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Hi folks,
I wonder if I could run a few things past the quantum gurus among you - I'm just not quite convinced of some of the results I've been deriving.
Consider the ground state of the simple harmonic oscillator at t = 0 with the normalised wave function [tex]\sqrt[4]{\frac{m\omega}{\pi\hbar}}e^{-\frac{m\omega}{2\hbar} x^2}[/tex].
If the position uncertainty is defined as [tex]\Delta x = \sqrt{<\widehat{x}^{2}>-<\widehat{x}>^{2}}[/tex] and the momentum uncertainty as [tex]\Delta p = \sqrt{<\widehat{p}^{2}>-<\widehat{p}>^{2}}[/tex], determine the product [tex]\Delta x . \Delta p[/tex]. Comment on the result.
I begin by observing that [tex]<\widehat{x}>[/tex] and [tex]<\widehat{p}>[/tex] are both zero, by symmetry arguments. Thus [tex]\Delta x = \sqrt{<\widehat{x}^{2}>}[/tex] and [tex]\Delta p = \sqrt{<\widehat{p}^{2}>}[/tex].
I go on to calculate [tex]<\widehat{x}^{2}>[/tex] and [tex]<\widehat{p}^{2}>[/tex].
I find that [tex]<\widehat{x}^{2}> = \frac{\hbar}{2m\omega}[/tex], a result I am pretty confident of (it was an easy integration).
My queries arise for [tex]<\widehat{p}^{2}>[/tex], which comes out as [tex]\hbar(m\omega - \frac{1}{2})[/tex], giving [tex]\Delta x . \Delta p = \sqrt{\frac{\hbar}{2m\omega}} . \sqrt{\hbar(m\omega - \frac{1}{2})} = \hbar\sqrt{1/2 - 1/4m\omega}[/tex] - not a very "nice" result. I'm suspicious.
I make [tex]<\widehat{p}^{2}>[/tex] to be
[tex]<\widehat{p}^{2}> = \sqrt{\frac{m\omega}{\pi\hbar}}\int^{+\infty}_{-\infty} e^{-\frac{m\omega}{2\hbar} x^2}.(-\hbar^{2}\frac{\partial^2}{\partial x^2}).(e^{-\frac{m\omega}{2\hbar} x^2}) dx = . . . = \sqrt{m^{3}\omega^{3}\hbar/\pi}.I_{2} - \sqrt{m^{3}\omega^{3}/\pi\hbar}.I_{1}[/tex]
as
[tex](\frac{\partial^2}{\partial x^2}).(e^{-\frac{m\omega}{2\hbar} x^2}) = ... = -\frac{m\omega}{\hbar}.e^{-m\omega x^2/2\hbar} + (\frac{m\omega}{\hbar})^2.x^2.e^{-m\omega x^2/2\hbar}[/tex],
where
[tex]I_{1} = \int^{+\infty}_{-\infty}x^2.e^{-m\omega x^2/\hbar} dx = ... = \sqrt{\frac{\pi\hbar^{3}}{4m^{3}\omega^{3}}}[/tex]
[tex]I_{2} = \int^{+\infty}_{-\infty}e^{-m\omega x^2/\hbar} dx = ... = \sqrt{\frac{\pi\hbar}{m\omega}}[/tex]
etc.
determining finally that [tex]\hbar(m\omega - \frac{1}{2})[/tex]
Has something gone wrong here, perhaps, or should I just accept the result?
Cheers!
I wonder if I could run a few things past the quantum gurus among you - I'm just not quite convinced of some of the results I've been deriving.
Homework Statement
Consider the ground state of the simple harmonic oscillator at t = 0 with the normalised wave function [tex]\sqrt[4]{\frac{m\omega}{\pi\hbar}}e^{-\frac{m\omega}{2\hbar} x^2}[/tex].
If the position uncertainty is defined as [tex]\Delta x = \sqrt{<\widehat{x}^{2}>-<\widehat{x}>^{2}}[/tex] and the momentum uncertainty as [tex]\Delta p = \sqrt{<\widehat{p}^{2}>-<\widehat{p}>^{2}}[/tex], determine the product [tex]\Delta x . \Delta p[/tex]. Comment on the result.
The Attempt at a Solution
I begin by observing that [tex]<\widehat{x}>[/tex] and [tex]<\widehat{p}>[/tex] are both zero, by symmetry arguments. Thus [tex]\Delta x = \sqrt{<\widehat{x}^{2}>}[/tex] and [tex]\Delta p = \sqrt{<\widehat{p}^{2}>}[/tex].
I go on to calculate [tex]<\widehat{x}^{2}>[/tex] and [tex]<\widehat{p}^{2}>[/tex].
I find that [tex]<\widehat{x}^{2}> = \frac{\hbar}{2m\omega}[/tex], a result I am pretty confident of (it was an easy integration).
My queries arise for [tex]<\widehat{p}^{2}>[/tex], which comes out as [tex]\hbar(m\omega - \frac{1}{2})[/tex], giving [tex]\Delta x . \Delta p = \sqrt{\frac{\hbar}{2m\omega}} . \sqrt{\hbar(m\omega - \frac{1}{2})} = \hbar\sqrt{1/2 - 1/4m\omega}[/tex] - not a very "nice" result. I'm suspicious.
I make [tex]<\widehat{p}^{2}>[/tex] to be
[tex]<\widehat{p}^{2}> = \sqrt{\frac{m\omega}{\pi\hbar}}\int^{+\infty}_{-\infty} e^{-\frac{m\omega}{2\hbar} x^2}.(-\hbar^{2}\frac{\partial^2}{\partial x^2}).(e^{-\frac{m\omega}{2\hbar} x^2}) dx = . . . = \sqrt{m^{3}\omega^{3}\hbar/\pi}.I_{2} - \sqrt{m^{3}\omega^{3}/\pi\hbar}.I_{1}[/tex]
as
[tex](\frac{\partial^2}{\partial x^2}).(e^{-\frac{m\omega}{2\hbar} x^2}) = ... = -\frac{m\omega}{\hbar}.e^{-m\omega x^2/2\hbar} + (\frac{m\omega}{\hbar})^2.x^2.e^{-m\omega x^2/2\hbar}[/tex],
where
[tex]I_{1} = \int^{+\infty}_{-\infty}x^2.e^{-m\omega x^2/\hbar} dx = ... = \sqrt{\frac{\pi\hbar^{3}}{4m^{3}\omega^{3}}}[/tex]
[tex]I_{2} = \int^{+\infty}_{-\infty}e^{-m\omega x^2/\hbar} dx = ... = \sqrt{\frac{\pi\hbar}{m\omega}}[/tex]
etc.
determining finally that [tex]\hbar(m\omega - \frac{1}{2})[/tex]
Has something gone wrong here, perhaps, or should I just accept the result?
Cheers!
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