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Hi,
I know that a Gaussian wavepacket has minimum uncertainty. The issue is, some sources are telling me that σxσp=ħ and others are telling me that σxσp=ħ/2. I am really confused. I think the latter is correct due to what I have been taught about the uncertainty principle, but then I don't understand what sources like the following are telling me:
http://oer.physics.manchester.ac.uk/QM/Notes/jsmath/Notesse39.html (see equation 10.17)
However, my own math (to my understanding) gives me a result similar to the link above. Working this out on my own, this is how I understand it:
A normalized Gaussian, where ##\left<x\right>=0## with a standard deviation ##\sigma_x## is
$$\psi (x) = \frac{1}{\sigma_x \sqrt{\pi}}e^{-x^2/2\sigma_x^2}$$
It's Fourier transform is
$$\tilde{\psi} (k) = \frac{\sigma_x}{\sqrt{\pi}}e^{-k^2\sigma_x^2/2}=\frac{1}{\sigma_k\sqrt{\pi}}e^{-k^2/2\sigma_k^2}$$
where the uncertainty in k is
$$\sigma_k=\frac{1}{\sigma_x}$$
This seems to yield
$$\sigma_x\sigma_k=1$$
or
$$\sigma_x\sigma_p=\hbar$$
Since the Gaussian minimizes uncertainty, the uncertainty principle would thus be
$$\sigma_x\sigma_p\geq \hbar$$
as the linked article suggests. To my knowledge, however, the uncertainty principle is
$$\sigma_x\sigma_p\geq\frac{\hbar}{2}$$
What is going on here?
I know that a Gaussian wavepacket has minimum uncertainty. The issue is, some sources are telling me that σxσp=ħ and others are telling me that σxσp=ħ/2. I am really confused. I think the latter is correct due to what I have been taught about the uncertainty principle, but then I don't understand what sources like the following are telling me:
http://oer.physics.manchester.ac.uk/QM/Notes/jsmath/Notesse39.html (see equation 10.17)
However, my own math (to my understanding) gives me a result similar to the link above. Working this out on my own, this is how I understand it:
A normalized Gaussian, where ##\left<x\right>=0## with a standard deviation ##\sigma_x## is
$$\psi (x) = \frac{1}{\sigma_x \sqrt{\pi}}e^{-x^2/2\sigma_x^2}$$
It's Fourier transform is
$$\tilde{\psi} (k) = \frac{\sigma_x}{\sqrt{\pi}}e^{-k^2\sigma_x^2/2}=\frac{1}{\sigma_k\sqrt{\pi}}e^{-k^2/2\sigma_k^2}$$
where the uncertainty in k is
$$\sigma_k=\frac{1}{\sigma_x}$$
This seems to yield
$$\sigma_x\sigma_k=1$$
or
$$\sigma_x\sigma_p=\hbar$$
Since the Gaussian minimizes uncertainty, the uncertainty principle would thus be
$$\sigma_x\sigma_p\geq \hbar$$
as the linked article suggests. To my knowledge, however, the uncertainty principle is
$$\sigma_x\sigma_p\geq\frac{\hbar}{2}$$
What is going on here?
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