MIT 8.04 Quantum Physics I, Spring 2016 View the complete course: ocw.mit.edu/8-04S16 Instructor: Barton Zwiebach License: Creative Commons BY-NC-SA More information at ocw.mit.edu/terms More courses at ocw.mit.edu
It needs to be included precisely because it works. As he showed from 3:25 to 5:00 when you take the second derivative you get something akin to satisfying the differential equation. Because cu^k exp(u^2/2) is **more general** than aexp(u^2/2)+bexp(u^2/2), it needs to be part of the general solution. Consider a second order equation like d^2 x + 5 d x + 6 x = 0. This is solved by A exp(-2t). But this does not qualify as a general solution because it does not capture the entire space of functions that satisfy the equation. The solution should be A exp(-2t) + B exp(-3t), which also satisfies the equation. The main difference is that the second form captures the entire solution space, whereas the first does not. That's my (rambling) attempt at explaining why the u^k is important.
