00:01
In this prime, you're looking at three functions, an associated operator, and you're being asked, is the function an eigen function of that operator? and if it is, what is the eigenvalue? now, before we get into the actual problems, if you have an eigenfunction, you have some operator acting on the function f, whatever that is, when you evaluate that, that can be written as some, i can value, some number times, the original function.
00:35
So, all there is to it.
00:36
So we calculate the left and see if we can write it in terms of a number times the original function.
00:42
If we can, f is an eigenfunction.
00:45
If we cannot, then it's not.
00:48
And if it is, then whatever that number is, is what we call the eigenvalue.
00:53
So let's look at our first part a.
00:56
We have a complex exponential here, i being the square root of negative 1.
01:02
Operator is the second derivative respect to x, respect to x so let's take the first derivative partial f respect to x with exponential it's easy we're always going to have the exponential back but we've got to take the the derivative of the argument so that will become minus 3i e minus i i 3x goes 2y now the second derivative we've got minus 3 i and then we just what we just wrote here is what we get again this really is minus 3 i, f.
01:51
So it's minus three i times the partial of f respect to x, which is this.
01:57
Minus three i, e minus i, three x, two y.
02:09
Minus three times minus three is nine.
02:12
I squared those minus one.
02:13
So it becomes minus nine, e minus i, three x, is two y.
02:24
So this is minus nine times f.
02:30
So f is, f is eigenfunction with eigenvalue minus 9.
02:58
So that's the first, part a.
03:01
Part b, we got a square root.
03:03
And here's our operator, a little more complicated, but not anything horrible.
03:10
All right, so we've got to take the derivative of the square root.
03:13
So first we bring down the power, one -half, and then this will become the power one -half minus one.
03:24
So minus one half.
03:30
So this is the one half minus one.
03:33
And then we've got to take the derivative of the inside.
03:41
Now the two is cancel out.
03:44
X cancels out.
03:48
So we get x first power.
03:52
So it's going to be x squared plus y squared.
03:57
First power minus the half power.
04:04
So that's the square root of x squared plus y squared, which is 1 times the square root of x squared.
04:15
Well, i don't need, let me not even write that again.
04:18
This is 1.
04:27
Let me just fix this.
04:32
Just 1 times f.
04:34
Just try it like that.
04:37
So, f again is eigen function with eigenvalue of 1.
05:00
Have all the x and the 2 canceled out.
05:03
So we've just left with that.
05:14
Now, our third in part c, we got a function of theta, sine theta times cosine theta, and here's our operator...