- #1
Hyperreality
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(a) In the infinite one-dimensional well, what is [tex]p_{av}[/tex]?
(b) What is [tex](p^2)_{av}[/tex]?
(c) What is [tex]\Delta p = \sqrt{(p^2)_av - (p_av)^2}[/tex]?
(d) Compute [tex]\Delta p \Delta x[/tex], and compare with the Heisenberg uncertainty relationship.
Here's my working:
(a) [tex]p_{av}=0[/tex].
I'm not so sure about this bit
(b)[tex](\frac{p^2}{2m})_{av} = E_{n} = \frac{\hbar^2\pi^2n^2}{2mL^2}[/tex].
There fore [tex](p^2)_{av}=(\frac{\hbar\pi^2n^2}{L})^2[/tex]
(c)Therefore,
[tex]\Delta p = \frac{\hbar\pi n}{L}[/tex].
(d)[tex]\Delta p\Delta x = \frac{\hbar}{2}\sqrt{2n^2\pi^2 -1}[/tex]
Part (d) it seems the most suspicious, that is, the uncertainty increases with n^2. Have I done anything wrong?
(b) What is [tex](p^2)_{av}[/tex]?
(c) What is [tex]\Delta p = \sqrt{(p^2)_av - (p_av)^2}[/tex]?
(d) Compute [tex]\Delta p \Delta x[/tex], and compare with the Heisenberg uncertainty relationship.
Here's my working:
(a) [tex]p_{av}=0[/tex].
I'm not so sure about this bit
(b)[tex](\frac{p^2}{2m})_{av} = E_{n} = \frac{\hbar^2\pi^2n^2}{2mL^2}[/tex].
There fore [tex](p^2)_{av}=(\frac{\hbar\pi^2n^2}{L})^2[/tex]
(c)Therefore,
[tex]\Delta p = \frac{\hbar\pi n}{L}[/tex].
(d)[tex]\Delta p\Delta x = \frac{\hbar}{2}\sqrt{2n^2\pi^2 -1}[/tex]
Part (d) it seems the most suspicious, that is, the uncertainty increases with n^2. Have I done anything wrong?
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