Question about Gaussian wave packet

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The discussion centers on solving a problem related to Gaussian wave packets, where the user believes they have successfully addressed part (a) of the homework but struggles with a complex integral. Concerns are raised about the accuracy of the calculations, particularly regarding the completion of the square in the exponential factor. Participants encourage careful verification of the work done so far. The complexity of the function is highlighted as a significant challenge in obtaining the correct solution. Overall, the thread emphasizes the importance of meticulous checking in mathematical problem-solving.
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Homework Statement


http://cafeptthumb2.phinf.naver.net/MjAxNzAzMTVfMjcw/MDAxNDg5NTY5MTEwOTk5.FfccOAaDqBZbb9wXwuUWda8EN1aVvzMdcXO0GvaWwTsg.EjdATfmPtj-DteqAXFu9urHMslV8JTeridwC4ZNoJ68g.PNG.nikel92/asfasf.PNG?type=w740

Homework Equations

The Attempt at a Solution


http://cafeptthumb2.phinf.naver.net/MjAxNzAzMTVfMTk5/MDAxNDg5NTY5MTExNTIw.4LJ_551zal_jTeWLSObHiqdLPvaKKsKjrYrDzjkqVrUg.20VhdFGF31Bvi_la0HWc-BCkAFfafevKhQCMn-EG20gg.JPEG.nikel92/KakaoTalk_20170315_181019347.jpg?type=w740

I think i solved the (a), but i can't calculate the integral because the function is to complex.
 
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check your work..., as I work thru this, I am getting something different, be careful with completing the square

So far so good...check your exponential factor
 
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The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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