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notes:physics:quantum-mechanics:harmonic-oscillator
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| notes:physics:quantum-mechanics:harmonic-oscillator [2026/09/30 18:57] – [The Number Operator] chris | notes:physics:quantum-mechanics:harmonic-oscillator [2026/09/30 19:21] (current) – [position and momentum operators] chris | ||
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| Line 599: | Line 599: | ||
| & | & | ||
| \end{align}$$ | \end{align}$$ | ||
| - | ==== position | + | ==== Position |
| + | The usefulness of the formulas above can be clearly seen while calculating the | ||
| + | expected values and the position and momentum dispersion. In a classical harmonic | ||
| + | oscillator the expected position should be the equilibrium point, $\left< x\right> | ||
| + | and the expected momentum should also be $\left< p\right> | ||
| + | $x$ and $p$ the oscillator is in, there is another //pair// state with $-x$ and $-p$. | ||
| + | The same happens in the quantum oscillator. | ||
| + | $$\begin{align} | ||
| + | \left< | ||
| + | &= \left< m\left|\hat x\right|n\right> | ||
| + | &= \sqrt{\frac{\hbar}{2m\omega}}\left< | ||
| + | &= \sqrt{\frac{\hbar}{2m\omega}}\left(\left< | ||
| + | &= \sqrt{\frac{\hbar}{2m\omega}}\left(\sqrt{n+1}\left< | ||
| + | &= \sqrt{\frac{\hbar}{2m\omega}} (0+0)=0 | ||
| + | \end{align}$$ | ||
| + | |||
| + | $$\begin{align} | ||
| + | \left< | ||
| + | &= \left< n\left|\hat p\right|n\right> | ||
| + | &= i\sqrt{\frac{\hbar m\omega}{2}}\left< | ||
| + | &= i\sqrt{\frac{\hbar m\omega}{2}}\left(\left< | ||
| + | &= i\sqrt{\frac{\hbar m\omega}{2}}\left(\sqrt{n+1}\left< | ||
| + | &= i\sqrt{\frac{\hbar m\omega}{2}}(0+0)=0 | ||
| + | \end{align}$$ | ||
| + | |||
| + | Unlike the expected position and momentum, we cannot //guess// their dispersion. Here we | ||
| + | actually need the creation and annihilation operators in order to avoid difficult integrals. | ||
| + | |||
| + | $$\require{cancel}\begin{align} | ||
| + | \Delta x_n^2 | ||
| + | &= \left< | ||
| + | &= \left< n\left|\hat x\hat x\right|n\right> | ||
| + | &= \frac{\hbar}{2m\omega}\left< | ||
| + | &= \frac{\hbar}{2m\omega}\left< | ||
| + | \end{align}$$ | ||
| + | |||
| + | The operator $\hat a^\dagger\hat a^\dagger$ will give us the state $\left|n+2\right> | ||
| + | and it has no contribution to the sum. Similarly, we can ignore the $\hat a\hat a$ term. The operator $\hat a^\dagger\hat a$ is | ||
| + | just the number operator $\hat N$ with eigenvalue $n$. The operator $\hat a\hat a^\dagger$ can be written a different way | ||
| + | with the help of the commutator, $\hat a\hat a^\dagger=\left[\hat a,\hat a^\dagger\right]+\hat a^\dagger\hat a=1+\hat N$, and | ||
| + | then also replaced by the eigenvalue $1+n$. | ||
| + | |||
| + | $$\begin{align} | ||
| + | \Delta x_n^2 | ||
| + | &= \frac{\hbar}{2m\omega}\left< | ||
| + | &= \frac{\hbar}{2m\omega}\left(2n+1\right)\left< | ||
| + | &= \frac{\hbar}{2m\omega}\left(n+\frac12\right) | ||
| + | \end{align}$$ | ||
| + | |||
| + | When calculating $\Delta p_n^2$ we can use the same //trick//: | ||
| + | |||
| + | $$\require{cancel}\begin{align} | ||
| + | \Delta p_n^2 | ||
| + | &= \left< | ||
| + | &= \left< n\left|\hat p\hat p\right|n\right> | ||
| + | &= i^2\frac{\hbar m\omega}{2}\left< | ||
| + | &= \frac{\hbar m\omega}{2}\left< | ||
| + | &= \frac{\hbar m\omega}{2}\left< | ||
| + | &= \frac{\hbar m\omega}{2}\left< | ||
| + | &= \frac{\hbar m\omega}{2}\left(2n+1\right)\left< | ||
| + | &= \hbar m\omega\left(n+\frac12\right) | ||
| + | \end{align}$$ | ||
| + | |||
| + | We find the least dispersion at $n=0$. We then have: | ||
| + | |||
| + | $$\Delta x\Delta p=\sqrt{\frac12\frac{\hbar}{m\omega}}\cdot\sqrt{\frac12\hbar m\omega}=\frac{\hbar}{2}$$ | ||
| + | |||
| + | This result very much reminds us of [[uncertainty-principle|Heisenberg' | ||
notes/physics/quantum-mechanics/harmonic-oscillator.1790794649.txt.gz · Last modified: by chris
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