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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!

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- Thread starter Thunder_Jet
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The infinite dimensionality is accounted for in the summation over n in the expectation value formula <X> = \sum_{n=0}^{\infty} x_n |\langle n | \psi \rangle|^2.In summary, the conversation discusses calculating <X>, <P>, <X^2>, etc. for a harmonic oscillator with eigenkstates |n>. It is not necessary to define a wavefunction in the |n> basis, as the operators X and P can be expressed as linear combinations of the raising and lowering ladder operators. The expectation values can be calculated using these ladder operators and the infinite dimensionality is accounted for in the summation over n.

- #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!

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Thank you by the way for the idea!

- #4

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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

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Thunder_Jet said:

Thank you by the way for the idea!

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

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