Find the fractional uncertainty p/p for a 1keV electronunder a variety of condit

Question

Find the fractional uncertainty p/p for a 1keV electronunder a variety of conditions: a) The electron is free (unconfined) b) The electron is confined to a region 1 m long c) The electron is confined to a region 1 cm long d) The electron is confined to a region 1 Angstrom long (aboutthe size of an atom)
I promiseto rate you as lifesaver if you SHOW ALL WORK, with calculationsand units. Find the fractional uncertainty p/p for a 1keV electronunder a variety of conditions: a) The electron is free (unconfined) b) The electron is confined to a region 1 m long c) The electron is confined to a region 1 cm long d) The electron is confined to a region 1 Angstrom long (aboutthe size of an atom)
I promiseto rate you as lifesaver if you SHOW ALL WORK, with calculationsand units. Find the fractional uncertainty p/p for a 1keV electronunder a variety of conditions: a) The electron is free (unconfined) b) The electron is confined to a region 1 m long c) The electron is confined to a region 1 cm long d) The electron is confined to a region 1 Angstrom long (aboutthe size of an atom)
I promiseto rate you as lifesaver if you SHOW ALL WORK, with calculationsand units.

Explanation / Answer

According to the Heseinberg's uncertainty principle wehave x * p = (h/2) or x * m * v = (h/2) or v = (1/x * m) * (h/2) --------(1) the energy of the electron is E = (1/2)m * v2 or v = (2E/m)1/2 ---------(2) the fractional uncertainty (p/p) is (p/p) = (m * v/m * v) = (v/v)----------(3) from (1),(2) and (3) we get (p/p) = ((1/x * m) *(h/2)/(2E/m)1/2) or (p/p) = (1/2E * m)1/2 * (h/2) *(1/x) ------------(4) a)when the electron is free (unconfined) we have x = from equation (4) we get (p/p) = (1/2E * m)1/2 * (h/2) *(1/) or (p/p) = 0 b)when the electron is confined to a region 1 m longthen x = 1 m or (p/p) = (1/2E * m)1/2 * (h/2) *(1/1) = (1/2E * m)1/2 * (h/2) E = 1 keV = 1 * 103 eV = 1 * 103 *1.6 * 10-19 J = 1.6 * 10-16 J m = 9.1 * 10-31 kg h = 6.63 * 10-34 J-s c)when the electron is confined to a region 1 cm longthen x = 1 cm = 1 * 10-2 m or (p/p) = (1/2E * m)1/2 * (h/2) * (1/1* 10-2) =  (1/2E * m)1/2 *(h/2) * 102 d)when the electron is confined to a region 1 Angstrom long(about the size of an atom) then x = 1 A = 1 * 10-10 m or (p/p) = (1/2E * m)1/2 * (h/2) * (1/1* 10-10) =  (1/2E * m)1/2 *(h/2) * 1010 d)when the electron is confined to a region 1 Angstrom long(about the size of an atom) then x = 1 A = 1 * 10-10 m or (p/p) = (1/2E * m)1/2 * (h/2) * (1/1* 10-10) =  (1/2E * m)1/2 *(h/2) * 1010

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