The path integral formulation is an alternative description
of quantum mechanics, equivalent to the traditional Schrödinger equation.
Whereas the latter is based on Hamiltonian mechanics,
the former comes from Lagrangian mechanics.
It expresses the propagatorK
as the following “sum” over all possible paths x(t)
that take the particle from the starting point (x0,t0)
to the destination (xN,tN):
K(xN,tN;x0,t0)=Aallx(t)∑exp(iS[x]/ℏ)
Where A is a normalization constant,
and S[x] is the classical action of the path x(t),
defined as shown below from the system’s Lagrangian L,
and whose minimization would lead to the
Euler-Lagrange equation
of classical Lagrangian mechanics.
Let x˙(t)=dx/dt:
S[x]≡∫t0tNL(x,x˙,τ)dτ
Note that K’s sum gives each path an equal weight,
even unrealistic paths taking bigs detours.
This apparent problem solves itself as follows:
paths close to the classical optimum xc(t)
have an action close to Sc=S[xc],
since S is stationary there.
Meanwhile, for paths far away from xc,
S gives very different values,
which change by a lot if a small change is made to x.
Because S[x] is inside a complex exponential,
paths close to xc therefore add more or less constructively,
while the others add destructively and cancel out.
Consequently, the “quantum path” is still close to xc(t).
An interesting way to think about this is by treating ℏ as a parameter:
as its value decreases, small action changes result in bigger phase differences,
which makes the quantum wavefunction stay closer to xc
for the aforementioned reasons.
In the limit ℏ→0, quantum mechanics simply turns into classical mechanics.
In reality, K’s sum is evaluated as an integral over all paths x(t),
hence this is called the path integral formulation.
The proof that the propagator K’s Schrödinger-picture definition
can be rewritten as such an integral is given below.
Time-slicing derivation
For a time-independent Hamiltonian H^,
we start from the definition of the propagator K,
and divide the time interval tN−t0 into N “slices”
of equal width Δt≡(tN−t0)/N:
Between the exponentials we insert identity operators
∫−∞∞∣x⟩⟨x∣dx,
and define xj≡x(tj) for an arbitrary path x(t),
where tj is the endpoint of the jth slice.
This is equivalent to splitting K
into a product of all slices’ individual propagators:
For sufficiently small time steps Δt (i.e. large N),
we can split the Hamiltonian
into its kinetic and potential terms H^=T^+V^.
Note that this is an approximation,
since T^ and V^ are operators that do not commute,
but it becomes exact in the limit Δt→0:
e−iH^Δt/ℏ≈e−iT^Δt/ℏe−iV^Δt/ℏ
We substitute V^=V(xj), and apply it directly to ∣xj⟩,
such that we can take it out of the inner product as a constant factor:
In order to evaluate the remaining inner product,
we insert the identity operator again,
this time expanded in the momentum basis ∫−∞∞∣p⟩⟨p∣dp,
and use T^=p^2/(2m) to get:
It is worth noting that there are N−1 integrals,
but N factors (−im/2πℏΔt)1/2
i.e. one for each slice.
According to convention, N−1 of those factors
are said to belong to the integrals,
and then the remaining one belongs to the process as a whole.
In the limit Δt→0 (or N→∞),
the sum in the exponent becomes an integral:
Where we have recognized the Lagrangian L=T−V
and hence the action S[x] of the path x(t).
We thus arrive at the following formula for the global propagator K,
known as Feynman’s path integral
or sometimes the configuration space path integral:
K=∫eiS[x]/ℏDx
Where we have introduced the following notation
to indicate an integral over all paths,
because writing the factor and all those integrals can become tedious:
∫Dx≡N→∞lim(2πℏΔt−im)N/2∫⋯∫dx1⋯dxN−1
It is worth stressing that this is simply an abbreviation;
in practice, calculating K in this way
still requires the individual slices to be taken into account.
References
R. Shankar,
Principles of quantum mechanics, 2nd edition,
Springer.
L.E. Ballentine,
Quantum mechanics: a modern development, 2nd edition,
World Scientific.