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Path integral in quantum mechanics

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Path integral in quantum mechanicsbased on S-6Consider nonrelativistic quantum mechanics of one particle in one dimension with the hamiltonian:P and Q obey:Probability amplitude for the particle to start at q’ at time t’ and end up at position q’’ at time t’’ iswhere and are eigenstates of the position operator Q. In the Heisenberg picture:and we can define instantaneous eigenstates:Probability amplitude is then:62=to evaluate the transition amplitude:let’s divide the time interval T = t’’ - t’ into N+1 equal piecesinsert N complete sets of position eigenstates63let’s look at one piece first:Campbell-Baker-Hausdorf formulacomplete set of momentum states64=to evaluate the transition amplitude:let’s divide the time interval T = t’’ - t’ into N+1 equal piecesinsert N complete sets of position eigenstates-important for general form of hamiltonianswith terms containing both P and Qin our case, it doesn’t make any differencewe find:65taking the limit we get:should be understood as integration over all paths in phase space that start at and end at (with arbitrary initial and final momenta)In simple cases when hamiltonian is at most quadratic in momenta, the integral over p is gaussian and can be easily calculated:prefactor can be absorbed into the definition of measure lagrangian66 for p and plugging the solution back to taking the limit we get:should be understood as integration over all paths in phase space that start at and end at (with arbitrary initial and final momenta)In simple cases when hamiltonian is at most quadratic in momenta:where is calculated by finding the stationary point of the p-integral by solving:67What is it good for?Consider e.g.:the result can be simply written using path integral as:Similarly:Time-ordered products appeared in LSZ formula for scattering amplitudes!time ordering is crucial!68Functional derivatives:they are defined to satisfy all the usual rules of derivatives (product rule, ...)“continuous generalization” of Consider modifying hamiltonian to:Dirac delta functionThen we have:And we find, e.g.:69after we bring down as many qs and ps as we want we can set more examples:and return to the original hamiltonian:70Finally, we want both initial and final states to be ground states and take the limits and :looks complicated, we will use the following trick instead:is the ground-state wave functionis wave function of n-th stateeigenstate of Hcorresponding eigenvaluelet’s replace with and take the limit :every state except the ground state is multiplied by a vanishing factor!71thus we have:Similarly, for the replacement picks up the ground state as the final state in the limit .we can integrate over q’ and q’’ which leads to a constant factor that can be absorbed into the normalization of the path integral.Thus, with the replacement we don’t have to care about the boundary conditions and we have:72Adding perturbations:we can simply write (suppressing the ): Finally, if perturbing hamiltonian depends only on q, and we want to calculate only time-ordered products of Qs, and if H is no more than quadratic in P and if the term quadratic in P does not involve Q, then the equation above can be written as:73equivalent toPath integral for harmonic oscillator based on S-7Consider a harmonic oscillator:ground state to ground state transition amplitude is:thus going to lagrangian formulation (integrating over p) we get:external force74and setting for simplicity, we getusing Furier-transformed variables:E!= −Eand thus:75it is convenient to change integration variables:then we get:a shift by a constantand the transition amplitude is:76But since, if there is no external force, a system in its ground state remain in its ground state.thus we have:or, in terms of time-dependent variables:where:using inverse Fourier transformation77Comment: is a Green’s function for the equation of motion of the harmonic oscillator:you can evaluate it explicitly, treating the integral as a contour integral in the complex E-plane and using the residue theorem. Make sure you are careful about closing the contour in the correct half-plane for t > t’ and t < t’ and that you pick up the correct pole.you should find:78Let’s calculate “correlation functions” of Q operators:for harmonic oscillator we find:For odd number of Qs there is always one f(t) left-over and the result is 0!79For even number of Qs we pair up Qs in all possible ways:in general:We can now easily generalized these results to a free field theory...80Path integral for free field theory based on S-8Hamiltonian density of a free field theory:similar to the hamiltonian of the harmonic oscillatordictionary between QM and QFT:classical fieldoperator fieldclassical sourcewe repeat everything we did for the path integral in QM but now for fields;we divide space and time into small segments; take a field in each segment to be constant; the differences between fields in neighboring segments become derivatives; use the trick: multiplying by is equivalent to replacing with which we often don’t write explicitly; ... eventually we can integrate over “momenta” and 81obtain path integral (functional integral) for our free field theory:path in the space of field configurationsComments:lagrangian seems to be more fundamental specification of a quantum field theorylagrangian is manifestly Lorentz invariant and all the symmetries of a lagrangian are preserved by path integral82to evaluate we can closely follow the procedure we did for the harmonic oscillator:Fourier transformation:83change of integration variables:a shift by a constant84But 85Thus we have:wherewe used inverse Fourier transformation to go back to position-functionsis the Feynman propagator, a Green’s function for the Klien-Gordon equation:integral over zero’s component can be calculated explicitly by completing the contour and using the residue theorem, the three momentum integral can be calculated in terms of Bessel functions86Now we can calculate correlation functions:we find:For odd number of there is always one J left-over and the result is 0!87For even number of we pair up in all possible ways:in general:Wick’s


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