We have used stationary phase to figure out the time dependence of the position of peaks...

 

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We have used stationary phase to figure out the time dependence of the position of peaks in wavepackets constructed from integral representations. More generally, the stationary phase approximation can help get the value of the integral itself. Consider the integral of a Gaussian peaked at x = 2 against a phase factor: ƒ(X) = 1₁ ²i(x,x), 6(x,x) 50(x-¹), AER. dx e-100(x-2)² = We want to confirm that f(A)| peaks at a value A selected by stationary phase and get the value of f(x). (a) What is the width A at half-maximum for the gaussian? In other words, what is the largest A for which for all x in x-2 ≤A the gaussian is larger than half-maximum? If you had to do the integral numerically, would it be safe to integrate from 1 to 3? Explain. (b) Use stationary phase to find the critical value A of A for which f(x) would have the largest magnitude. For A, write (A., x) as a Taylor expansion around x = 2 up to and including terms quadratic in (x - 2). (c) What is the excursion of the phase (A.,x) for x2 <A? Your result, expressed in units of 7, should imply that it is a decent approximation to ignore the phase variation at the critical A. Do so and then perform the resulting integral analytically. The answer is a complex number. Write your answer in terms of a phase times the magnitude. (d) Perform the integral analytically using the quadratic approximation for the phase. Write your answer in terms of a phase times the magnitude. (e) Perform the integral numerically as a function of A for the interval λ = [0, 1]. Plot the absolute value f(A)]. What is the value of f(A) for the critical X? Compare with your previous estimates. What is the value of A that leads to the largest f(x)? We have used stationary phase to figure out the time dependence of the position of peaks in wavepackets constructed from integral representations. More generally, the stationary phase approximation can help get the value of the integral itself. Consider the integral of a Gaussian peaked at x = 2 against a phase factor: ƒ(X) = 1₁ ²i(x,x), 6(x,x) 50(x-¹), AER. dx e-100(x-2)² = We want to confirm that f(A)| peaks at a value A selected by stationary phase and get the value of f(x). (a) What is the width A at half-maximum for the gaussian? In other words, what is the largest A for which for all x in x-2 ≤A the gaussian is larger than half-maximum? If you had to do the integral numerically, would it be safe to integrate from 1 to 3? Explain. (b) Use stationary phase to find the critical value A of A for which f(x) would have the largest magnitude. For A, write (A., x) as a Taylor expansion around x = 2 up to and including terms quadratic in (x - 2). (c) What is the excursion of the phase (A.,x) for x2 <A? Your result, expressed in units of 7, should imply that it is a decent approximation to ignore the phase variation at the critical A. Do so and then perform the resulting integral analytically. The answer is a complex number. Write your answer in terms of a phase times the magnitude. (d) Perform the integral analytically using the quadratic approximation for the phase. Write your answer in terms of a phase times the magnitude. (e) Perform the integral numerically as a function of A for the interval λ = [0, 1]. Plot the absolute value f(A)]. What is the value of f(A) for the critical X? Compare with your previous estimates. What is the value of A that leads to the largest f(x)?