In quantum physics, when you have the eigenstates of a system, you can determine the allowable states of the system and the relative probability that the system will be in any of those states.
The commutator of operators A, B is [A, B] = AB – BA, so note that the commutator of
![image0.png](https://www.dummies.com/wp-content/uploads/398056.image0.png)
is the following:
![image1.png](https://www.dummies.com/wp-content/uploads/398057.image1.png)
This is equal to the following:
![image2.png](https://www.dummies.com/wp-content/uploads/398058.image2.png)
This equation breaks down to
![image3.png](https://www.dummies.com/wp-content/uploads/398059.image3.png)
And putting together this equation with the Hamiltonian,
![image4.png](https://www.dummies.com/wp-content/uploads/398060.image4.png)
Okay, with the commutator relations, you’re ready to go. The first question is: if the energy of state | n > is En, what is the energy of the state a | n >? Well, to find this, rearrange the commutator
![image5.png](https://www.dummies.com/wp-content/uploads/398061.image5.png)
Then use this to write the action of
![image6.png](https://www.dummies.com/wp-content/uploads/398062.image6.png)
like this:
![image7.png](https://www.dummies.com/wp-content/uploads/398063.image7.png)
So a | n > is also an eigenstate of the harmonic oscillator, with energy
![image8.png](https://www.dummies.com/wp-content/uploads/398064.image8.png)
not En. That’s why a is called the annihilation or lowering operator: It lowers the energy level of a harmonic oscillator eigenstate by one level.
So what’s the energy level of
![image9.png](https://www.dummies.com/wp-content/uploads/398065.image9.png)
You can write that like this:
![image10.png](https://www.dummies.com/wp-content/uploads/398066.image10.png)
All this means that
![image11.png](https://www.dummies.com/wp-content/uploads/398067.image11.png)
is an eigenstate of the harmonic oscillator, with energy
![image12.png](https://www.dummies.com/wp-content/uploads/398068.image12.png)
not just En — that is, the
![image13.png](https://www.dummies.com/wp-content/uploads/398069.image13.png)
raises the energy level of an eigenstate of the harmonic oscillator by one level.
So now you know that
![image14.png](https://www.dummies.com/wp-content/uploads/398070.image14.png)
You can derive the following from these equations:
![image15.png](https://www.dummies.com/wp-content/uploads/398071.image15.png)
C and D are positive constants, but what do they equal? The states |n – 1> and |n + 1> have to be normalized, which means that <n – 1|n – 1> = <n + 1|n + 1> = 1. So take a look at the quantity using the C operator:
![image16.png](https://www.dummies.com/wp-content/uploads/398072.image16.png)
And because |n – 1> is normalized, <n – 1|n – 1> = 1:
![image17.png](https://www.dummies.com/wp-content/uploads/398073.image17.png)
But you also know that
![image18.png](https://www.dummies.com/wp-content/uploads/398074.image18.png)
the energy level operator, so you get the following equation:
< n | N | n > = C2
N | n > = n | n >, where n is the energy level, so
n < n | n > = C2
However, < n | n > = 1, so
![image19.png](https://www.dummies.com/wp-content/uploads/398075.image19.png)
This finally tells you, from a | n > = C | n – 1 >, that
![image20.png](https://www.dummies.com/wp-content/uploads/398076.image20.png)
That’s cool — now you know how to use the lowering operator, a, on eigenstates of the harmonic oscillator.
What about the raising operator,
![image21.png](https://www.dummies.com/wp-content/uploads/398077.image21.png)
First you rearrange the commutator
![image22.png](https://www.dummies.com/wp-content/uploads/398078.image22.png)
Then you follow the same course of reasoning you take with the a operator to show the following:
![image23.png](https://www.dummies.com/wp-content/uploads/398079.image23.png)
So at this point, you know what the energy eigenvalues are and how the raising and lowering operators affect the harmonic oscillator eigenstates.