8. In class, I mentioned that the uncertainty principal \"does things\". Here is

Question

8. In class, I mentioned that the uncertainty principal "does things". Here is what I mean: We deco mposed the particle in a box wavefunctions shown below (A & B) into momentum wave eigenstates (i.e. ek) as shown below: 90% 2% 1% 190 1% 1% Recall that, as the "A" state has more uncertainty in position, it can be decomposed into just a few momentum waves. However, the "B" state requires more momentum waves, perhaps ~100 of them. Also look at the handout for a definitive example that was done with computer analysis. a) How does the energy of the momentum waves change? Better yet, just tell me which momentum wave below has more energy and why: VS. b) Given your answer in pt. a, which of the two wavefunctions (the delocalized state "A" or more localized state "B") have more kinetic energy and why? Hint: I have made up a table of components l5 that have respective energies of 1 J->5 J, and the percent that each contributes to states A and B. Use these data to find the average values of energies for states A and B, which should give you some insight into how to answer this problem. Energy | %A | %B 90% 30% 5% | 20% 20% 090 | 20% 090 | 10% 5% 4 This should help you understand how increasing the percent of higher energy states will affect the total energy

Explanation / Answer

The wavefunction in the right has more energy and momentum because it is localized or uncertainty in position is very low, hence its momentum uncertainty is too high. As the wave moves from delocalized state to localized state, its energy increases, and uncertainty also.

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