For a free particle (PE = PE_o = constant): Show that Psi (x, t) = A e^i(4ks - Z

Question

For a free particle (PE = PE_o = constant): Show that Psi (x, t) = A e^i(4ks - Zwt) is a solution to the Schrodinger equation Show that the solution from (a) gives a constant value for PE that only depends on k and m. (assuming the de Broglie relationship between momentum and wavelength holds) This particle is an electron. It enters a region with an magnetic field B^bar = 14 T z cap and an electric field E^bar = 2.2 N/C x cap while traveling in y cap. If it's position can be measured to a precision of 2.6 times 10^-8 m, how precisely can it's speed be known? How precisely can the electromagnetic force (the Lorentz force) that the electron experiences be known?

Explanation / Answer

a. the shrodinger's wave equation is given by

d^(psi)/dx^2 + 8*pi^2*m(E - V)psi/h^2 = 0

now, E - V = KE

for a free particle, V = constant = Vo

also, given psi = A*e^i(4kx - 2wt)

putting this in the schrodingter's equation we get

-16k^2*A*e^i(4kx - 2wt) + 8pi^2*m(E - V)A*e^i(4kx - 2wt)/h^2 = 0

16k^2 = 8pi^2*m(E - V)/h^2

2k^2*h^2/pi^2*m = E - v = KE = mv^2/2

4k^2*h^2/pi^2*m^2 = v^2

v = 2k*h/m*pi

k = mv*pi/2h [ for the given psi to be a solution of a shrodinger's equation]

b. now, KE of particle = mv^2/2

energy of particle = mv^2

so, PE = E - v = mv^2/2 = m(4k^2*h^2/2m^2*pi^2) = 2k^2*h^2/m*pi^2

hence PE just depends on mass and k

c. given, precision in position measurement

dy = 2.6*10^-8 m

then, from heisenbergs uncertiuanity principle

dy*m(dv) = h/4pi [ where dv is uncertianity in measurement of velocity]

now for electron, m = 9.1*10^-34 kg

so, dv = h/4*pi*m*dy = 2229.9181 m/s

d. lorentz force on a moving particle in the electric and magnetic field is

F = qE + qvB

so uncertianity in force dF = dv*q*B = 4.99501*10^-16 N

( given B = 1.4 T and q = 1.6*10^-19 C for el;ectron)

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