operator. Once this is done, the time evolution
of the system (of its wave function) is given by the Schrödinger equation,

(9) 
This leads to a set coupled differential equations that can be, in
principle, solved once the initial conditions =0) are
known. One thus obtains the complete past and future history of the
system . The rest is just a technicality
;) of how to solve differential equations.
Quantum mechanics also adds a pivotal element to our understanding
of how our world works, namely, the probabilistic interpretation.
In classical mechanics, once our EulerLagrange equations give us
the set of coordinates at time , the experiment performed
at time to find the system at point yields one possible
answer: the system is there. In quantum mechanics, the same experiment
yields the answer that the system is within volume around
with probability
and the answer that
it is not there, with probability
.
Hélas, it seems that the world is just like that, nothing is
certain any more. However, at least the probabilities of obtaining
given experimental answers can be rigorously calculated.
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Up: Quantum Field Theory in
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Jacek Dobaczewski
20030127