Feynman Path Integrals in Quantum Mechanics

can be a solution for the kernel as well. From periodicity follows the condition A n+1 = e iδ A n ,δ ∈ R. With A 0 = 1 we get A n = e inδ and can then write:∞∑K(ϕ 2 , t 2 ; ϕ 1 , t 1 ) = A n K n (ϕ 2 , t 2 ; ϕ 1 , t 1 ), (39)n=−∞where K n (ϕ 2 , t 2 ; ϕ 1 , t 1 ) is the kernel for all paths with n loops. For each K n we can carrythe Lagrangian 13 from S 1 to R and, for a free particle, use 14 (11). Inserting this in (39)we obtain the kernel for a free particle moving on a circle:K(ϕ 2 , t 2 ; ϕ 1 , t 1 ) =∞∑n=−∞(I2πi(t 2 − t 1 )7 Statistical Mechanics) 1/2exp[inδ +]iI2(t 2 − t 1 ) ((ϕ 2 − ϕ 1 ) − 2nπ) 2In this last section we just want to see, how Statistical Mechanics and path integralsin Quantum Mechanics are related. The central object in Statistical Mechanics is thepartition function. It is defined by(40)Z := ∑ je −βEj= ∑ j〈j|e −βĤ|j〉 = Tre −βĤ, (41)where β = 1/k B T and E j is the energy of the state |j〉. Recall the alternative definitionfor the kernel: K(x 2 , t 2 ; x 1 , t 1 ) = 〈x 2 |e −iĤ(t2−t1)/¯h |x 1 〉. Let’s see what happens, if wemake the substitution (t 2 − t 1 ) := −iβ¯h, where β is real:K(x 2 , −iβ¯h; x 1 , 0) = 〈x 2 |e −iĤ(−iβ¯h)/¯h |x 1 〉 = 〈x 2 |e −βĤ ∑ j|j〉〈j||x 1 〉 (42)= ∑ je −βEj 〈x 2 |j〉〈j|x 1 〉 = ∑ je −βEj 〈j|x 1 〉〈x 2 |j〉 (43)where we used the completenessrelation ∑ j |j〉〈j| = 1. Putting x = x 1 = x 2 and integratingover x, we get:∫K(x, −iβ¯h; x, 0)dx = ∑ ∫e −βEj 〈j| dx|x〉〈x| |j〉 = Z (44)j } {{ }=1This equation tells us how the propagator evaluated at negative imaginary time isrelated to the partition function! This can be of practical use: It gives us a tool to easilycalculate the partition function of a physical system of which we know the path integral(kernel).8 SummaryThis was a short introduction in the path integrals in Quantum Mechanics. I hope it gavea vague overview over the field even if the work is not complete at all. Especially thetreatment of measurements and operators in the path integral formalism was completelyleft out. But still the basic ideas and some applications were presented. In quantum fieldtheory path integrals will play an even more important role and I hope (for myself) thatthe acquired knowledge will be of use! In the last section it was shown how different fieldsof physics, such as statistical mechanics and Quantum Mechanics, can be linked by pathintegrals. A similar relation can be made between statistical field theory and quantumfield theory.13 We can do that, since we can define a smooth mapping from R to S 1 . For details consult [5]!14 Since L = I ˙ϕ 2 /2, we just set m = I.8