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1. The problem statement, all variables and given/known data

Use the propagator equation for a free particle

[tex]U(t) = \exp\left(\frac{i}{\hbar}\left(\frac{\hbar^2t}{2m}\frac{d^2}{dx^2}\right)\right) = \sum_{n=0}^{\infty}\frac{1}{n!}\rleft(\frac{i\hbar t}{2m}\right)\frac{2^{2n}}{dx^{2n}}[/tex]

The initial state of the wave packet is

[tex]\psi(x',0)= \frac{exp(-x^2/2)}{(\pi)^{1/4}}[/tex]

Find psi(x,t).

2. Relevant equations

Hint 1: Express the initial wave function as a power series:

[tex]\psi(x',0) = (\pi)^{-1/4} \sum_{n = 0}^{\infty}{\frac{(-1)^nx^{2n}}{n!(2)^n}}[/tex]

Hint 2: Find the action of a few terms

[tex] 1, \left( \frac{i\hbart}{2m}\right) \frac{d^2}{dx^2}, \frac{1}{2!}\left( \frac{i\hbar t}{2m} \frac{d^2}{dx^2}\right)^2 [/tex]

3. The attempt at a solution

I am stuck on the second hint. How do you find the action when you do not have a Lagrangian?

Please do not solve the entire thing--just help me with this hint. Thanks.

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# Homework Help: Another Way to do the Gaussian Problem

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