00:01
In this problem, we are asked to find the value of a, alpha, and the energy -aggin value for the wave function of the harmonic oscillator in the first excited state, and this is the wave function.
00:19
And we're asked to use the method outline in section 5 .5.
00:23
The method outline in the section 5 .5 is to simply stick this into the schrodinger equation and solve for a and alpha.
00:31
And the schrodinger equation is time -in -time -independent.
00:38
Shoting -e equation is basically equal to this.
00:48
We're not going to worry about a at the moment.
00:51
We're going to figure that out by requiring that the wave function is normalized.
00:58
So to stick this into the schrodinger equation, we have to first figure out the second derivative of the wave function.
01:02
The first derivative is easy.
01:05
We can say that by using the product rule, this is equal to a e to the alpha x squared where we have taken the derivative of x plus a x and now we're to take the derivative of the exponential minus 2 of x 2 x 2 to the alpha x squared and we can simplify this a bit to write it down as a e to the minus alpha x squared 1 minus 2 alpha x squared.
01:45
We had to take the second derivative as well.
01:49
D squared 5, dx squared.
01:53
We're going to use the.....