An introduction into the Feynman path integral by Grosche C.

By Grosche C.

During this lecture a quick creation is given into the idea of the Feynman course critical in quantum mechanics. the final formula in Riemann areas can be given in response to the Weyl- ordering prescription, respectively product ordering prescription, within the quantum Hamiltonian. additionally, the idea of space-time adjustments and separation of variables should be defined. As undemanding examples I speak about the standard harmonic oscillator, the radial harmonic oscillator, and the Coulomb capability.

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1038] √ 2 xyt 1 1+t t−α/2 Iα exp − (x + y) 1−t 2 1−t 1−t ∞ 1 tn n! e− 2 (x+y) = (xy)α/2 Ln(α) (x)Ln(α) (y). 78) 2mω Γ( N−l 2 + 1) · N+l+D ¯hr D−2 Γ( 2 ) = l+ D−2 2 mω 2 r ¯h exp − mω 2 (l+ D−2 ) mω 2 r L N −l 2 r . 79) The path integral for the harmonic oscillator suggests a generalization in the index l. This will be very important in further applications. 80) The functional measure corresponds to a potential Vl = h ¯ 2 l(l+1) in the Schr¨odinger 2mr2 equation. Assuming that we can analytically continue in l → λ with ℜ(λ) > −1, then λ2 − 1 we get for an arbitrary potential Vλ (r) = h ¯ 2 2mr24 na¨ıvely inserted into the radial path integral r(t′′ )=r′′ Dr(t) exp r(t′ )=r′ i ¯h t′′ λ2 − 41 m 2 m r˙ − ¯h2 − ω 2 r 2 dt .

Separation of Variables Let us assume that the potential problem V (x) has an exact solution according to x(t′′ )=x′′ i Dx(t) exp ¯h x(t′ )=x′ t′′ t′ m 2 x˙ − V (x) dt = 2 dEλ e− i Eλ T /¯h Ψ∗λ (x′ )Ψλ (x′′ ). 1) Here dEλ denotes a Lebesque-Stieltjes integral to include discrete as well as continuous states. Now we consider the path integral K(z ′′ , z ′ , x′′ , x′ ; T ) z(t′′ )=z ′′ x(t′′ )=x′′ d′ f d (z) = i=1 z(t′ )=z ′ ×exp  i  ¯h t′′ t′  d gi (z)Dzi (t)  m  2 k=1 x(t′ )=x′ d′ Dxk (t) d gi (z)z˙i2 + f 2 (z) i=1 k=1 ′ d N−1  x˙ 2k  −    V (x) + W (z) + ∆W (z)  dt  f 2 (z) d ND m 2 (j) (j) dxk f d (z (j) ) gi (z (j) )dzi = lim N→∞ 2π i ǫ¯ h i=1 j=1 k=1    ′ d i N m (j)   × exp gi (z (j−1) )gi (z (j) )∆2 zi + f (z (j−1) )f (z (j) )  ¯h 2ǫ j=1 i=1 d k=1  (j) ∆2 xk    V (x ) − ǫ 2 (j) + W (z (j) ) + ∆W (z (j) )  .

In particular the (time-dependent) radial harmonic oscillator will be exactly evaluated. 4) The Coulomb potential. 1. The Free Particle For warming up we calculate the path integral for the free particle in an D-dimensional Euclidean space. From the representation K(x′′ , x′ ; T ) x(t′′ )=x′′ im 2¯ h Dx(t) exp = x(t′ )=x′ = lim N→∞ m 2π i ǫ¯h ND 2 N−1 j=1 t′′ x˙ 2 dt t′ ∞ −∞  dx(j) exp  im 2ǫ¯h N j=1  2 x(j) − x(j−1)  . 1) it is obvious that the various integrations separate into a D-dimensional product.

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