# 2.4. The path integral: from quantum mechanics to classical mechanics¶

In the previous section we have considered how we can arrive from classical mechanics at the Schrödinger equation. This formalism can be generalised in the sense that for each system for which we can write down a Lagrangian, we have a way to find a quantum formulation in terms of the path integral. Whether a Schrödinger-like equation can be found is not sure: sometimes we run into problems which are beyond the scope of these notes. In this section we assume that we have a system described by some Hamiltonian and show that the time-evolution operator has the form of a path integral as found in the previous section.

The starting point is the time evolution operator, or propagator, which,
for a time-independent Hamiltonian, takes the form
The matrix element is difficult
to evaluate -- the reason is that the Hamiltonian which, for a particle
in one dimension, takes the form
is the sum
of two *noncommuting* operators. Although it is possible to evaluate the
exponents of the separate terms occurring in the Hamiltonian, the
exponent of the sum involves an infinite series of increasingly
complicated commutators. For any two noncommuting operators
and we have
This is the
so-called Campbell--Baker--Hausdorff (CBH) formula. The cumbersome
commutators occurring on the right can only be neglected if the
operators and are small in some sense. We can try to arrive at
an expression involving small commutators by applying the time slicing
procedure of the previous section:
Note that no CBH commutators occur because commutes
with itself.

Having this, we can rewrite the propagator as (we omit the hat from the
operators)
Now that the operators occurring in the exponents can be made
arbitrarily small by taking very small, we can evaluate the
matrix elements explicitly:
The last matrix element can be evaluated by inserting two unit operators
formulated in terms of integrals over the complete sets
:
We have seen that
.
Realising that the operator
is
*diagonal* in space, we find, after integrating over :
All in all we
have
Note that we
have evaluated matrix elements of operators. The result is expressed
completely in terms of numbers, and we no longer have to bother about
commutation relations. Collecting all terms together we obtain
The expression in the exponent is the
discrete form of the Lagrangian; the integral over all intermediate
values is the sum over all paths. We therefore have shown that the
time evolution operator from to is
equivalent to the sum of the phase factors of all possible paths from
to .

In conclusion, we have seen that the idea that the probability to a find a particle starting off from at at at time is given by a sum of all the phase factors corresponding to all paths from the starting to the end point, gives us the quantum theory. On the other hand, once we have the Hamiltonian of a quantum theory, we can reformulate this as a path integral involving the corresponding Lagrangian.