- #1

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Given that a harmonic oscillator has eigenkstates |n> where n = 1,2,3,..., how can we calculate <X>, <P>, <X^2>, etc. Is there a need to define a wavefunction in the |n> basis?

Thanks!

- Thread starter Thunder_Jet
- Start date

- #1

- 18

- 0

Given that a harmonic oscillator has eigenkstates |n> where n = 1,2,3,..., how can we calculate <X>, <P>, <X^2>, etc. Is there a need to define a wavefunction in the |n> basis?

Thanks!

- #2

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- #3

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Thank you by the way for the idea!

- #4

- 649

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[tex]

\hat{a}=\frac{1}{\sqrt{2}}(\hat{x}+i\hat{p})

[/tex]

and

[tex]

\hat{a}^{\dagger}=\frac{1}{\sqrt{2}}(\hat{x}-i\hat{p})

[/tex]

You can use these to write [itex]\hat{x}[/itex] and [itex]\hat{p}[/itex] in terms of [itex]\hat{a}[/itex] and [itex]\hat{a}^{\dagger}[/itex]. Then you know

[tex]

\hat{a}|n\rangle = \sqrt{n}|n-1\rangle

[/tex]

and

[tex]

\hat{a}^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle

[/tex]

You have the tools to take the expectation value.

- #5

Matterwave

Science Advisor

Gold Member

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You don't express the kets as ladder operators acting on the vacuum, you express x and p as ladder operators.

Thank you by the way for the idea!

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