# Evaluating annihilation and creation operators

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.

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dextercioby
Homework Helper
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).

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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.

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
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?