Evaluating annihilation and creation operators

[tombrown9g2]

1. Evaluate the following (i.e. get rid of the operators):

$\hat{a}^{+}\left|5\right\rangle,~~~\hat{a}\left|5\right\rangle,~~~(\hat{a}^{+})^{3}\left|2\right\rangle~~~\hat{a}^{3}\left|2\right\rangle,~~~(\hat{a}^{+}\hat{a}\hat{a}^{+}\hat{a}\left|1\right\rangle,~~~\hat{a}^{+}\hat{a}^{+}\hat{a}\hat{a}\left|1\right\rangle$

Homework Equations

$\hat{a}\left|n\right\rangle=\sqrt{n}\left|n-1\right\rangle,~~~\hat{a}^{+}\left|n\right\rangle=\sqrt{n+1}\left|(n+1)\right\rangle$

The Attempt at a Solution

The first one is $\sqrt{6}\left|(6)\right\rangle$ and second one $\sqrt{5}\left|(4)\right\rangle$

However I'm unsure how to evaluate for the others using the equations given. Could someone please point me in the right direction?

Thanks.

 

[dextercioby]

What does the cube of an operator mean ? And lose the round brackets inside the bra and ket | and <> symbols (the pun is not intended).

 

[tombrown9g2]

What does the cube of an operator mean ? And lose the round brackets inside the bra and ket | and <> symbols (the pun is not intended).

I have no idea how powers affect operators. Can't find any examples in my lecturers notes nor when searching the internet. Really struggling to grasp quantum mechanics as the mathematics I know doesn't seem to apply.

 

[strangerep]

I have no idea how powers affect operators.

It's just like ordinary algebra, e.g., $x^2 := x \, x$.

So can you evaluate $(a^+)^2|n\rangle$ and $a^2 |n\rangle$ now?

Also, you should be able to evaluate $a^+ a|n\rangle$ with the "relevant equations" you already written down.

 

[tombrown9g2]

Ahh I see, you just use the operators one by one. Thanks.
There is one more thing.. what does a ket of a number actually mean?
I understand that for example $\hat{p}\left|\psi\right\rangle~=~ h/i*d/dx~\psi(x)$ but I don't see how this relates to actual numbers?

 

[strangerep]

There is one more thing.. what does a ket of a number actually mean?
I understand that for example $\hat{p}\left|\psi\right\rangle~=~ h/i*d/dx~\psi(x)$ but I don't see how this relates to actual numbers?

You didn't specify the context in your original post, so I can only give a broad answer. For the quantized harmonic oscillator, one denotes the ground state (i.e., state of lowest energy) by $|0\rangle$. The higher energy (eigenstates) are then successively numbered by 1,2,3... etc. The creation and annihilation operators are an example of so-called Ladded Operators which map from one such eigenstate to another. The Wiki page has more info about other contexts.