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Evaluating annihilation and creation operators

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

[itex]\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[/itex]


Homework Equations



[itex]\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[/itex]

The Attempt at a Solution



The first one is [itex]\sqrt{6}\left|(6)\right\rangle[/itex] and second one [itex]\sqrt{5}\left|(4)\right\rangle[/itex]

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

Thanks.
 

Answers and Replies

  • #2
dextercioby
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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).
 
Last edited:
  • #3
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.
 
  • #4
strangerep
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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.
 
  • #5
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 [itex]\hat{p}\left|\psi\right\rangle~=~ h/i*d/dx~\psi(x)[/itex] but I don't see how this relates to actual numbers?
 
  • #6
strangerep
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There is one more thing.. what does a ket of a number actually mean?
I understand that for example [itex]\hat{p}\left|\psi\right\rangle~=~ h/i*d/dx~\psi(x)[/itex] 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.
 

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