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(a) For a function ƒ(x) that can be expanded in a Taylor series, show that f(x + xo) = ei Pxo/n f(x) (where xo is any constant distance). For this reason, §/ħ is called the genera- tor of translations in space. Note: The exponential of an operator is defined by the power series expansion: e =1+ Ô + (1/2)Q² + (1/3!)ĝ³ +... .....

(a) For a function ƒ(x) that can be expanded in a Taylor series, show that f(x + xo) = ei Pxo/n f(x) (where xo is any constant distance). For this reason, §/ħ is called the genera- tor of translations in space. Note: The exponential of an operator is defined by the power series expansion: e =1+ Ô + (1/2)Q² + (1/3!)ĝ³ +... .....

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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*Problem 3.39 (a) For a function ƒ(x) that can be expanded in a Taylor series, show that f(x + xo) = e' Pxo/h ƒ (x) %3D (where xo is any constant distance). For this reason, §/ħ is called the genera- tor of translations in space. Note: The exponential of an operator is defined by the power series expansion: eº =1+ Ô + (1/2)Q² +(1/3!)ĝ³ + .... (b) If ¥(x, 1) satisfies the (time-dependent) Schrödinger equation, show that ¥(x,t + to) = e (where to is any constant tìme); –Ĥ /ħ is called the generator of translations in time. :) Show that the expectation value of a dynamical variable Q(x, p, t), at time t + to, can be written34 (Q)+» = (¥(x, t)le'Hta/h Ô (f. §, 1 + to)e¯iim/^|¥(x,1)). Use this to recover Equation 3.71. Hint: Let to = dt, and expand to first order in dt. %3D
Transcribed Image Text:*Problem 3.39 (a) For a function ƒ(x) that can be expanded in a Taylor series, show that f(x + xo) = e' Pxo/h ƒ (x) %3D (where xo is any constant distance). For this reason, §/ħ is called the genera- tor of translations in space. Note: The exponential of an operator is defined by the power series expansion: eº =1+ Ô + (1/2)Q² +(1/3!)ĝ³ + .... (b) If ¥(x, 1) satisfies the (time-dependent) Schrödinger equation, show that ¥(x,t + to) = e (where to is any constant tìme); –Ĥ /ħ is called the generator of translations in time. :) Show that the expectation value of a dynamical variable Q(x, p, t), at time t + to, can be written34 (Q)+» = (¥(x, t)le'Hta/h Ô (f. §, 1 + to)e¯iim/^|¥(x,1)). Use this to recover Equation 3.71. Hint: Let to = dt, and expand to first order in dt. %3D
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