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## Homework Statement

Does the n = 2 state of a quantum harmonic oscillator violate the Heisenberg Uncertainty Principle?

## Homework Equations

$$\sigma_x\sigma_p = \frac{\hbar}{2}$$

## The Attempt at a Solution

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I worked out the solution for the second state of the harmonic oscillator.

$$\frac{1}{\sqrt{2}}(\frac{mw}{\pi\hbar})^{1/4}(\frac{1}{\hbar})(e^{\frac{-mwx^2}{2\hbar}}(2mwx^2 - \hbar)$$

Should I be solving for standard deviation of position and momentum? Or is there another way to do this problem. I'm not sure what the next step is. Thank you for the guidance.