Phy5645/HO Virial Theorem: Difference between revisions
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We can now write the average potential for the <math>n^{\text{th}}</math> state of the harmonic oscillator as | We can now write the average potential for the <math>n^{\text{th}}</math> state of the harmonic oscillator as | ||
<math> \langle V \rangle = \frac{k}{ | <math> \langle V \rangle = \frac{\hbar k}{4m\omega}\langle n|(\hat{a} + \hat{a}^\dagger)^2|n \rangle </math> | ||
<math> = \frac{k}{ | <math> = \frac{\hbar k}{4m\omega}\langle n|(\hat{a}^2 + \hat{a}^{\dagger 2} + \hat{a}\hat{a}^\dagger + \hat{a}^\dagger\hat{a})|n \rangle </math> | ||
<math> = \frac{k}{ | <math> = \frac{\hbar k}{4m\omega}[\langle n|\hat{a}^2|n \rangle + \langle n|\hat{a}^\dagger|n \rangle + \langle n|(\hat{a}\hat{a}^\dagger + \hat{a}^\dagger\hat{a})|n \rangle] </math> | ||
The first two terms are zero because | |||
<math> \langle n|n-2 \rangle = \langle n|n+2 \rangle = 0 </math> | <math> \langle n|n-2 \rangle = \langle n|n+2 \rangle = 0 </math> | ||
Revision as of 16:56, 8 August 2013
The average potential energy is given by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \hat{V} \rangle = \tfrac{1}{2}k\langle \hat{x}^2 \rangle.}
Recall from a previous problem that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{x}=\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}+\hat{a}^{\dagger}),}
or
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{1}{2}k\hat{x}^2=\frac{\hbar k}{4m\omega}(\hat{a}+\hat{a}^\dagger)^2.}
We can now write the average potential for the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{\text{th}}} state of the harmonic oscillator as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle V \rangle = \frac{\hbar k}{4m\omega}\langle n|(\hat{a} + \hat{a}^\dagger)^2|n \rangle }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{\hbar k}{4m\omega}\langle n|(\hat{a}^2 + \hat{a}^{\dagger 2} + \hat{a}\hat{a}^\dagger + \hat{a}^\dagger\hat{a})|n \rangle }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{\hbar k}{4m\omega}[\langle n|\hat{a}^2|n \rangle + \langle n|\hat{a}^\dagger|n \rangle + \langle n|(\hat{a}\hat{a}^\dagger + \hat{a}^\dagger\hat{a})|n \rangle] }
The first two terms are zero because
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle n|n-2 \rangle = \langle n|n+2 \rangle = 0 }
and the operator in the third term can be written like:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}\hat{a}^\dagger + \hat{a}^\dagger\hat{a} = 1 + 2\hat{N} \text{ where } \hat{N} = \hat{a}^\dagger\hat{a} }
since
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}\hat{a}^\dagger |n \rangle = \hat{a} (n+1)^{\frac{1}{2}}|n + 1 \rangle = (n+1)|n \rangle }
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{N}|n \rangle = n|n \rangle }
So, now we have that:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle V \rangle = \frac{k}{4\beta^2} \langle n|(1 + 2\hat{N}|n \rangle = \frac{k}{4\beta^2}(2n + 1)\langle n|n \rangle = \frac{k}{2\beta^2}(n + \frac{1}{2}) }
And, replacing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta^2 = \frac{m\omega_0}{\hbar} } , we find that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle V \rangle = \frac{\hbar\omega_0}{2}(n + \frac{1}{2}) }
And can check that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle T \rangle = \frac{1}{2m} \langle \hat{p} \rangle = \frac{1}{2} \langle E \rangle = \frac{\hbar\omega_0}{2}(n + \frac{1}{2}) }
Which shows rather nicely that the Virial Theorem holds for the Quantum Harmonic Oscillator.
(See Liboff, Richard Introductory Quantum Mechanics, 4th Edition, Problem 7.10 for reference.)