T2: Path integrals in QM Flashcards
Define the Feynman kernel W(q’,t’; q,t)
The amplitude that a particle at position q and time t will be at position q’ at time t’.
How can we express the wf Ψ(q’,t’) using the Feynman kernel?
Ψ(q’,t’) = INT dq W(q’,t’;q,t) Ψ(q,t)
Briefly describe how we determine the generic Feynman kernel W(q’,t’; q,t)
Divide the time interval (t,t’) into N points such that t’-t = nε.
Write W(q’,t’; q,t) =⟨q’,t’|q,t⟩ and insert completeness N -1 times
Consider a general braket in Schrodinger and taylor expand exponential out
State the canonical Hamiltonian
p^2/2m + V(q)
State the boundary conditions for the generic Feynman kernel W(q’,t’; q,t).
Why are there none on momentum?
q(t) = q
q(t’) = q’
Heisenberg uncertainty :)
State the canonical form of the Feynman kernel W(q’,t’; q,t)
= N INT [Dq] INT [Dp] exp[iS/ℏ]
State the canonical form of the action S
S = INT_ti ^tf dτ [m/2 (dq/dτ)^2 - V(q)]
Define the Feynman kernel (as a path integral)
A sum over all the paths between (q,t) and (q’,t’) weighted by the phase exp(iS/ℏ)
How does the Feynman kernel change when we pop an operator between the states?
The operator acts on the corresponding state and the corresponding variable pops out in front of the exponential
Define the time ordering T{q(ti)q(tj)}
= Θ(ti - tj)q(ti)q(tj) + Θ(tj - ti)q(tj)q(ti)
Define the time-ordered Green’s function G_F(t1, t2)
= ⟨0|T{ q(t1) q(t2) }|0⟩
Give the integral form of the Green’s function G_F(t1,t2)
N: INT [Dq] q(t1)q(t2) exp(iS/ℏ)
D: INT [Dq] exp(iS/ℏ)
State the Green’s function form of the Lagrangian S
S = INT _-∞^∞ dt L(q, dq/dt)
Define the generating functional Z[J]
(vacuum to vacuum amplitude)
Z[J] = N INT [Dq] exp[iS/ℏ]
S = INT _-∞^∞ dt L(q, dq/dt) - iℏ Jq
What is Z[J=0]
= 1
