How do ladder operators generate energy values in a SHO?

[maximus123]

Hello,

I am currently studying ladder operator for a simple harmonic operator as a method for generating the energy values. This seem like a simple algebra question I am asking so I do apologize but I just can't figure it out. Here are my operator definitions,

$a_+=\frac{1}{\sqrt{2}}(-\frac{\partial}{\partial y}+y)$ and $a_-=\frac{1}{\sqrt{2}}(\frac{\partial}{\partial y}+y)$​

My notes then say that

$a_+a_-=\frac{1}{\sqrt{2}}(-\frac{\partial}{\partial y}+y)$$\frac{1}{\sqrt{2}}(\frac{\partial}{\partial y}+y)=\frac{1}{2}(-\frac{\partial^2}{\partial y^2}+y^2+1)$​

I see where every resulting term comes from but it seems like there is a cross product missing, when $y$ multiplies $\frac{\partial}{\partial y}$ ie the second term in the left hand brackets times the first term in the right hand bracket. It seems to have multiplied to equal zero.
Could anyone explain where I'm going wrong? Thanks

 

[maximus123]

Sorry, I've actually just figured this out, apply the operators one at a time to an arbitrary function and this works.

 

[dauto]

Sorry, I've actually just figured this out, apply the operators one at a time to an arbitrary function and this works.

That's exactly right. The operators make no sense by themselves. They must be applied to a wave function. That wavefunction is often absent to reduce clutter but it really is there all the time.