00:01
Are the wave functions eigen functions of the kinetic energy operator, and we have the kinetic energy operator given to us.
00:10
So our first wave function is e raised to the 2ix.
00:22
So we need this operator to act on this wave function.
00:31
So this is equal to negative h -bar squared over 2m times the second derivative with respect to x, which i wrote as a partial derivative, but it's really a total derivative, but that doesn't, it should not make difference here.
00:56
And we're taking the second derivative of e to the 2ix power.
01:04
So we can break this up a little bit to make this easier.
01:09
So we're going to write this as the derivative with respect to x.
01:16
I'm going back to total derivative notation here.
01:19
And we'll take these derivatives one at a time.
01:29
So when differentiating e to the 2x, e to the 2ix with respect to x, we're left with 2i times e to the 2ix.
01:51
So we can rewrite everything out here.
01:55
So we have negative hbar squared over 2m times the derivative of 2 .m.
02:07
2i, e to the 2i x.
02:14
So now we can take this derivative, which when we differentiate this, we're left with 4 i squared e to the 2ix.
02:27
We know i squared is equal to negative 1.
02:32
So this becomes negative hbar squared over 2m times negative 4e to the 2ix.
02:46
Now what we can do here is we can notice that these negatives will cancel out.
02:55
So we can combine things a bit here, and we can move this 4 to the numerator, and we can cancel it out with the 2 here.
03:08
So we're left with 2h bar squared in the numerator, and an m in the denominator.
03:18
Now, as we can see, the kinetic energy operator acting on the wave function is equal to 2h bar squared over m times.
03:34
E to the 2 ix.
03:38
We can make one simple substitution here.
03:41
What is e to the 2ix? well, that's just our wave function.
03:47
So we can put psi of x here.
03:51
So as we can see, the operator, acting on the wave function, is giving some linear scalar times the wave function.
04:05
So yes, the wave function is an eigenfunction of the kine.
04:30
Kinetic energy operator.
04:42
Now we'll move on to the next one, which is si of x is equal to a cosine 2x over l...