00:01
So the wave function is given as a times e to the minus absolute value of x divided by a.
00:07
And we have a basic graph of the wave function right here.
00:10
You see that on the right hand side, it looks like a decaying exponential function.
00:15
And on the left hand side, you essentially have a mirror image of the function on the right.
00:20
So this is actually an important observation to make about this wave function, because when you carry out integrals, the mirror image property about the, the vertical axis is important to take into consideration.
00:35
So that's actually what we're going to be doing next in getting a normalized wave function and determining the value of a.
00:43
So one thing to notice is i have plotted the function and made note where the values of a occur along the x -axis and also the maximum value of the function.
00:57
As you'll see, when we get the normalized wave function, you'll notice that the maximum value occurs at the origin at x equals 0, and the maximum value of the wave function is actually 1 divided by the square root of lowercase a.
01:14
So let's go ahead and get the normalized wave function, and we'll start with the usual.
01:23
We'll say that the integral from minus infinity to positive infinity of the probability density is equal to 1.
01:39
And in doing that, this means that here we have a squared times negative infinity to positive infinity of e to the minus 2 times the absolute value of x divided by a.
01:57
Now, we don't want to have to integrate an absolute value inside of an exponential, but again, we can use the property of the symmetry of this function, and this will allow us to rewrite.
02:10
Our integral.
02:13
So note that the area under this half of the curve is equal to the area under this half of the curve.
02:20
So we can rewrite the integral in terms of a single exponential function utilizing this symmetry property of the area.
02:29
And remember that a definite integral in this case for a one -dimensional function is simply the area under the curve.
02:37
So going forward, this integral can then be written as two times the integral.
02:44
So this two means that one of these areas multiplied by two will give us the entire area under the entire curve.
02:55
So we write this as zero to infinity of e to the minus x over there there's a two right here.
03:05
Let me make this clear.
03:12
E of the minus two x over a times dx.
03:19
I again, this two just tells us that, whereas this integral is the area under the curve on the right side, the two allows us to get the entire area, because this area on the right equals the area on the left.
03:35
So going forward now, we have minus 2a squared times lowercase a over two, times the function in brackets, because it's being evaluated from a definite integral.
04:03
This will give us negative capital a squared times lowercase a.
04:11
And we here have e to the minus 2 times x approaching infinity divided by a.
04:18
So we see that this is going to be zero because as e to the minus x, with x going to infinity gets incredibly large, e to the minus an incredibly large number must decay to zero.
04:31
Here we have minus e to the negative 2 times 0 over a, and e to the minus 0 is 1.
04:42
So we have these two terms, and going forward now, we have capital a squared times lowercase a.
04:51
We have 0 minus 1, and this gives us capital a squared times lowercase a.
04:59
That's a multiplication.
05:01
And remember this must equal one because we're normalizing.
05:06
So we see then that capital a squared equals 1 over lowercase a, and therefore capital a must equal 1 over the square root of lowercase a.
05:21
So 1 equals a square root of lowercase a.
05:26
And one other thing we're going to do is calculate a probability in the interval from minus a to a to positive a.
05:40
And this is going to equal capital a squared minus infinity, or rather minus a.
05:49
Remember, we're just looking at a finite interval now.
05:53
Minus a to positive a, and we have our, the square of our wave function.
06:04
And this is written as so...
