By I. T. Todorov, D. Ter Haar
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Additional resources for Analytic Properties of Feynman Diagrams in Quantum Field Theory
One unnormalized energy eigenfunction is $, = (2~’ - 32) elcp (-x2/2). Find two other (unnormalized) eigenfunctions which are closest in energy to Ga. a = m$h+1, n+i tiw, ( > n=0,1,2 ,.... a+$, = (7% + 1) q& ) Basic Principles and One-Dimensional Motions 51 AS G+&,=; (z+&) (z-%> (2x3-3x)e-+ 1 =- cc + $ (424 - 12x2 + 3) e-+ 2 ( > =4(2x3-3+-$=(3+1)+,, we have n = 3. Ja = 5 ( > x + 2 (2x3 - 3x) cz2/2 N (2x2 - 1) P2/2, ljzp28+~a=i 2x4 ( > z - $ (2x3 - 3x) e-x2/2 w (424 - 12x2 + 3) cz2/2 ) where the unimportant constants have been omitted.
Berkeley) Problems and Solutions on Electromagnetism 52 Solution: (a) The normalization condition ($(z, O), ti(x, 0)) = I A I2 C (1/2)(m+n)‘2 (6x7 tim) =21A12=1 = IAl2 gives A = l/d, taking A as positive real. (b) The time-dependent wave function is T/(x:, t) = e--ifit’Q(x, = ah> 0) n+l e--i++i) & @). n (c) The probability density is 1$(x, 0 *+I t)l”=C f mn e-i”“‘“-“‘~n(x)~~(x). Note that the time factor exp [-iw(n - m)t] is a function with period A, the maximum period being 27r /w . (d) The expectation value of energy is Noting cO” n=O 1 X n=X-_17 54 Problems and Solutions on Electromagnetism The energy of the particle can be estimated using the uncertainity principle px N fi/2b, where b= JG.
It is sufficient to take time average over one period T = 27r/w. Let (A) and A 48 Problems and Solutions on Electromagnetism denote the time average and ensemble average of an operator A respectively. As vl$)+. Yz211/1) =f rW~z,&~z$&) e+ =f Tw @~a,($$+l) + + 1)) e--iw(n+Wt we have +g&4+2anl n=O J (n+l)(n+2) cos(2wt+6,), where 6, is the phase of a;L+1 a,, . Averaging v over a period, as the second term becomes zero, we get (V) = ; jT0 Vdt= ;rll&,,, (n+;) . n=O Basic Principles and One-Dimensional Motiona 49 On the other hand, and (E) = I?.
Analytic Properties of Feynman Diagrams in Quantum Field Theory by I. T. Todorov, D. Ter Haar